T H E THEORY OF SEMISETS
PETR VOPENKA Department of Mathematics. Charles University, Prague
and
PETR HAJEK Mathematical Institute of the Czechoslovak Academy of Sciences, Prague
1972
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM
LONDON
All rights resermd. No part of this book may he reproduced in any form, by photostat, microfilm, re?rieral sjstem, or any other means, without written permission from the publishers
0
Petr VopEnka, Petr Hkjek, 1972
Library of Congress Catalog Card Number 77-157005 ISBN 0 7204 2267 1
PUBLISHERS: NORTH-HOLLAND PUBLISHING COMPANY
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AMSTERDAM
ACADEMIA - PUBLISHING HOUSE OF THE CZECHOSLOVAK ACADEMY OF SCIENCES
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PRAGUE
INTRODUCTION
This book on the theory of semisets presents an attempt to create a theory whose universe of discourse extends that of set theory; thus the new theory admits the existence of certain objects which cannot exist from the point of view of set theory. Semisets are subclasses of sets; whereas in set theory the comprehension axiom ensures that every subclass of a set i s a set, the theory of semisets admits the existence of semisets which are not sets (proper semisets). Although the reader of this book must be able to deal with abstract generalizations, no particular mathematical knowledge is required for its study. In the present introduction we compare the theory of semisets with classical set-theoretical systems. This comparison should serve to explain the reasons for the existence of the theory of semisets. The notion of set is of c o m e a very familiar one. Cantor “defined” the notion of set as follows: “Unter einer Menge verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unserer Anschauung oder unseres Denkens (welche die Elemente von M genannt werden) zu einem Ganzen”. (“A set is a collection into a whole of definite, distinct objects of our intuition or of our thought. The objects are called the elements (members) of the set.”) However since intuitive Cantorian ideas on sets are not free from contradiction - in particular the idea that any property determines a set consisting of precisely the objects having that property - it was necessary to create an axiomatic theory of sets. Let us mention two such axiomatic systems which are widely considered today. (a) Zermelo-Fraenkel set theory has as its basic notions “set” and “elementhood”. Since it is provable in Zermelo-Fraenkel set theory that there is no set of all sets, it is not always possible to “collect into a whole” all objects having certain properties. In other words there are properties P (the property of being a set, for example) for which there is no set whose elements are precisely all objects having the property P . 7
INTRODUCTION
(b) However the assumption of the existence of ‘‘large” collections of elements (e.g. the collection of all sets) is not contradictory. provided we suppose that these large collections are not sets and in particiilar that they cannot themselves be elements of any set. Bernays-Godel set theory has as its basic notions “class” and “elementhood”; sets are defined as those classes which are elements of classes. Both systems of set theory are closely related; namely, any statement which concerns sets and elementhood alone (and which is therefore meaningful in both systems) is provable in one system if and only if it is provable in the other. The known proofs of contradictions in intuitive set theory cannot be reproduced in Zermelo-Fraenkel or in Bernays-Godel set theory. Nevcrtheless the question remains whether one can prove a contradiction in these systems in another way. It follows from the above that there are only two possibilities; either both systems are consistent or both are contradictory. According to a famous result of Godel (1931), there can be no absolute proof of consistency for any “sufficiently strong” theory; this applies in particular to any theory, like set theory, in which Peano arithmetic can be developed. Thus it is impossible to establish the consistency of any such theory without using stronger means that those available in the theory itself. In view of Godel’s result, the most that can be hoped for is a demonstration of relative consistency: if a certain theory S is consistent then so is another theory T. If we regard S as in some sense trustworthy and T as questionable, then a demonstration of relative consistency would tell us that T is in fact just as trustworthy as S; insofar as one believes in the consistency of S one must also believe in the consistency of T. Convxsely if we suspect S to be inconsistent then the relative consistency result reduces the problem of finding a contradiction in S to the perhaps easier problem of finding a contradiction in T.Let us give three examples which have been important i n the history of the subject. (A) I n 1922 A. Fraenkel showed that if set theory is consistent then it remains consistent upon the addition of certain statements contradicting the axiom of choice. Fraenkel’s method was later placed on a firm logical foundation by Mostowski. A number of statements contradicting the axiom of choice have since been shown to be consistent using Fraenkel’s method. However these results all have a certain unsatisfactory character in that the sets for which the axiom of choice fails always have “irregular” elements; thus for example the method cannot be used to prove the consistency of the hypothesis that the set of real numbers has no well-ordering. If we assume as an axiom of set theory that there are no “irregular” elements (one of the S
INTRODUCTION
so-called regularity axioms), then Fraenkel’s method gives no information as to the independence of the axiom of choice (cf. sub. (C)). (B) In 1938 (and in a more detailed form in t940), K. Godel established the consistency of the axiom of choice and the generalized continuum hypothesis with the axioms of set theory (including the axiom of regularity) by means of his famous A-model. Other important consistency results have been obtained by a generalization of this construction (e.g. Hajnal). (C) In 1963 P. J. Cohen established the independence of the continuum hypothesis and the axiom of choice in set theory with the axiom of regularity. Cohen’s method was reworked independently by Scott-Solovay and by VopEnka into the so-called method of Boolean-valued models. We note that all the above-mentioned consistency results will be established in the course of this book. In his consistency proof Godel defines the notion of “constructible class” and shows that if the axioms of Bernays-Godel set theory are interpreted by replacing the notion of “class” by the notion of “constructible class’’ (so that in particular “set” is replaced by “constructible set”), then the interpreted axioms are provable in Bernays-Godel set theory and in addition the interpreted version of the axiom of choice is provable. The constructible sets form a class L and the constructible classes are certain subclasses of L. It is important that in Godel’s interpretation classes are interpreted as constructible subclasses of L and not as arbitrary subclasses. If classes are interpreted as arbitrary subclasses of L then the sets in the sense of this new interpretation are still precisely the elements of L but we can no longer prove the comprehensivity axiom in this interpretation. Indeed, if a is an element of L and x I S a subset of a then ,Y is a subset of L, and hence a semiset from the point of view of our interpretation; however we cannot prove that x is an element of L, i.e. that x is a set in our interpretation. Although we cannot verify all theorems of set theory in the new interpretation, we can nevertheless verify many of them. From the above remarks we see that one cannot verify the statement “every semiset is a set” in the interpretation, and so this interpretation allows us to “imagine” proper semisets. Many other constructions invented for consistency proofs can be modified in a similar way, interpreting the notion of class more widely than in the original construction. In each case we obtain a n interpretation in which proper semisets may (or do) exist. The question therefore arises whether it is possible to axiomatize this general situation in an appropriate way, i.e. 9
INTRODUCTION
to choose appropriate statements verified in such interpretations and to take these as the axioms for a new theory. Such axioms will be given in this book. Here we shall outline the intuitive content of the axioms in order to explain why they were chosen as they were. The basic ideas on classes, sets and semisets are as follows: 1. We study classes and their elements; we consider the membership relation between elements and classes. 2. Classes are identical iff they have the same elements. 3 . Certain classes may be elements of certain other classes. Those classes which are elements of other classes are called sets. We do not consider elements of classes other than sets. A class is called a semiset iff it is a subclass of a set. 4. For any two sets there is a pair set. 5 . A n y “reasonable” property determines a class, namely the class of all sets having that property. (It suffices to assume that each of seven particular “reasonable” properties determines a class.) 6. There exists an infinite set. 7 . “Small” cannot be mapped one-one onto “large”. More precisely, given a one-one correspondence between certain sets and certain semisets, the number of sets is “small” iff the number of semisets is “small”; the precise definition of a one-one correspondence between sets and semisets and of the notion “small” will be found in Chapter I. 8. “Reasonable” semisets are sets.
On specifying the word “reasonable” in items 5 and 8 we obtain the theory of semisets; all of the assumptions 1-8 will hold in the interpretations mentioned above. If in item 8 we specify the word “reasonable” in another way we obtain a theory of sets equivalent to the theory consisting of Godel’s axiom groups A, B, C. As far as items 1-7 are concerned, the theory of sets and the theory of semisets are identical; the difference between the two theories is to be found in item 8. In the theory of sets all semisets are considered to be “reasonable” and so the notion of semiset coincides with that of set. On the other hand, in the theory of semisets the “reasonable” semisets are those determined by “reasonable” properties in the manner referred to in item 5. As i n 5 it suffices to restrict ourselves to six “reasonable” properties and to postulate that the semisets determined by these are sets. We shall investigate both the theory of semisets and the theory of sets. In addition we shall consider various extensions of these theories; this is 10
INTRODUCTION
because in some proofs one cannot dispense with axioms of regularity and/or choice (axioms which, as we shall show, are consistent with both theories). However in this introduction we shall not distinguish between the theory of semisets (sets) and extensions obtained by adding regularity and/or choice axioms. The reader will find precise formulation in the body of the text. What i s the relation between the theory of semisets and the theory of sets?
(1) As we have seen, the theory of sets is an extension of the theory of semisets obtained by adding the axiom “all semisets are sets”. Moreover this extension is conservative w.r.t. statements concerning sets alone; i.e. such statements are provable in the theory of sets iff they are provable in the theory of semisets. (2) The theory of semisets is thus a generalization of the theory of sPts obtained by giving up the assumption “all semisets are sets”. In this sense the universe of discourse of the theory of semisets extends the universe of discourse of the theory of sets. (3) It follows from (1) that we have only two possibilities for the three theories - Zermelo-Fraenkel set theory, Bernays-Godel set theory and the theory of semisets: either all three are consistent or all three are contradictory. Moreover, any statement about sets alone is consistent with one iff it is consistent with all. (4) As a n application, the theory of semisets yields an axiomatization of Cohen’s method in the following sense. We formulate certain additional axioms on semisets, called the axioms of support; these axioms are consistent with the theory of semisets. With each statement cp which concerns sets alone, we associate its “semiset version” @ which concerns sets and semisets alone. This is done in such a way that: (i) The statement cp holds in all Boolean-valued models if and only if the statement @ is provable in the theory of semisets with a general axiom of support. (ii) The consistency of cp with set theory can be proved by means of a particular Boolean-valued model if and only if @ is provable in the theory of semisets with a particular axiom of support. Hence, having proved @ in the latter theory we have established the consistency of cp with set theory.
( 5 ) Axioms of support extend the theory of semisets conservatively w.r.t. statements which concern sets alone. Hence, having proved a statement of this kind in the theory of semisets with an axiom of support, we know that 11
INTRODUCTION
it is provab!e i i i the theory of semisets alone (and hence also in set theory). The reader will find applications of this method in the book. (6) There are also other axioms on semisets which extend the theory of scmisets conservatively w.r.t. statements which concern sets alone. For example, both the axiom of standardness and its negation extend the theory of semisets In this way. The axiom of standardness asserts that every noneimty semiset of ordinals has a first element; it follows from “reasonable” support axioms. Thc theory of semisets with the negation of the axiom of standardiiess will not be studied in the present work, but I t can be used in the study of non-standard analysis for example.
In presenting the theory of semisets the authors hope to make some contribution to the task of b r e a l h g through the bars of the prison in which mathematicians find themselves. This prison is set theory and the authors believe that mathematicians will escape from it just as they escaped from the prison of three-dimensional space. The book is devoted to mathematical and metamathematical investigations of the notion of semiset and related notions such as the nction of support. Particular consistency results will play the role of examples; for example the consistency and independence of the axiom of choice, of the generalrzed continuum hypothesis, G f the axiom “every set is definable”, the results of Jech and Sochor ctc. To say that we shall be concerned with both mathcmatical and metamat hcinatical results means that we shall riot only prove theorems within tlic theory of semisets but we shall also coilcern ourselves with results about theories. Thus we shall consider what statements are or are not provable in a theory; in particular we shall be concerned with relative consistency results. The statement “the theory T remains consistent upon the addition of the axiom q” is not a statement of T but a statement about T, a meta-statement. In this book, any statement about sets, semisets, classes, relations, etc. is a mathematical statement or, more strictly, a statement formulated within the theory of semisets or some similar theory: to prove such a statement we appeal to the axioms of the theory in question. On the other hand, statements about the statements of a theory, about provability and unprovability, are metamathematical statements; to demonstrate such statements (i.e. to convince ourselves of their truth) we shall use essentially weaker assumptions. In the metamathematical investigation of a theory we shall entirely ignore the usual mathematical meaning of the concepts of the theory and concern ourselves solely with the structure of statements and proofs; in other words, we shall be concerned only with syntactical aspects of theories. The possibility of abstracting from the mean12
INTRODUCTION
ings of concepts in this way we learn from mathematical logic. The aspects of mathematical theories which interest us here may be reduced to the consideration of symbols and their configurations; in a sense a theory is to be regarded as a purely formal game with symbols. We shall exclude certain methods of proof, such as proof by contradiction, from metamathematical arguments. If we assert for example that a certain proof exists then we shall always give instructions for constructing that proof. If we speak of a mapping in a metamathematical context then we shall give instructions which enable us to construct the image of any given object. In metamathematics we deal only with finite objects, such as formulas and proofs for example, and so our method may be said to be finitistic. It does not seem to be necessary to specify in advance exactly what methods are t o be regarded as finitistic; it will be clear from our arguments which methods are being used. We shall establish the relative consistency of a theory T w.r.t. another theory S by giving instructions which would enable us to convert any proof of a contradiction in the theory T into a proof of a contradiction in the theory S. In this book we shall always understand the notion of model in this sense; that is, a model of one theory in another theory consists of directions for converting proofs in the first theory into proofs in the second theory. We usually model one theory in another by interpreting the basic notions of the first theory as certain notions of the second theory; if this interpretation is suitably defined then statements and proofs in the first theory may be converted in a natural way into statements and proofs in the second ,.theory. Since we are concerned only with syntactical aspects of proofs, the models which we consider may be called syntactical models, in contradistinction to semantical models based on Tarski’s notions of satisfaction and truth. As particular examples of syntactical models we have the relative interpretations of Tarski. Many results and ideas of other authors are used in this book. Above all should be mentioned the classical results of axiomatic set theory, due to Zermelo, Fraenkel, von Neumann, Bernays, Godel, Mostowski, Specker, Ltvy and many others, as well as the results and ideas obtained in the application of Cohen’s method, due to Scott, Solovay, McAloon and many others, including the authors’ colleagues, Balcar, Bukovsk?, HrbZek, Jech, Polivka, Sochor and StEppBnek. The authors are very grateful to the latter for many fruitful discussions concerning the content of the book and for 13
INTRODUCTION
several improvements; in particular many proofs were simplified and corrected by B. Balcar and J. Polivka. In developing the new theory the authors had to adapt the ideas of other authors for the purpose. In particular we mention McAloon’s proof of the independence of the axiom of constructibility from the axiom of definability and the method of Jech and Sochor for using consistency proofs of the Fraenkel-Mostowski type to obtain consistency results for set theory with the axiom of regularity. The authors were not able to assign every particular notion or result to its true author and for this reason references have been omitted altogether. The bibliography at the end contains only a selection of important works more or less related to the topics treated in the present book. Thus this book should be regarded as a result of the work of all the mathematicians mentioned above and of the many others whose results have become indispensable for anyone concerned with set theory. The authors would also like to express their gratitude to T. Jech and G. Rousseau who prepared the translation of the book into English and to K. Cuda, Mich. Chytil, J. MIEek, J. Polivka and A. Sochor for their help in reading proofs. P. V. and P. H.
14
CHAP. I SEC. 11
1102
THEORY OF CLASSES
CHAPTER I SECTION 1
The theory of classes
In the present Section we introduce the fundamentai fragment of the various (godelian) theories of sets and consider a very general example of the notion of a model. The treatment in this Section will not be too formal. The reader should find the ideas natural, but on the other hand we will see the necessity for a systematic foundation. (Places where such a systematic foundation is necessary will be indicated by the sign Systematic foundations will be laid in Section 2; the reader will then see that the considerations of Section 1 are consistent with the general metamathematical conception developed there; on the other hand, the considerations of Section 1 will serve to illustrate the general concepts introduced in Section 2. We shall come to theories dealing specifically with sets and semirets in Section 4.
[!I.)
a ) Axioms of the theory of classes 1101. The fundamental concepts of the theory of classes are the concept of class and the concept of membership. We will denote arbitrary classes by the letters X , Y, 2, ... (with additional indices if necessary); i.e., the letters X , Y, 2, ... will be used as variables f o r cZasses. The statement “ X belongs to Y” (or, “ X is an element of Y”) will be written as X E Y ; we call the symbol E the membership predicate. We shall use logical connectives &, v , 3 , =, 1 (and, or, implies, if and only if, not) and quantifiers V, 3 (for every, for some). The formula X = Y will stand for “ X is equal to Y” (“X is the same class as Y”). For the time being, we shall treat these symbols as a standard vocabulary for expressing mathematical statements by formulas; e.g. the formula (VZ) (2 E X = 2 E Y) will be considered to stand for the statement “for every 2, Z belongs to X if and only if Z belongs to Y”. 110.2. AXIOM (FL) ( 3 X ) (3Y) ( X E Y ) .
hIOM(F2) (VZ)(ZEX
=ZEY)
E X
=
Y. 15
I103
THEORY OF CLASSES
[CHAP. I SEC. 1
DEFINITION (dl) M(X) = (3Y)( X E Y ) . DEFINITION (d2) [(qx) (x = X ) = M(X)] & (Vx) ( 3 X ) (x = X ) .
Axiom (Fl) states that there exist classes X and Y such that X belongs to Y. Axiom (F2) (the axiom of extensionality) states that X is equal to Y if and only if X has the same elements as Y.One part of this axiom is obvious if X is the same as Y then for any Z , Z E X iff *) Z E Y.This is a general property of the identity. The other part is nontrivial; if X and Y have the same elements then they are equal (i.e., different classes are distinguished by some element). The theory having F1 and F2 as its axioms is called the fuiidarnental E-theorq? T E (or, the fundamental theory of the epsilon predicate). The predicate M defined in (dl) stands for “. .. is a set”; a set is defined as a class which js a n element of some other class. Definition (d2) serves to introduce a new sort of variable; these are the set-variables (variables for sets), for which we use lower case letters; X is equal to some x iff it is a set. An immediate consequence of axiom (Fl) is the statement ( 3 X ) M(X), i.e. there exists at least one set.
[!I;
1103. AxIoM((A1) (Vx, y ) ( 3 z ) (Vu) (U E z =. u = x v tl = y ) . This is the pairing axiom; for any two sets x, y there exists a set whose elements are just x and y . It follows from the axiom of extensionality that given two sets x and y, there can be at most one set having x and y as its only elemcnts, i.e. [(VU)(UEZ &(VU)(UE
w
E
= .u = x .u
v
tl =
=x v u =
y)]
y)& +
z
=
w.
Thus we have proved from (Al) and (F2) that for each x and y there exists a unique z such that
if the symbol 3! is used to stand for “there exists a unique”, then we may write this as follows:
( v x , y) ( 3 ! z )(vu) (u E z
= .u
*) “iff” is an abbreviation for “if and only if ”.
16
=x
v u = y) .
CHAP. I SEC. 11
1107
THEORY OF CLASSES
We denote this unique z by {x, y } [!] (the pair of x, y), i.e. we define the pairing operation as follows: (d3)
(U =
u E {x, y }
x v u = y).
We shall write {x} instead of {x, x}. 1104. The ordered pair and the ordered triple are defined in the usual way as follows: (d4)
<x, Y >
(d5)
<x, Y , z> = (x, ( Y , z>> *
= {{x}, {X’
Y}}
1105. LEMMA. (x, y ) = ( u , u>
= (x
= u & y = u)
.
Proof. Exercise. 1106. (d6) Re1 (X)
= (Vz)
( z E X + (3u, u ) ( z
=
(u, u ) )
.
X is a (binary) relation if all its elements are ordered pairs. (d7)
Rel, (X)
= (Vz) ( z E X 3 (3u, u, w) ( z
=
(u, u, w)))
.
X is a ternary relation if all its elements are ordered triples.
1107. We shall now present seven axioms which ensure the existence of certain classes. As we shall show later, these axioms enable us to prove for any “reasonable” property the existence of a class whose elements are just the sets with this property. (B1)
(32) (vx) (x E
(B2)
( V X ) (32)(VX) (X E Z
(B3)
(VX, Y ) ( 3 2 )(Vx) (x E 2
E
( 3 ~U,) (X E
=
(u,
V)
x E X &x 4 Y )
= ( 3 ~( ()Y , x> EX)) (B5) (VX, Y ) (32) (Vx) (X E 2 = (Yu, u) (X = (u, (B4) (VX) (3Z) (vx) (x
E
2
(B6)
(VX) (32) (VX)
2
E
( 3 ~U,) (X
(B7)
(VX) ( 3 2 ) (Vx) (x E Z
E
(3u,
(X E
& u E o& x E X))
=
U)
& u E Y & x EX))
(u, u ) & ( u ,
U , W ) (x =
(u,
V,
U>
W) &
EX))
( v , W , u) EX)) .
Axiom (Bl) ensures the existence of the class of all sets (the universal class or universe). Axiom (B2) ensures for any X the existence of the class whose elements are all ordered pairs (u, u ) E X such that u is an element of u; this class is called the representation of E on X. Axiom (B3) is the axiom 17
1108
THEORY OF CLASSES
[CHAP. I SEC. 1
of complement, and (B4) is the axiom of domain (or projection). (B5)is the axiom of restriction; the elements of Z are the ordered pairs ( u , v) belonging to X such that v E Y. Axioms (B6) and (B7) are the axioms of converse. It follows immediately that in each of these 7 cases, the class 2 is determined uniquely by X in cases 2, 4, 6 and 7, and by X and Y in cases 3 and 5. Hence all of these axioms may be strengthened by writting (3!Z) instead of ( 3 2 ) ; these strengthened statements follows from axioms (Bl) to (B7)and the axiom of extensionality. Hence the following definitions are justified:
1108. (d8) (Vx) (x E V) V is a constant denoting the universal class. & (Vu, u) ( ( a , v ) E E(X)= (u,v} E X & u E u) (d9) Re1 (E(X)) (the representation of membership on X)
(dtO) X E X - Y X E X & X $ Y (the complement of Yrelative t o X ) (dl l ) x E D (x)= ( I Y ) ( ( Y , x> EX) (the domain of X) (d12) R e i ( x r Y ) & ( v u , v ) ( ( u , u ) E x r Y ~ ( (the restriction of X to Y) (d13) Re1 (Cnv ( X ) ) & (Vu, u) ((u, u ) (the converse of X )
E
Cnv (X)
(d14) Rel, (Cnv, ( X ) ) & (Vu, v , w) ((u , v, w) (the ternary converse of X).
u , U ) E ~ & U E ~ )
= (u, u ) EX)
E Cnv,
(X)
= ( 0 , w, u } E X)
1109. Axioms F1, F2, Al, B1 -B7 (together with definitions (dl)-(d14)) are the axioms of the theory of classes TC. We have defined one constant V and a number of operations, e.g. pairing, erepresentation, complementation, domain, restriction and two converses. The operations just mentioned are called the (fundamental) godelian operations and we shall sometimes denote them by F,, ..., F,.
1110. A number of useful definitions follows:
(15) ClS(X)
=X
=X
( X is class; every class satisfies CIS (x)) (d16a) X 18
c Y = (VZ)(ZEX + ZEY)
CHAP. I SEC. 11
1112
THEORY OF CLASSES
( X is a subclass (a part) of Y, X is included in Y, Y contains X); E is the predicate of inclusion (d16b) X c Y = . X c Y&X += Y (X is a proper subclass of Y); X 4 Y means the same as 7 (X
= Y)
(d17) 0 = V - V (the empty class)
(d18) -X = V - X (the complement o f X ) (d19) X n Y = X - (X - Y) (the intersection o f X and Y) (d20) X u Y = - ( - X n (the union of X and Y)
-Y)
(d21) x x Y = (v r y ) n c n v ( v (the product ofX and Y)
rx)
(d22) E = E(V) (the representation of membership) (d23) W (X) = D (Cnv (X)) (the range of X) (d24) C (X) = D (X) u W (X) (the field of X). 1111. LEMMA(TC).1 ( ~ x ) ( x E O ) ; xEXnY = .XEX&XEY; X E Ex g x ; XEXUY E .XEX v X E Y ; Rel(X x Y)&(Vu,u)((u,u)EX x Y = u E X & u E Y ) ; Re1 (E)& (Vu, u ) ( ( u , u ) E E E u E u) ; XEW(X) E(3Y)(<X,Y)EX).
-x
Proof. Exercise. 1112. Finally we define
19
1113
[CHAP. I SEC. 1
THEORY OF CLASSES
(this definition gives instructions for defining quadruples, quintuples etc.).
v’
=
v,
V”+l = v x
(vn)
(V” is the class of all n-tuples).
b) Representability of formulas In this book we shall be dealing not only with the theory TC, but also with certain other theories which have additional axioms (in certain cases we shall also consider theories with fewer axioms). Each of these theories will include the axioms FL, F2; in other words we will be concerned with extensions of the fundamental €-theory TE. Each such theory has the predicates E and = and the class-variables; however there may be additional predicates, operations and constants present, as well as partial variables (ranging over some of the classes). In fact we have already introduced for example the predicates M. Rel, Rel,, =, E,c ,the constants V, 0, E, the operations E, D, Cnv, Cnv,, { ), ( ), -, and the set-variables x, y , z, ...
r
1113. Given a theory T, a formula cp i s said to be a formula of T or a T-formula if cp is formulated in the language of T. Each variable occurring in q is either free or bound in q. See Section 2 for the details; here we only stress the fact that, given two T-formulas cp and $, cp & $ is a (meaningful) formula (and a T-formula) iff no variable free in one of the formulas cp, $ is bound in the other. (Cf. the definition of normal formulas given below.) Similarly for q v $, cp -+ $, cp = $. Expressions such as D ( X ) ,X , W ( X n n Cnv ( Y ) ) are called terms (“descriptions for classes”). By an extension of a theory T we mean a theory which has among its axioms all axioms of T (thus T is an extension of itself). We promised above to furnish a proof that for any “reasonable” property the class of all sets having this property exists. To specify which properties we have in mind we introduce the metamathematical concept of normal T-formula. We simultaneously define what it is for a variable to be free or bound in a normal T-formula.
1114. METADEFINITION. Let T be an extension of TE. (a) If x is a set variable and Y is an arbitrary variable or constant of T different from x (the last condition is automatically satisfied if Y is not a set 20
CHAP. I SEC. 11
1115
THEORY OF CLASSES
variable) then x E Y is a normal T-formula; x isfree in s E Y ; if Yis a variable then Yis free in x E no variable is bound in x E Y. (b) If cp and $ are normal T-formulas and if no variable is free in one formula and bound in the other, then cp & $ and 7cp are norinal T-formulas? a variable is free (or bound) in cp & $ if it is free (or bound) in either cp or $; a variable is free (bound) in 1cp if it is free (bound) in cp. (c) If x is a variable not bound in a normal T-formula cp then (3x) cp is a normal T-formula; x is bound in (3x) cp; other variables occurring in cp are f r e e (bound) in ) .3( cp if they are free (bound) in cp. (d) Every normal T-formula is obtained by a finite number of applications of rules b) and c), beginning with normal T-formulas described sub a). Remark. If the theory T is fixed or is clear from the context we say “normal formula” of “NF” instead of “normal T-formula”. E.g. the formula l ( 3 z ) (z E
x & z 6 Y ),
which is equivalent to the formula X c is a NF. Notice that no classvariables are bound in NFs. We shall now give the promised existence proof. For each normal formula we prove the existence of the corresponding class; moreover we describe this class. 1115. METADEFINITION. Let T be an extension of T C (e.g., T C itself) and let cp be a T-formula. (1) A sequence x,,. .., x, of distinct set variables is said to befree for cp iff no x i (1 5 u 1. n) is bound in cp. (2) Let xl, . .., x, be free for cp. We say that cp is representable in T w.r.t. x,, . .., x, if there is a term T of T such that the following is provable in T:
(*)
(Vx,, ..., x,) ((x,,
..., x,)
ET
= cp)&
T c V” ,
T is said to represent cp in T w.r.t. x,, . . ., x,. (3) cp is said to be fully representable in T if it is representable in T w.r.t. any sequence free for cp. Remark. It is not required that the xiactually occur in cp, but only that they should not be bound in cp. On the other hand it is possible that cp contains some free variables not occurring among the x;s; let these be Yl, . .., Y, (not necessarily class variables). Then we understand (*) as saying the same as (VY,, .. ., Y,) [(Vx,, ..., x,) ((x,,
. .., x,)
E
T
= cp)&
T
c V”] .
21
1116
THEORY OF CLASSES
[CHAP. I SEC. 1
(Cf. the axioms of group theory: one says ‘‘x -t- ( y + z ) = (x + y ) + z is an axiom of group theory” meaning “(Vx) (Vy) (Vz) (x + ( y + z ) = (x + + y ) + z ) is an axiom of group theory”.) In order to demonstrate that every N F is fully representable in TC [and also for other purposes) we demonstrate some Metalemmas. I n these Metalemmas T is an arbitrary extension of TC.
1116. METALEMMA. If cp is fully representable in T then 1cp is fully representable in T. Demonstration. If T represents cp w.r.t. x l r . .., x, then V“ - T represents l c p w.r.t. x l , ..., x,. 1117. METALEMMA. If cp, t+b are fully representable in T then cp& JI is fully representable in T. Demonstration. If TI and T, represent cp and t+b respectively w.r.t. xl, . .. ..., x, then T, n T, represents cp& t+b w.r.t. xl, ..., x,. 1118. METALEMMA. If cp is fully representable in T and x is a set variable not bound in cp then (3.x) cp is fully representable in T. Demonstration. Let xl, . .., x, be a sequence free for (3x) cp. Then x,xl, ... ~. ., x, is free for cp and, by the full representability of cp, we have a term T such that (V ...) ((x, xl,..., x,) E T E cp)& T E Vn+’ is provable in T. Consequently, (V . . .) ((x,, ..., x,> E D (T) E (3x) cp) & D (T) E V” is provable in T. Remark. (V ...) q denotes the closure of cp, i.e. (V ...) cp is an abbreviation for (Vx) . . . (Vy) cp where x,..., y are all the free variables of cp. 1119. METALEMMA Let x be a set variable and let Yan arbitrary variable or constant of T distinct from x.Then x E Y is fully representable in T. Demonstration. The demonstration is technically difficult but the reader is recommended to read it in order to become more familiar with operations we have introduced. Let xl,. . ., x, be a sequence of distinct set variables. First suppose that this sequence does not contain Y. (This is automatically the case if Y is not a set variable.) (a) Consider the “pathological” case where x does not occur in xi, ..., x,. Then we are looking for a term T such that the following is provable: T = 0 if x 4 Y, T = V” if x E Y. Show that the term D ((Y n (x)) x V”) has this property. (b) Now let x be x i ; we are looking for a term T such that (<xl, . .., x,) E E T = xi E Y) & T G V“ is provable. If n = 1 take Y for T. If n > 1 and 22
CHAP. I SEC. 11
1120
THEORY OF CLASSES
i = 1 t a k e Y x V"-lfor T . I f n > 1 a n d i = n t a k e V x (V x n-
x Y ) ...) f o r T . I f 1 < i < n tak eV x (V x ... x (V x (Y x for T. i - 1 times
... x (V
x
i times V-I))...)
Now let Y be xi. (c) If x does not occur in x,, ..., x, then we are looking for a term such that the following is provable: T is the class of all n-tuples such that the j-th element contains x. For n = 1we must find a term S defining the class of all sets containing x (as an element). Show that W ((Cnv (E)) {x}) has this property. For n > 1continue as in (b) above writing W ((Cnv (E)) instead of Y.
r
r (XI)
(d) Finally, let x be xi and let Y be xi. We have assumed that x and Y are distinct, so that i 9 j. If i < j let TI be the constant E; if i > j let TI be the term Cnv (E). Put i = min ( i , j), j = max (i, j). It is provable in TC that (xi, xi) E TI = xi E xi. If j = n let T, be T,; otherwise let T, be the term Cnv, (Cnv, ((V x ... x V) x Tl)); it is provable that
-
(n- j ) times
(xi, xi, xi+ 1, ..., x,) E T, E x iE xi. If j = 5 notice first that the following is provable: (xi(xi, ..., x,))
E
T2 = ( ( x i , ..., xn), xi) E Cnv (T,)
zz <xi-l, <xi, ..., x,), xi)
= (xi,
+ 1, let T, be T,
xi, . .., x,)
E
E
V x Cnv (T,)
if j 9 i
+1
=
=
Cnv, (V x Cnv (T,)) .
We have succeeded in putting the variable xi-l inside. If we repeat this device ( j - i - 1) times, we obtain a term T, such that the following is provable:
(xi,
. .., x,)
~ i + l ,
E
T,
xi E x j .
Hence we let T be the term V x (V x ... (V x T,)). This completes the demonstration. ( i - 1 ) times
1120. METATHEOREM. If T is any extension of T C then every normal T-formula is fully representable in T. This is an immediate consequence of the preceding lemmas. In fact, we have shown even more: if each variable free in q is one of the x i then the operation { ] is not used in the construction of T. Thus, define: 23
1121
THEORY OF CLASSES
[CHAP. I S E C 1
1121. METADEFINITION. Let T be an extension of TC. (a) Every variable and constant of T is a godelian term of T. (b) If T,, T, are godelian terms of T then E (T,), TI - T,, D (T,), T, T,, Cnv (Tl), Cnv, (TI) aregodelian terms of T. (Briefly, godelian terms.) In the following corollaries T means an arbitrary extension of TC.
r
If cp is a normal T-formula and x,, ..., x, are all the 1122. COROLLARY. free variables of cp then there is a godelian term representing cp in T w.r.t. XI,
...) x,.
Remark. If = $ is provable in T and cp is fully representable in T then obviously $ is fully representable in T. Let T be an extension of T C and let $ be a T1123. METADEFINITION. formula. I) is said to be normal in T if there is a normal T-formula cp such that cp = $ is provable in T. Every formula normal in T is fully representable 1124. COROLLARY.
in T. Example. Show that formulas x
= y,
x
=
Yare normal in TC.
1125. The term which represents a N F cp(x,, ..., x,, X I , ..., X,) w.r.t. the sequence x,, . .., x, of its free set variables will often be denoted by the expression
{<x,*...> x,>;
(P("1,
..., Xn,X1, . . ' , X , ) } ~
In what follows we shall work in the theory TC. 1126. METALEMMA. Let T be a n extension of TC; let cp be a normal Tformula, and let Y be a variable free in cp. If G is a godelian term of T such that no variable occurring in G is bound in cp, then the result of replacing all occurrences of Y in cp by G is a formula normal in T. Demonstration. Y can occur either in the context x E Y or, if Y is a set variable, in the context Y E 2. The last formula is equivalent to (3u) (u = = Y& u E 2) where u is a set variable not occurring in cp, and u = Y is equivalent to (Vu) (u E u E u E Y). Hence it suffices to show that u E G is a formula normal in T. This is done by induction. (a) If G is a variable or a constant then u E G is normal. (b) Let us show that the formula (zl,z,) E R is normal in TC. This formula is equivalent (in TC) to (3x) (x = (z,, z,) & x E R), hence it suffices to show that x = (z,, z,) is normal. But (in TC) x = (zl,z,) = 24
CHAP. I SEC. 11
1128
THEORY OF CLASSES
= x = {{q},{zl, z,}} = (3.4v) (x = (24’ v} & u = {zl, z,} & 0 = {Zl,z2}, hence it suffices to showthat x = {u, u) is normal. Now (in TC), x = { u , u ) = = ( V y ) ( y ~ x3 y = u v y = v) and the right hand side is normal because y = u is normal. Analogously one can show that ( z , , z2, z 3 ) E R is normal. Moreover, if t is a term and t E R is normal then t = R is also normal (for it is equivalent to (Vv) (v E t = u E R)). Now, let G,, G, be godelian terms of T and suppose that u E G,, u E G, are normal. Then u E E (Gl), u E (GI - G,) etc. are also normal as may be seen from the following equivalences provable in T:
.
= (guy W ) ( U = (u, w)& u E W & G -~ G2 = U E G ~ & ~ $ G etc. ,,
u EE(G,)
U E
u E GI).
1127. A number of useful definitions follows:
(d25) U(X) = ( z ; (3Y) ( z E Y 8z Y E X ) > (the sum o f X (the union o f X ) , sometimes denoted by UX) (d26) P(X) = ( z ; z c X> (the power-class of X ) (d27) X”Y = w ( X
rY )
=
iu; (go
Y ) (+, v)
(d28) U n ( X ) = R e l ( X ) & ( V u , v , w ) ( ( v , u ) E X & ( w , u ) E X . - , r (X is a function (or mapping))
= w)
(d29) U n ( X , Y , Z ) = . U n ( X ) & D ( X ) = Y & W ( X ) s Z ( X is a mapping of Y into Z)
.
(d30) Un, ( X ) = Un ( X ) & Un (Cnv (X)) ( X is a one-to-one mapping)
.
(d31) Un, ( X , Y, Z) = Un (X, Y, Z) & Un (Cnv ( X ) , 2, Y ) ( X is a one-to-one mappings of Y onto Z) (d32) F’y = z if Un(F) F’y = 0 otherwise ( z is the image of y in F )
and
(2,y)~F
c ) A model 1128. We shall now be concerned with the study of relations. 25
1129
[CHAP. I SEC. 1
THEORY OF CLASSES
(d33) If R is a relation and x
EdR ( x ) = { y ; (the extension of x in R ) (
~
X> E
7
C ( R ) , then R)
E
(d34) Sat, ( X ) E X E C ( R ) & (VX,y =EX~~(Y).+~EX) ( X is a saturated part of C ( R ) ) .
E
C ( R ) )( X
E X & EdR (x) =
(d35) ForX E C ( R )
SAT, ( X ) = { z E C ( R ) ;(3x E X )(EXfR(x) = EXfR (2))) (the saturation o f X ) . 1129. LEMMA (TC).If R is a relation and X and Yare saturated parts of its field then X n X u Y and X - Yare also sarurated. The saturation of a subclass of C ( R ) is saturated. 1 130. DEFINITION (TC).
(d36) A relation R is extensional (Extl ( R ) ) if
(x, Z)
E
R & EdR( x ) = E d R( y )
--+
(y,
Z) E
R
for all x , y , z in C ( R ) .
1131. LEMMA (TC).R is extensional if and only if for each x E C ( R ) the extension of x in R is saturated. 1132. DEFINITION (TC).
(d37) Prg ( R ) E (VX, y ( R is pairing)
E
C ( R ) )(32 E C ( R ) )(Ed,
(2) =
SAT,
((X,
Y}))
+
(d38) Elk ( R ) = Re1 ( R )& R 0 & Extl ( R ) & Prg ( R ) . Thus a non-empty relation R is E-like if it is extensional and pairing. (d39) SExtl ( R ) ( V X , y E c ( R ) (X 4. y ( R is strongly extensional).
-+
Ed,
(X)
* EXtR ( y ) )
1 f33. LEMMA (TC).(a) Any strongly extensional relation is extensional. (b) A relation R is strongly extensional iff every part of C ( R ) is saturated. Proof. (a) If a relation R is strongly extensional, then EXtR ( x ) = EXtR ( y ) implies x = y , and hence (x, z ) E R implies ( y , z ) E R . (b) If x =k y and 26
CHAP. I SEC. I ]
1138
THEORY OF CLASSES
Ext, (x)
= Ext, ( y ) then {x> is not saturated. Hence if every part of C ( R ) is saturated then R is strongly extensional; the converse is obvious.
1134. LEMMA (TC). E is an E-like relation.
Proof. Since there exists at least one set x and we have (x, {x>> E E by A l , the relation E is non-empty. E is strongly extensional, since Exte (x) = x for any x. Finally, E is pairing, since for z = {x, y] we have Ext, ( z ) = = {x, Y > = SAT,({x, Y > ) . 1135. In what follows we denote by the constant R some fixed (arbitrarily chosen) E-like relation. By that we enlarge T C in an inessential manner [!]; in the remainder of this Section we shall denote by TC, the theory TC with
the constant R and with the assumption Elk (R) (and possibly with further definitions in which the constant R is involved). 1136. DEFINITION (TC,).
( q X * ) (X* = X )
= Sat,
(X);
the variables X * , Y*, . .. range over saturated parts of C (R);
X*
E*
Y*
3
( 3 E~Y * ) (X* = Ext, (z)) .
1137. Remark. 1) X * = ExtP(z) means that X * is the extension of z (in R). We shall also say that z is a code of X * (or, precisely, and R-code of X * ; this phrase will be often used). Then the preceding definition reads: X * is an *-element of Y* iff there is a code of X * which is an element of Y*.
Thus R “codesyythe new membership.
2) If R = E then the variables X * range over all classes and X equivalent to X E Y. (Prove.)
E* Y
is
We have introduced new variables X * and a new predicate E*; we shall regard the X * as new classes and E* as a new membership predicate. We may ask whether we are justified in doing so, whether the axioms of TC hold in the new sense. We shall show that this is indeed the case. Moreover, in the course of the verification we shall make clear the precise meaning of the phrase “the axioms hold in the new sense”. 1138. LEMMA(TC,).(3X*, Y * ) ( X * E* Y * ) .
(Axiom (Fl) in the new sense.)
1139
[CHAP. I S E C . 1
THEORY OF CLASSES
Proof. Since R is nonempty there exists some (x, y ) E R and hence x E ExtR(J.). The extensions of both x and J.' are saturated, since R is extensional; hence Ext, (x) E* Ext, (4'). 1139. LEMMA (TC,). (VX*, Y * )( X * = Y* E (VZ") ( Z * E* X * = Z* E*Y*)). (Axiom (F2) in the new sense.) Proof. The implication from left to right is trivial; hence suppose that X * $: Y*, e.g. that X * - Y* 0. Let z E X * and z 4 Y*; we have Exta (2) E* E* X * ; if Ext, (L) E* Y*, then Ext, ( z ) = ExtR ( u ) for some u E Y*, so that z E Y* since Y* is saturated; this is a contradiction, so we have Z* IS*X * and Z* $* Y*, where Z * = Ext, (z). Hence l(VZ*) (2" E* X * = Z* E* Y*).
+
1140. DEFINITION (TC,). M*(X*)
= (3Y*) ( X * E* Y*) .
(Definition (dl) i n the new sense.) 1141. LEMMA (TC,). M*(X*)
(32 E C (R)) ( X * = Ext, (z)). Proof. The implication from left to right follows from the definition of E*; suppose that X * = Ext, ( z ) for some z E C (R). Since C (R) is saturated we have X* E* C (R)and (3Y*) ( X * E* Y*); hence M*(X*). 5
1142. DEFINITIOX (TC,). (VX*) [ ( 3 ~ * )(.x* = X * )
M*(X*)] & (VX*) ( X f * )(x*
=X*)
1143. LEMMA (TC,).
(Vx*, y*) ( 3 z * ) (vu*) (ti* €* z*
E
. 1L*
=
x* v u*
=
y*) .
(Axiom A t in the new sense.) Proof. Suppose that x* = Ext, ( p ) , y* = Exta ( 4 ) and let z be such that Ext, ( 2 ) = SAT, ( { p , 4)) (such a z exists since R is pairing). For z* = = Ext, ( z ) we have u* E* z* = . u* = x* v u* = y*. 1144. DEFINITION (TC,).
-, * E * ! ,x* ,y*)*
. z*
= x* v z* = y*
.
1145. METADEFINITION. If cp is a formula of TC then let 'p* be the formula which is obtained by adding asterisks to all predicates other than = and to 28
CHAP. I SEC. 11
1148
THEORY OF CLASSES
all operations and variables; i.e. we replace E by E*, X by X * , x by x*, M by M* etc. T o prove (Bl)*, . .., (B7)* in TC, we need the following 1146. METALEMMA. If cp(x,,
(*I
(Z1 E
0 , XI, 0 )
c (R) & 0 ) -+
is a N F then
V*(EXtR ( Z l ) ,
0,
x:,
0)
is normal in TC,. (The formula cp*(ExtR(zl), 0, X : , 0 ) is the result of substituting of Ext, (zl), 0 for x:, 0 in q*(x:, 0, X : , 0). xl, o is a n abbreviation for a finite sequence of variables, say, x,, ..., x,. Analogously in other cases.) Demonstration. For each N F cp we shall find a formula rfi normal in TC, and such that (2, E
C (R)&
0)
-+
[cp*(Ext, (z,), 0 , X:,
0)
= $(zl,
0 , X:,
0,
R)]
is provable in TC,.Then the formula (*) is equivalent to (zl
E
C (R)&
0 ) -+
IcI(zl,
0 , X:,
0,
R)
which is normal in TC,.If cp is x, E x2 take ( z , , z t ) E R for $; if cp is xl E X , take z1 E X : for rfi. The induction step for & and 1 presents no difficulties. Let cp(y, x,, 0 , X , , 0 ) be a NF and suppose (w -+
E
C (R)& z1 E C (R)& 0 )
[cp*(EXtR ( W ) , EXtR (Zl), 0 ,
is provable in TC,where
I)
x:,
0)
-+
tb(W, Z1. 0 ,
x:, 0 , R)]
is normal i n TC,.Then
is provable in TC, and the formula (3w normal in TC,.
E C (R)) $(HI,
z,,
0,
X?,
0, R)
is
1147. Remakk. The normality of the formula (zlr z 2 ) E R in TC, can be established as in the demonstration of 1126. 1148. METATHEOREM. (Bl)* to (B7)* are provable in TC,. Demonstration. a) (Bl)*. We can prove in TC, that C(R) is saturated and that (Vx*) (x* E* C (R)); hence (3Z*) (Vx*) (x* E* Z"). 29
I149
THEORY OF CLASSES
[CHAP. I SEC. 1
b) (B2)* - (B7)*. Axioms (B2) to (B7) have the form (VX,, ...) ( 3 2 ) (Vx) E pi(x, X , , 0 ) ) where the cpi are normal in TC. Hence we may argue as follows in TC,: For any X T , 0 let z € 2 = z E C ( R ) & & pT(ExtR(z), X T , 0 ) ; such a class 2 exists in view of 1124 and 1146. From the defining property of Z we see that if E d R(z) = E X t R (w), then z E Z implies w E Z ; hence Z is saturated. Thus we may set Z* = Z , so that x* E* Z* = cp:(x*, X r , . . .). We have the following (x E Z
.
1149. METATHEOREM. For any axiom cp of TC, p* is provable in TC,. 1150. We see therefore that the sort X * and the predicate E* furnish a model of TC in TC,. We assigned to each formula cp of TC its counterpart cp* (the same formula in a new sense). Evidently [!] the counterparts of provable formulas are provable; for we have shown that the axioms hold in the new sense and hence each proof may be performed in the new sense. I n this way we obtain a number of provable statements. Notice thar if tp is a formula of TC then cp* tells us something about a fixed E-like relation R. If we carry out the development in some extension T of the theory TC,, everything proceeds in the same way: we introduce the asterisked concepts and prove in T all asterisked axioms of TC. We may of course impose on R stronger conditions than those mentioned above; everything proceeds in the same way as before, provided that we guarantee (in T) the existence of some R which satisfies these requirements. It may turn out that, in addition to the asterisked axioms of T, we can also prove in T some p* for which it is not known whether is provable in T. It may then be asserted that if T is consistent then it remains consistent on the addition of cp as a new axiom. For if a contradiction would be proved in this extended theory, then an analogous proof could be carried out for the asterisked concepts, and so a contradiction could be proved in the theory T (in which cp* is provable). This example gives us an indication of what models are and of the significance of constructins models; the construction of a model serves at least as a means for generating new statements (and notions) and as a method for providing relative consistency proofs. The next section will provide a formal foundation for the developments carried out so far; however the results of the present section serve not only as an illustration but play an important role throughout the book. After reading Section 2 the reader may return to Section 1; he will then see that the present section is consistent with the formal conception to be presented in Section 2.
[!I,
30
CHAP. I SEC. 21
LOGICAL FOUNDATIONS
1201
SECTION 2
Logical foundations
1201. In this section we shall provide a formal foundation for certain informal considerations, including those of the preceding section. We shall analyse precisely various concepts such as “theorem”, “proof”, “definition”, “model”; thus we shall not take these notions for granted but shall derive them from simpler concepts and so reduce to a minimum the number of nctions which are taken to be intuitively known. In doing this, we shall try to observe two requirements: the requirement of exactness - we shall for example define precisely and in an entirely abstract way the notion of a model - and, on the other hand, the requirement of adequacy - we shall require that the defined notions adequately represent the intuitive notions. It turns out that essentially, it suffices to take as a basis for our considerations the notion of symbols and their finite configurations. The developments of Section 1serve as an illustration; on the other hand, these developments satisfy the formal requirements of the present section. In what follows, a large number of important concepts will be introduced. To avoid confusion we shall first indicate the most important of these. Firstly, the notion of an axiomatic theory is defined; each theory has its language (basic concepts) and axioms (concerning these concepts). Further, we define what it is for a statement of the language (a formula) to be provable in the theory. A model of one theory in the other assigns to each formula of the first theory a formula of the second theory in such a way that provability is preserved (in a certain well-defined sense). The concept of afaithful model is of special importance; a few criteria for models to be faithful will be given. Direct models are a very simple type of model which so to speak only change the notation of basic concepts. Definitions are axioms of a certain type which introduce a new basic concept; it follows that definitions do not enrich the theory, i.e. a definitional extension of a theory does not make any new 31
I202
LOGICAL FOUNDATIONS
[CHAP. I SEC. 2
formulas of the original theory provable. The theory of classes and other theories investigated in this book are built up from two concepts - membership relation and class - by adding axioms which either do not enrich the language or are definitions (or “quasi-definitions”). Such theories are called €-theories. It turns out that for any €-theory T,it suffices to “model” (in some theory S) only the basic concepts mentioned; this already induces a direct model of T in a certain definitional extension of S. We shall now carry out in detail the programme outlined above. If the notions informally explained in the outline are already sufficiently clear to the reader, he may omit the rest of the section, returning as occasion demands,
a ) Formulas, Axioms, Theories
1202. The spmbols which will appear in our work are the following: a) Variables of various sorts (e.g. class-variables XI, X 2 , ..., set-variables
xl, x2, . ..); a sort is assumed to be an integer; for each pair of integers p
and q there is a unique variable which is called the q-th variable of sort p . b) Predicates of various arity (e.g. the unary (1-ary) predicate M, “. .. is a set”, the binary (2-ary) predicate E “... is a n element of ...”); for each pair of integers p > 0 and q , there is a unique predicate called the q-th p-ary predicate. c) Operations of various arity (e.g. the 0-ary operation V, “the universal class”, the unary operation D, “the domain o f . . .”, the binary operation n, “the intersection of . . . and . . .”); for each pair of integers p and q there is a unique operation, called the q-th p-ary operation; 0-ary operations are called constants. d) Connectives: binary &, v , +, valent to”) and unary I (“not”).
=
(“and”, “or”, “implies”, ‘‘.1s equi-
e) QuantiJiers: V, 3 (universal, existential).
f ) EquaEity predicate all predicates sub b)).
= :a
particular binary predicate (different from
g) Parentheses (,). It is assumed further that all these symbols are distinct, that none of them is a n integer and that for each symbol it can be decided to which of the 32
CHAP. I SEC. 21
LOGICAL FOUNDATIONS
1205
groups a) to g) it belongs; if a symbol belongs to one of the groups a), b), c) then the corresponding integers p and q are uniquely determined. 1203. A language is any finite sequence of distinct predicates, operations and sorts; we assume that this sequence does not contain the equality predicate and that it contains at least one sort. In referring to the language we shall often represent a sort by some variable of this sort, e.g. by the first one. For simplicity it is sometimes assumed that each predicate precedes each operation and each operation precedes each sort. A language L, is a sublanguage of a language L, if each predicate (operation, sort) of L, is a predicate (operation, sort) of L,. Two languages L,, L, are similar if they have the same length and if, for each integer p, the p-th symbol in L, is a k-ary predicate (k-ary operation, sort) if and only if the p-th symbol in L, is a k-ary predicate (k-ary operation, sort). For example, the language (E,X , ) is a sublanguage of (E, 5 ,X , , x,); the languages (E, X , ) and ( E,x,) are similar. Now we define recursively the concepts of term and formula of a language.
1204. (a) Each variable of each sort in L is an atomic term of the language
L;each constant in L is an atomic term of L.
(b) If F is a p-ary operation in the language L and t,, . .., t, are terms of L, then the configuration F(t,, . .., t,) is a term of L. (c) There are no other terms than those obtained from atomic terms by applying the rule (b) finitely many times. 1205. (a) If t, and t, are terms of L then the configuration t, = t, is an atomic formula of the language L; each variable occurring in this formula is free and no variable is bound in this formula. If P is a p-ary predicate in L and t,, ..., t, are terms (of L) then the configuration P(t,, . .., t,) is an atomic formula of L; each variable occurring in this formula is free in P(t,, . .., t,) and no variable is bound in this formula. (b) If q and $ are formulas of L and if no variable is simultaneously free in 40 and bound in $ or conversely, then each of the following configurations is a formula of L:
a variable is free (bound) in such a formula if it is free (bound) either in q or in $. 33
1206
LOGICAL FOUNDATIONS
[CHAP. I SEC. 2
(c) If cp is a formula of L then so is 19;a variable is f r e e (bound) in 1cp if it is free (bound) in cp. (d) If cp is a formula of L and x is a variable of a sort in L and if x is not bound in cp, then (Vx) cp and ).3( cp are formulas of L; x is bound in this formula and each variable other than x is f r e e (bound) in this formula, it if is free (bound) in cp. 1206. Remark. If P is a binary predicate and if t, and tz are terms, then we often write t, P tz instead of P (tl, tz) (e.g. tl E t, instead of E (t,, t2)); similarly for binary operations (we write X n Y instead of n ( X , Y)).
1207. A formula is closed if no variable is free in it. We shall use lowercase Greek letters 9,$, . .. to denote formulas; if a, is a formula such that x,..., y include all variables free in cp then cp may be written as q ( x , ..., y ) . 1208. Remark. The notion of a formula (in particular, a closed formula) adequately represents the intuitive notion of “statement” or “proposition”. We shall now define the notions of proof and provability. We follow here the natural intuitive idea of a proof; i.e. a succession (sequence) of statements which are either evident or evidently follow from their predecessors.
1209. LOGICAL AXIOMS. I. If cp, $, x are formulas then the following configurations are axioms, provided they are formulas
is an axiom ( ~ ( yis) the result of substituting y for all occurrences of x in cp). 34
CHAP. I SEC. 21
1212
LOGICAL FOUNDATIONS
b) If cp, $ are formulas then the configuration
is an axiom (provided it is a formula; i.e. provided x does not occur in cp). c) If (3x) cp is a formula then (IX) ‘p
=
l(VX) l ’ p
is an axiom. 111. If t, s, r, t,, s1, ... are terms, while P is a predicate and F is an operation, then the ‘following formulas are axioms: t =t t =s+s=t (t = s & s = r ) - + t = r (t, = s,& 0) -+ (P(tl, 0) = P(s,, 0 ) + F(t1, 0 ) = F(s,, (ti =
0)) 0)
(if we suppose that P and Fare p-ary then t, = sI & 0 is an abbreviation for t, = s, & ...& t, = sp, and similarly, (t,, 0) is an abbreviation for (tl, ... * .., t,)). 1210. RULESOF INFERENCE I. Detachment Rule. Let cp -+ $ be a formula. Infer $ from cp and (‘p
11. Generalization Rule. Let (Vx) cp be a formula. Infer (Vx)
‘p
-+
$).
from
‘p.
1211. AXIOMSYSTEM; AXIOMATIC THEORY If L is a language then any finite (possibly empty) sequence of formulas of L is called an axiom system in L. An (axiomatic) theory T consists of a language L and of an axiom system in L. The language L is called the language of the theory T and the members of the axiom system are called the (nonlogical) axioms of T. Instead of “a formula of the language of the theory T” we often say briefly “a formula of the theory T ’ or “a T-formula” .
1212. If T is a theory then a sequence cp,, ..., ‘ p p of T-formulas is a proof in the theory T if for each i = 1, . .., p , ‘piis either a logical axiom or a n axiom of T or is inferred from some of the formulas cpl, . . ., ‘pi- by a rule of inference. A T-formula cp is provable in T
35
1,713
LOGICAL FOUNDATIONS
[CHAP. I SEC. 2
if there exists a proof in T which has cp as its last element. Example. Let t be a term, let x be a variable of a sort s and let ~ ( x be ) a formula such that no variable occurring in t is bound in q. Denote by cp(t) the result of substituting t for all occurrences of x in cp. Let y be a variable of the sort s and distinct from all variables occurring in t. The formula (3j,) ( t = y) can be read “t is of the sort s”. It is a matter of routine to show that the formula
(*>
((v.y)
& (34.1 (t = 4’)
i
f
~(t)
is provable in each theory T such that (*) is a T-formula. The metamathematical definition stated above constitutes one possible formalization of the notion of provability. Since the number of accepted “evident statements” and “evident inferences” is minimized, the formal proofs are very lengthy,; however, the methods used in informal proofs are mirrored by a number of metatheorems on formal provability; this convinces us of the following 1213 . AtlequLicj.Ilypothesis:
The formal notion of provability adequately represents the intuitive notion. i.e. a mathematical proof is correct if and only if each of its parts can be fornialized. We shall use this hypothesis in two ways. Firstly in proving some mathematical statement (c.g. in the set theory) we shall proceed informally (in words): having proved the statement informally we deduce that a formal proof exists and that it would be only a matter of time to find it. On the other hand we shall demonstrate some metathcorems on formal provability; we deduce from them the rules for informal mathematical proof. 1214. Variables of a sort i are denoted (if necessary) by x i , yi,x i etc. If T is a thecry and i, j are sorts in its language, then i is subordinate to j in T if T k (Vx;) (3yj) (.xi = y j ) . The sort i is unioersul in T if for each term t of T and each variable x i not occurring in t we have
T k ( 3 ~ (~x i) = t) ; it follows that all sorts in the language of T are subordinate to i. 1215. LEMMA.Every sort is subordinate to itself; if a sort i is subordinate to a sort j and j is subordinate to a sort k, then i is subordinate to k. (Obvious.) 36
CHAP. I SEC. 2:
LOGICAL FOUNDATIONS
I219
1216. LEMMA.Let T be a theory and let the sort i be subordinate to the sort j in T ; let cp(y') be a T-formula and suppose that x i is not bound in cp. Then T k ( V y j ) cp(yi) -+ (Vx') cp(x'). (Exercise.)
1217. If T and S are theories, then T is stronger than S if every S-formula provable in S is provable in T.
(Corollary: the language of S is a sublanguage of the language of T, i.e. each symbol of the language of S is in the language of T.) In particular, T is an extension of S if the sequence of axioms of S is an initial segment of the sequence of axioms of T. The extension of S by an axiom cp is denoted by (S, cp) or (S cp) or simply S,cp. T is equivalent to S if T is stronger than S and S is stronger than T.
+
1218. METATHEOREM (on deduction). Let T be a theory. If cp a formula and cp is closed, then
-, $
is
Demonstration. If T t- cp --f $ then T, cp I cp -+ I) and T, cp t. cp; hence T, cp t- $ by Detachment Rule. Conversely, if T, cp t- $ then one can verify T k cp -+ by induction on the length of the proof of cp. (Technicalities omitted .)
+
b) Models
1219. Let T and S be theories. A mapping*) 9.N which assigns to each T-formula cp an S-formula cpm is called a (syntactic) model of T in S if the following conditions hold: 1)
preserves axioms; i.e.
whenener cp is a logical axiom or an axiom of T, 2 ) IDZ preserves implication, quantifiers and negation, i.e., for any T-formulas cp and $ and variable x, *) When we speak of a mapping in metamathematics we assume always that the value
to be assigned to each argument is prescribed explicitly.
I220
(We suppose that (cp
LOGICAL FOUNDATIONS
.+
$) and (Vx)
[CHAP.I SEC. 2
cp are formulas.)
1220. T is contradictory if there exists a T-formula cp such that both cp and 1cp are provable in T. T is consistent if it is not contradictory. 1221. LEMMA. T is contradictory if and only if every T-formula is provable in T. (Exercise.) 1222. The Provability and Consistency Principles. Let YJl be a model af T i n S. 1) For any T-formula cp, T I- cp implies S I- cpm; moreover, for any T-formula $, T,$ I- cp implies S, $" I- cp". (The Provability Principle.) 2) If T is contradictory then so is S; i.e. if S is consistent then so is T. (The Consistency Principle.) These are simple but fundamental principles. Dentonstration. 1) Let cpl, . .., cp,, be a proof of cp in (T, $); we show by induction that S, $'" t- pm. If cpi is a logical axiom, an axiom of T or the formula $, then S, $'IR t- cp? by the definition of a model. If cpy, ..., have been proved in (S, $m) and cpi immediately follows from cpj, cpk by the Detachment Rule cpk being 'pi --+ qi,( j , k < i) then S, $" t- cpy, since '2R preserves implication; if 'pi immediately follows from pi by the Generalization Rule (cp, being (Vx) cpj, j < i) then S, $" t- cp? since YJl preserves the quantifier. 2) Suppose T t- cp, T t- 1 cp. Then, by l),S t- cp" and S I- (1 v)"; hence S t- (V ...) (cp") and S t- l ( V ..) (cpm), since 92 preserves the negation. Unprovability Principle. Let 9 2 be a model of T in S, let cp be a T-formula such that S I- l(cpm). If S is consistent then cp is not provable in T. Indeed, if T I- cp then we would have S t- cpln and so S would be contradictory.
"
.
Remark. If (1q ~ is )S-provable ~ then 9R is a model of (T, 19)in S and, if S is consistent, cp cannot be T-provable by the Consistency Principle. But it is possible that ~ ( c p " ) is S-provable and (7~)" is not; moreover, it is possible that both i ( c p m ) and l ( ( 1 c p ) " ) are S-provable. In these cases we obtain the unprovability of cp in T by the unprovability principle. (Note that in most cases we are then able to construct another model W of (T, 1cp) in S.) 38
CHAP. I SEC. 21
LOGICAL FOUNDATIONS
1229
1223. LEMMA. If 1111 is a model of T in S and cp is a T-formula, then 1132 is a model of (T, 'p) in ( S , cp=). 1224. Let 2132 be a model of T in S and let cp be a T-formula; we say that
cp holds (is valid) in 2132 if S t. cpm.
1225. Let T be a theory. The mapping which assigns to each T-formula cp the formula cp itself is called the identical model of T and is denoted by 3b,. 1226. Let T, S, U be theories; suppose that 2132, is a model of T in S and that 2132, is a model of S in U. The mapping which assigns to each T-formula cp the U-formula (cpsnl)wz is a model of T in U; it is called the composition of the models %R,, 1132, or the model 2132, constructed within the model 1132,; it is denoted by 1111, * !IN2. 1227. We see that theories and models behave as objects and morphisms in the sense of category theory. We shall not study categorical properties of syntactic models here; but sometimes we shall use diagrams to describe relations among various models. E.g. Lemma 1223 means that in the diagram
the upwr arrow is a model (and obviously the diagram is commutative). Double arrows denote identity mappings. 1228. LEMMA.Let 1132 be a model of T in S and let % be a model of S in U. If a T-formula cp is valid in 1111 then it is also valid in 2132 * %. Demonstration. Let cp hold in 2132, i.e. S !- (pm. Then U I- (cpm)", i.e. cp holds in 2132 * %. 1229. Let T and S be theories and let s%R1and 1111, be models of T in S. We say that 1111, is weaker than !Ill2 (%R1 < %TI2)*) if for any T-formula cp, qm' is provable in ( S , qml). is equivalent to %TI2,if 2132, < %TI2and 9x2 < 1132, .
m, as a model of T in S is weaker than m2since %IT,and W , can be models of other theories as well; see the following lemma.
*) To be exact, we should say that
39
1230
LOGICAL FOUNDATIONS
[CHAP. I SEC. 2
1230. LEMMA. a) S is stronger than T iff the identical mapping 3b, of T-formulas is a model of T in S. b) Let 9X be a model of T in S and let S be stronger than S; then W is a model of T in S. c) Let 911 be a model of T in S and let T be stronger than To;then the restriction of %ll to To-formulas is a model of To in S. d) Let 9Xl, %TI2 be models of 7 in S and let %TIl < %ll,; then each T-formula valid in 9X1 is valid in %I2. e) Let %R, and %R2 be models of T in S such that (YJ, < W2;if q is a T-formula then YX, as a model of (T, q) in (S, 'psnl) is weaker than W 2 as a model of (T, 'p) in (S, qml). The demonstrations follow from the definitions; e.g. 91 as mapping of T-formulas into 5-formulas is the composition of two arrows in the diagram
where both arrows are models; hence YJl is a model of T i n S.
1231. Let 9 2 be a model of T in S and % a model of S in T. We say that % is an inuerse of 911 if 9 3 * 92 is equivalent to the identical model 3b,. Theories T, S are isoinorphic if there are models W (of T in S) and 'iJl (of S in T) such that each is inverse to the other. 1232. Let 911 be a model of T in S. We say that (JJZ is faithful if for any T-formula q, S I- qmimplies T t q.If, for every T-formula q,the model %TI considered as a model of (T, 'p) in (S,qm)is faithful, then we say that W is essentially faithful. If a model %ll of T in S is faithful, then only provable T-formulas are valid in and hence instead of proving a T-formula q in T we may prove (pm in S; in many cases this is easier. This notion is also of importance in proving theories to be consistent; for if W is a model of T in S and S is consistent then so is T. We may ask whether we obtain more by the construction of W,whether $92is a model of some extension (T, q) of T where cp is unprovable in T. The answer is negative if the model $92of T in S is faithful; the model 9Jl yields us no other consistency results relative to S; in other words, the model has been exploited to the full. 1233. LEMMA. Let YJl be a model of T in
S,% a model of S in U. (1) If 9X, % are (essentially) faithful then 'Dl * 'iJl is (essentially) faithful. 40
CHAP. I SEC. 21
LOGICAL FOUNDATlONS
1237
(2) If 91 * W is (essentially) faithful then
Iuz is (essentially) faithful. and W is faithful then S t cpw; if im
Demonstration. (1) If U t (cp")" is also faithful, we obtain T I- cp. (2) If S t cpm then U I- (cpm)" T t cp. Similarly for essentially faithful models.
and therefore
a,
1234. LEMMA.Let '9 be models of T in S. (1) If 9 1 < W and W is faithful then 91 is also faithful, (2) If 912 is equivalent to '3 and W is cssentially faithful then '3N is aIso essentially faithful. Demonstration. (1) If cp is a T-formula and S t cp" then S I- q" and so T t cp. (2) If cp, $ are T-formulas such that S, cpm t $" then S, cpa t cpm, S, cpw t- $m, S, $% I- $'; hence S, cp' t $" and T, cp t $ since W is essentially faithful.
1235. LEMMA.Let lllt be a model of T in S and let 'illbe an inverse of '9X; if cp is a T-formula then '9 is a model of (S, cpW) in (T, cp) and W is an inverse of Iuz considered as a model of (T, cp) in (S, 9"). Demonstration. To verify that W is a model of (S, cp") in (T, cp) it sufSince Iuz * W is equivalent to 3b, we have fices to prove (T, cp) t (cpm)". T, cp I- (cpm)'. It remains to verify that for any T-formula $ we have T, cp, $ t $m*" and T, cp, $m*' k $; this follows from the fact that T, I,$~*' t $ and T, )I t $m*w.
1236. LEMMA.Suppose that 93 is a model of T in S. (1) If '9X has an inverse then it is essentially faithful. (2) If T is stronger than S and if the model Iuz considered as a model of T in T is (essentially) faithful then %l is (essentially) faithful as a model of T in S . Demonstration. (1) The identical model is faithful; if % is an inverse of 91 then %l * W is faithful and Iuz is faithful by Lemma 1233. Moreover, if cp is a T-formula and if Iuz has an inverse, then 93 considered as a model of (T, cp) in (S, cpm) has an inverse by Lcmma 1235 and is therefore faithful. (2) The identical model of S is a model of S in T; since Iuz* 3b, considered as a model of T in T is faithful, the model %l considered as a model of T in S is also faithful.
1237. METATHEOREM. Suppose that the languages L and L* are similar, where L is (Pi, 0 , F1, 0 , sl, 0 ) and L* is (P?, 0 , F:, 0 , sr, 0 ) . If A , (AL.) is the theory with the empty axiom system and the language L (L*), then AL is isomorphic to A,,.. 41
1238
LOGICAL FOUNDATIONS
[CHAP. I SEC. 2
Demonstration. Assign to each L-formula cp an L*-formula cp* simply by adding asterisks to all predicates, operations and sorts with the exception of the equality predicate. This mapping is one-to-one from L-formulas onto L*-formulas, assigns logical axioms to logical axioms and preserves +, V, 1. The inverse mapping has the same properties and hence each of these models is an inverse of the other.
1238. If L and L* are similar languages then the mapping defined in the course of the preceding demonstration is called the direct translation of L-formulas onto L*-formulas; we denote it by Dir,, 1239. LEMMA. Suppose that L is the language of a theory T, that L* is a sublanguage of the language of a theory S and that L and L* are similar. Then the direct translation of L-formulas onto L*-formulas is a model of T in S if and only if, for any axiom q of T, q* is provable in S. (This follows from Metatheorem 1237).
1-740. If the direct translation of T-formulas onto formulas of some sublanguage of the language of S is a model of T in S,then we call it a direct inodrl of T i n S . 1241. LEMMA. (1) The composition of direct models is a direct model. ( 2 ) If a direct model is faithful then it is essentially faithful. Deinonstration. (1) is obvious. (2) Let 9.JL be a direct model of T in S and suppose 911 to be faithful. Let cp be a closed T-formula. If S, 9%t- $% then S t- qm .+ Grn by the Deduction theorem; hence S t- (q + $)" because qm ---t t,bm is the same as (q -,$)" for direct models. Since is a faithful model of T in S it follows that T I- cp 4 $ and so T, cp I- $; thus 93 is a faithful model of T, q in S, q". Note that it really suffices to consider closed cp; for ((V . ..) q)m is the same as (V . ..) cp", so that if S, qrnt- $% then S, (V . . .) q% t- $%,hence T, (V . ..) cp I- $ and T, q t- $.
c ) Conservative extensions
1242. Let S be an extension of a theory T. S is said to be a conservative extension of T if the identical model of T in S is faithful. (Commentary). Note that 3b, considered as a model of T in S is faithful iff each T-formula cp provable in S is provable in T. We shall be interested in certain extensions which enrich the language; from the point of view of the original theory, new symbols can be treated as a means for simplifying formulas and proofs. 42
CHAP. I SEC. 21
1244
LOGICAL FOUNDATIONS
1243 CONVENTION.In the rest of the book we restrict ourselves to theories having a universal sort. In each theory T we fix a sort which is universal in T and call it the universal sort of T. 1244. We shall consider now four particular methods for obtaining conservative extensions of a given theory T: a) Let cp(x,, ..., x,) be a T-formula having free variables xl, ..., x, of arbitrary sorts (where n > 0) and let P be an n-ary predicate not in the language of T. The formula
(whereX,, 0 are variables of the universal sort and not occurring in cp(x,, 0)) is called the definition of P in T b y q. b) Let cp(y, xl, ..., x,) be a T-formula having free variables y , xl, ..., x, of arbitrary sorts (where n 2 0) and suppose that
let F be an n-ary operation not in the language of T. The formula
(where X , , 0 are variables of the universal sort and not occurring in q(y. x i , 0 ) ) is called the definition ofF in T b y q. c) Let q(x') be a T-formula having one frce variable xi and let j be a sort not in the language of T; suppose that T k (3xi) c p ( x ' ) .
The formula
(vx') [ ( j y ' ) (xi
=
y')
E
q(x')] & ('dy') (3x9 (xi
=
y')
is called the definition of the sort j in T b y q. 43
1245
LOGICAL FOUNDATIONS
[CHAP. I SEC‘. 2
d) Let cp(x,, ..., x,,) be a T-formula having free variables x,, arbitrary sorts (where n > 0) and suppose that
..., x, of
let A,, . . ., A,, be constants not in the language of T. The formula
is called thefixing of constants A , , i n T b y cp. Remark. Each definition is a conjunction of two parts. The first part is the interesting part of the definition and tells us what the new symbol means for variables of sorts we are interested in. The second part is the securing part; in the case of a predicate and of a function it defines the meaning of the new symbol for each tuple of objects containing an element of a sort uninteresting w.r.t. the definition, and in the case of a sort the securing part says that the sort j is subordinated to the sort i. Similarly, each fixing consists of two parts; the first (interesting) part tells us the property that the new constants have, and the second (securing) part says that the new con$tants are objects of appropriate sorts. The securing part will be abbreviated by sec. Hence we write e.g. a definition of a predicate i n the form (Vxl,
0)
(P(xl
0,)
=
‘p(s,, 0 ) ) & sec
Note that sec is empty if we define a 0-ary function, i.e. a constant. The part sec will be often omitted at all, in particular if we will formulate a definition informally. The string of quantifiers in the first part of a definition will be usually also omitted. (Hence we write e.g. only P(x,, 0 ) = q(x,, o).)
1245. METATHEOREM. (1) Let T be a theory and let S be an extension of T obtained by adding the definition of a precidate, operation or sort. Then T and S are isomorphic; there is a model 93 of S in T such that 93 and 3 b , (as a model of T in S) are inverse each to the other. (2) Let T be a theory and let S be an extension of T obtained by adding the fixing of some constants. Then Sh, as a model of T in S has an inverse. (3) Consequently, the extension of a theory T obtained by adding a definition or fixing is conservative. The tcchnical details of the demonstration will be omitted; we give here an outline for the case of a predicate and of constants. 44
CHAP. 1 SEC. 21
LOGICAL FOUNDATIONS
1247
(a) To each S-formula $ we assign a T-formula $' in such a way that P(t,, 0 ) is replaced by (3x,, 0 ) (tl = x1 & 0 & cp(xl, 0 ) ) ; note that it is necessary to rename some of the variables. It can be seen that $' is a T-foris a model of S in T, and 3 b , mula and that S I- $ = $'. Moreover, and % are inverse each to the other. (It follows that both models are essentially faithful.) (b) Let S be the extension of T by the fixing of constants A,, ..., A,, by cp(x,, ..., x,). If $(A,, ..., A,,, y , , ..., y,) is an S-formula then we define $' as the T-formula cp(xl, ..., x,) --* $ ( x l , ..., x,, y,, ...,y,) (some variables must be renamed). Then '93 is a model of S in T and is inverse to 3,. It is easy to see that 93 is a faithful model of T in S, i.e.
Thus we observe that the fixing of constants corresponds to the usual mathematical reasoning: "there exist xl,..., x,, such that 'p(x,, ..., x,,); choose sotne such x,, ..., x,, and denote them by A,, ,.., A,,". If we prove something for fixcd but arbitrarily chosen A , , ..., A,, then we corclude that we have proved it for all xlr ..., x, such that cp(x,, . .., x,,). Note that '93 is not necessarily essentially faithful.
1246. LEMMA. a) Let S be the extcnsion of T by the definition of a sort j subordinate to the sort i by cp(x'). If $ ( x i ) is an S-formula and y' is not bound in $(xi), then
s I- (v$) $($) = (v2) (cP(x') $ ( x i ) ) , s I- (3y') $(y') = @ x i ) (cp(X')& $ ( x i ) ) . -+
2247. LEMMA. Let YR be a direct model of T in S and let T be the extension of T(a) by the definition of a predicate P by cp(x,, . .., x,),(b) by the definition of an operation F by q ( y , x,, . .., x,), (c) by the definition of a sort j by cp(xi), or (d) by the fixing of constants by cp(x,, . .., x,). Then in S, we can use the formula cpgn (a) to define an n-ary predicate by cpm, (b) to defice an n-ary operation by cpm, (c) to define a sort subordinate 10 i* by (p"', or (d) to fix constants by qw. Demonstration. For (a) it is sufficient to note that the number of free variables in cpgn is the same as in cp and that their sorts correspond. For(b) it is sufficient t o observe that if T t ( 3 ! y ) q(y, xl, 0 ) then S I( 3 ! y * ) cp*(y*, x:, 0 ) . Cases (c) and (d) are treatcd similarly. 45
1248
LOGICAL FOUNDATIONS
[CHAP. I SEC. 2
We are led to introduce the following concepts: 1248. Let a i r , L* be a direct model of T in S and let ^T be the extension of T (a) by the definition of an n-ary predicate P by p, (b) by the definition of an n-ary operation F by p, (c) by the definition of a sort j by p, or (d) by the fixing of constants A,, 0 by cp. Let p* be the image of p in IDirL,L.. Suppose (a) that P* is an n-ary predicate not in the language of S,(b) that F* is an n-ary operation not in the language of S, (c) that j * is a sort not in the language of S or (d) that AT, 0 are constants not in the language of S. The extension 5 of S (a) by the definition of P* by cp*, (b) by the definition of F* by p*, (c) by the defin'tion o f j " by p* or (d) by the fixing of A:, 0 by cp* is called the extension of S induced b y the model IDirL,L*and the extension ?. The direct translation (a) of (L, P) onto (L*, P*), (b) of (L. F) onto (L*, F*), (c) of (L,j ) onto (L*,j * ) or (d) of (L,A,, 0 ) onto (L*,AT, 0 ) is called the extension of the model !DirL,L*induced b y the extension ? and the theory S . The extension of air,, L* induced by f and S is a model of of S induced by airL, and T. Demonstration. We denote the extension of 3irL,L*by $. Clearly 5 is a model of T in S;to prove that it is a model o f f , it suffices to verify the validity in 5 of the added definition (fixing of constants). This follows from the construction of indeed, the translation of the definition (fixing of constants) is implied by the definition (fixing of constants) which is added to s. 1249.
LEMMA.
T in the extension
s;
d ) Epsilon-theories
=) and denote (the membership predicate). Further, we choose a sort with variables X , Y, . .. which we call class-variables. The language ( E , X ) is called the jundamental language, and each language similar to the fundamental language is called F-like. Formulas of the fundamental language will be called fundamental formulas (FF). The fundamental €-theory TE is the theory having the fundamental language and the following axioms: it
2250. We now choose a binary predicate (different from
E
(F1) (% y ) (xE Y ) (F2) (VZ) ( Z E X 46
3
2E Y)
X
=
Y.
CHAP. I SEC. 21
LOGICAL FOUNDATIONS
I252
Let T be a n extension of the fundamental etheory and Ict cp be a T-formula. The formula cp is calledfundamental in T if there exists some fundamental y o such that T t-. cp I cpo. The concept of an E-theory is defined recursively: 1) The fundamental etheory is an €-theory. 2) If T is a n etheory and cp a definition (of predicate, operation or sort) or a fixing of constants in T,then (T, cp) is an etheory. 3) If T is an etheory and cp is a T-formula fundamental in T, then (T, cp) is an E-theory.
1251. Let cp be an axiom of an E-theory T. If cp is one of the axioms (Fl), (F2) or if cp is formulated in the €-theory whose axioms coincide with the sequence of axioms of T preceding cp then cp is called a proper axiom of T. In the opposite case cp is a definition or fixing of constants and is called a n improper axiom of T. When we speak of the language LT of an etheory T we shall assume that the symbols in LT are ordered according to their first occurrences in the axioms of T, i.e. LT begins with X , E, and continues with notions introduced by successive improper axioms of T. It can be said that in €-theories we are interested mainly in the fundamental formulas, i.e. in the properties of the membership-predicate. The defined symbols play a n auxiliary role, although they greatly simplify the development. Theoretically, we could in fact restrict ourselves to theories having the fundamental language: 1252. LEMMA. If T is an etheory then there exists an E-theory To having the fundamental language and such that T is equivalent to some conservative extension of To. Demonstration. By induction. 1) The lemma is obvious if T is the fundamental etheory. 2) Let T be an etheory. Let To be an etheory having the fundamental language and suppose that T&is a conservative extension of To such that Tois equivalent to T. If cp is a definition or a fixing of constants then (T& cp) is a conservative extension of To and is equivalent to (T, cp). 3) Suppose that T, To and T&are as before. Let cp be a T-formula fundamental in T ; i.e. there exists a fundamental cpo such that T I- cp E cpo and so T A I- cp = cpo. Denote by TI the theory (To,cpo) and by Ti the theory (T&40). Then Ti is a conservative extension of TI and is equivalent to (T, cp). To prove this it suffices to show that Ti t- cp and that (T, cp) t- q0; these facts however are immediate.
1253
LOGICAL FOUNDATIONS
[CHAP. I SEC. 2
1253. Lct S be some fixed E-theory and suppose that L is a n F-like language ( X * , E*) which is a sublanguage of the language of S. Given a n E-theory T we ask if there is a conservative extension S of S such that there is a direct model of T in S translating X as X * and E as E*. If L, is the language of T and L, is a language similar to L, and beginning with X * , E*, then we can add to S translations of all improper axioms of T by air,,,L2 and we obtain an extension S;we also have the direct translation 3irL,,Lzof T-formulas into S-formulas. But two questions arise: (1) whether S is a conservative extension of S and, moreover, whether S is an E-theory; (2) whether air,,,Lz is a model of T in S. This leads to the following definjtions. 1254. Let L be an F-like language and let T be an €-theory with language L,. A derioation of T w.r.t. L is the sequence of translations of all improper axioms of T by airLl, Lz where L, is any language similar to L, and beginning with L. An arbitrary derivation of T w.r.t. L is denoted by dT/dL. The language L, wed for the construction of dT/dL is uniquely determined by L and dT/dL; hence we denote D)irL,,L2 by a i r (dT/dL) and call it a direct translation giwn b y L and T.
1255. Lct dT/dL be a n arbitrary derivation of T w.r.t. L. We say that dT/dL is compatible with S if the language L, used for the construction of 8T/2L has no symbols in common with the language of S except those of L. Suppose that T has n axioms; T,,, denotes the sequence consisting of the first 111 axioms of T.
Given a derivation dT/dL of T w.r.t. L we denote by dT,/dL the segment of ZT/i?Lwhich is a derivation of T, w.r.t. L. 1256. We say that L determines a model of T i n S if there is a derivation of T w.r.t. L compatible with S such that, for every m < n such that (P,+~ is a proper axiom of T, qm+ holds in air (dT,/dL) as a model in (S, dT,,,/dL).
,
To justify the definition we prove the following 1257. LEMMA.If L determines a model of T in S then, for any derivation dT/2L compatible with S, (S, dT/dL) is a conservative extension of S (in addition, an E-theory) and IDir (dT/dL) is a model of T in (S, aT/dL). Moreover, if (dT/dL),. (dT/dL), are two derivations of T w.r.t. L compatible with S and arrows 1, 2 in the diagram 1258 are the corresponding direct translations then there isadirect model 3 of S, (dT/dL), in S, (dT/dL), which is an isomorphism and which makes the diagram commute.
48
CHAP. I SEC. 21
1260
LOGICAL FOUNDATIONS
1258.
1259.
Demonstration. Let (dT/dL), be a derivation of T w.r.t. L compatible with S. Further let (aT/dL), be another derivation of T w.r.t. L compatible with S. Then for each m j n the languages of S, (dT,/dL)>, and of S, (dT,/dL), are similar and we have direct translations 3" = BirLn(,), L,(l, and 3''' = DirL,cZ,,L,cl, (Lc) is the language of S, (dT,/dL),). Evidently, these translations are models of S,(dT,/dL), in S, (dT,/dL), and of S, (dT,/dL), in S, (aT,/aL), respectively and their compositions are the corresponding identities. Hence J is the inverse of 3 and 3is the inverse of J; the diagram 1258 commutes. Hence it remains to demonstrate that S, (dT/aL), is a conservative extension of S and that the arrow 1 in Diagram 1258 is a model of T. Write dT/dL instead of (dT/dL),. We use induction on rn. Set m = 2. Then T, is the fundamental €-theory and so all of its axioms hold in Bir (dT,/dL) as a model in S because they are proper axioms. (Note that in this case dT,/dL is empty). Suppose now that S, (dT,/dL) is a conservative extension of S (and an €-theory) and that Bit (dT,/dL) is a model of T, in S, (dT,/dL). Take T,,, = (T, cp). If cp is a proper axiom then dT,+,/dL is the same as dT,/dL; by assumption cp holds in Bit (dTm/JL) as a model in S, (dT,/dL); hence S, (dTm+,/dL) is a conservative extension of S, and Bir (dT,+ ,/dL) is a model of Tmnfl in S, (dT,+>,/dL). Finally if cp is an improper axiom then S, (dT,+,/dL) is the extension of S, (dT,/dL) induced by T,+, and Bit (dT,/dL), while air (dT,, @L) is the extension of Bit (dT,/dL) induced by T,, and S, (dT,/dL). Hence, by Lemma 1247, S, (dT,+ ,/dL) is a conservative extension of S (and an €-theory) and Bit (dT,, ,/dL) is a model of T,+, in S, (dT,+,/dL). Thus the lemma is proved.
,
1260. We illustrate the step-by-step construction of S, (dT/aL) and the decision whether L determines a model of T in S by the following example (cf. Diagram 1261):
49
1261
1261
LOGICAL FOUNDATIONS
[CHAP. I SEC. 2
Suppose that the axioms of T are the axioms of TE followed by adefinition, two proper axioms, a definition, etc. Given the language L (a sublanguage of the language of S) we have a i r = 9 i r F , L (F being the fundamental language); this is certainly a model of A , in S (arrow 1). If we succeed in proving the translations of the axioms of TE (= T,) in S we know that L determines a model of T, in S (arrow 2). T, is a conservative extension of T, by a definition, hence it induces together with 9 i r a conservative extension S‘ of S and a direct model 9ir’ of T, in S’ (arrow 3); moreover, the identity 3 b , as a model in S’ has an inverse (arrow a); the composition of arrows 3 and a is a model of T, in S. T, i s an extension of T, by a proper axiom; if we show that this axiom holds in 9ir‘ we know that Dir’ is a model of T, in S’ (arrow 4). The same can be said for arrow 5. The explanation of arrows 6 and b is thc same as that for arrows 3 and a. Finally we see that Dir” (i.e. a i r (dT,/dL), arrow 6) is a model of T, in s” (i.e. in s, dT,/dL); if we want we can compose the model 93’ with arrows b and a and obtain a model of T, in S. This justifies our term “determines a model in S”;but it is more convenient to consider the model 9 i r (dT,/dL) as a model in s, (dT,/aL) itself, keeping in mind the theoretical possibility of obtaining mutatis ittutaizdis a model in S. More generally, suppose that we have T and S and we want first to definc a sort and a binary predicate in S and then ask if the F-like language so defined determines a model of T in S (extended by our definitions). Consider the general form of a definition of an F-like language. 1262. A pair of formulas
(I) ( V X ) [ ( I x *(x* ) = X ) = K(x)] & sec, (2) (VX*, Y*)( X * E* Y*
E
E(x*, Y*)) & sec
is a dcfinitiori of a n F-like language ( ( X * , G*) by means of x, E ) if (a) x is a formula with exactly one free variable X (class variable) and not containing any variable of the sort of X* nor containing the binary predicate E*, (b) E is a formula with exactly two free variables X * , Y* and not containing the predicate E*. 50
CHAP. I SEC. 21
LOGICAL FOUNDATIONS
1266
Every definition of an F-like language (briefly an F-dejnit ion) is uniquely determined by the formulas x, E and by the language ( X * , E*). Conversely, given a n F-definition A , the formulas x , E and the language LA = ( X * , E*) are uniquely determined.
x
1263. Given an F-definition A we say that A is an F-deJinition in S if (a) is an S-formula such that S F ( 3 X ) x ( X ) , (b) E is an (S, (1))-formula and (c) neither X * nor E* belong to the language of S.Then obviously ( I ) isadefinition of a sort in S and (2) is a definition of a binary predicate in (S,(l)). In this case (S, A) is an €-theory provided that S is; furthermore, in (S, A; we have the language LA = ( X * , E*) and may ask if it determines a model of T in (S, A). We write dT/dA instead of (A, dT/dL,) and say that dT/dA is compatible with S if A is a n F-definition in S and if 6T/dL, is compatible with (S, A). We also say that A determines a model of T in S instead of saying that A is an F-definition in S and that L, determines a model of T in (S, A). (Draw a diagram similar to 1261 !) Finally we write Dir (dT/dA) instead of Dir (dT/dL,). The last translation is often denoted by A if T and S are evident from the context. 1264. The notion “A is an F-definition in S” can be g e n e r a k d i n the following way: We say that n: is a speciJicution f o r A in S if there are constants A,, 0 occurring in x and/or in E such that n I S a fixing of A,, 0 i n S and if A is an F-definition in (S, n); in this case we also say that A with n is a parametric F-dejnition in S . We say that A with the specification n determines a model of T in S if n is a specification for A i n S and if A dctermines a model of T in (S, n) (i.e. n is a fixing of A,, 0 in S, A is an F-definition in (S, A) and LA determines a model of T in (S, n, A)). If it is the case thcn a i r (dT/6A) is called the direct mode1 of T determined b y A . 1265. Given S and A we can have various specifications for A in S. Jf nl,n, are two such specifications then we say that n 1 is stronger thun n2 in case S, n, F n,. We have the following useful facts: 1266. LEMMA.(a) If A with the specification n, determines a model of T in S and if n, is a specification for A in S stronger than n2,then A with the specification n1 determines a model of T in S; every formula which holds in air (dT/dA) as a model in S, n,, dT/aA holds also i n Bir(dT/dA) as a model in S,n,, dT/aA. (b) Let 7c1, n2 be specifications for A in S and suppose that n1 is stronger than n,. If Dir (dT/aA) is a model of T in S, n,, dT/aA and if it is a faithful model of T in S, n,, dT/dA then it is also faithful as a model in S, n,, aT/aA. 51
1267
[CHAP. I SEC. 2
LOGICAL FOUNDATIONS
Demonstration. (a) follows from the fact that S, nl,aT/ad is stronger than S, n,, dT/dA so that all arrows in the following commutative diagram arc models:
(b) follows from this diagram by Lemma 1233.
1267. (Absoluteness.) Let T, S be two €-theories such that the language of T is a sublanguage of the language of S. Let 3 be a direct model of T in S and let q ( U , 0 ) be a T-formula (with free variables U , 0 of arbitrary sorts). Denote [cp(U, o)]” by cp*(U*, 0 ) and suppose that the sort of U* is subordinate (in S) to the sort of U and similarly for other free variables of cp. (The last condition is trivially satisfied for every variable U which i s a class variable.) Then we say that (a) cp is absolute in S w.r.t. B if
s t (c- = U * &
0) +
[cp*(v*,0 ) = cp(u,o)]
;
(b) cp is absolute fro171 above in S w.r.t. 9 if
s t (U
=
U*&
0)
-+ [& 0I ), -+ q*(U*, o)] ;
(c) v, is absolute from below i n S w.r.t. 3 if
s t ( u = U * & 0 ) + [cp*(u*,0 ) -+ cp(U, o)] . 1268. LEMMA.(a) If cp(U, 0 , V ) is absolute from below then so i s 0 , V ) is absolute from above then so is
(3’) (cp(U, 0 , V ) ) ; if cp(U, (VV)(v(U, 0 , V ) .
(b) q ( U , 0 ) is absolute iff it is absolute both from below and from above. (Obvious.) We close this Section with a useful test for the faithfulness of certain direct models. 52
CHAP. I SEC. 21
LOGICAL FOUNDATIONS
1269
1269. LEMMA.Let T, S be €-theories such that S is a conservative extenE*) of the language of S sion of T. Let L be an F-like sublanguage (X*, determining a model of T in S. Suppose that there is an S-formula p ( X , X * ) with exactly two free variables such that
(3)
s I- (vx)(3! x*)p(x, x*);
(4)
s F (vx*)(3! x)p(x,x*);
(5)
s F p(x,X*)& p(Y, Y * ) . + [xE Y = x* €* Y*].
If all improper axioms of T are definitions (no fixing of constants) then Bir (aT/dL) and 3b, are equivalent as models in S, aT/aL (consequently, Pir (dT/aL) is a faithful model of T in S, aT/aL). Demonstration. We proceed by induction on the “segment theories” T,. Set n = 2; then all T,-formulas are fundamental formulas and it follows easily by induction on the structure of a n arbitrary T,-formula that the following is provable in S:
Hence S, cp F cp* and S, cp* F cp so that Pir(aT,/aL) is equivalent to 3bT, as a model in S. Now suppose that for some n we already know that (6) is provable in S, aT,/aL for every T,-formula cp. If (P,,+~ is a proper axiom of T then T,+,-formulas are the same as T,-formulas so that the induction step is trivial. If cp,,+ is a definition in T,, by means of t,h then we have
which has the following consequences: (a) If cp,+
is a definition of a predicate P then
(b) If cp,+
is a definition of a function F then
53
1270
LOGICAL FOUNDATIONS
[CHAP. I SEC. 2
(c) If qn+ is a definition of a new variable u then
Now it follows by induction on the structure of an arbitrary formula that ( 6 ) is provable in S, dT,, ,/dL for each T,+,-formula. Hence the lemma i\ demonstrated. 1270. Remar-ks.(1) We also see the difficulty with fixings: if the induction assumption holds for T, and if q,, is a fixing of constant A, say, by means of x ( X ) then in S, dT,+,/dL we cannot deduce the formula p(X1,XT) -+ -+ [X, = A = X y = A*] (or, equivalently, p(A, A*)) from the formula ,u(X1,X:) -+ [ x ( X , ) 5 x*(XT)] (which is certainly provable in S, dT,/dL) because we do not know if ( 3 ! X ) x ( X ) is provable in S, dT,/dL. However p(A, A*) is consistent with S,aT,/.'L since it is a fixing of the constant A*, and hence 9i.r (dT/dL) is a faithful model of T,, in S, dT,/dL, p(A, A*), the latter theory being stronger than S, 2Tn+l/dL. (2) If p ( X , A'*) is X = X * then (3) and (4) reduce to S F (VX) ( 3 X * ) ( X = x*).
54
CHAP. I SEC. 31
THEORY OF CLASSES CONTINUED
1302
SECTION 3
The theory of classes continued
The theory of classes was described in Section 1; in the present Section we shall investigate this theory from the point of view of Section 2. We shall show among other things that T C really is an ethcory and that the model described in Section 1 is given by a certain F-definition, called the normal F-definition. We shall investigate some of the properties of this F-definition and we shall introduce another related F-definition, called the standard F-definition. Finally, we shall consider absoluteness in models determined by these F-definitions with certain specifications; of particular interest here are the restricted formulas. The general results obtained in this Section will be used throughout the book. u)
TC as an €-theory
1301. It follows immediately from t,.e definition 1 50 of a n €-theory that T E (i.e. the theory with axioms (Fl, F2)) is an €-theory. Moreover, (dl) is a definition in T E and (d2) is a definition in (TE, dl), so that (TE, d l , d2) is an ethcory. If we alrcady know that a theory T stronger than (TE, d l , d2) is an etheory and want to add an axiom cp as a proper axiom we must verify that it is fundamental in T. The following Metalemma is sometimes useful. 1302. METALEMMA. Let T bc an etheory stronger than (TE, d l , d2). If cp is a formula normal in T whose only free variables are class variables and which does not contain any constants then cp is fundamental in T.
Demonstration. We show by induction that for every normal T-formula 0 , X , 0 ) whose only free variables are class and/or set variables and
cp(x,
55
1303
THEORY OF CLASSES CONTINUED
[CHAP. I SEC. 3
which does not contain any constants there is a fundamental formula $(Z, 0 , X , 0 ) such that (TE, d l , d2) t-
(X
= Z&
0 ) -+ [cp(x, 0 , X
,
$(Z,
0)
0,X
, o)]
.
If cp is x 1 E x2 or s1 E X , then for Ic/ take 2 , E Z 2 or 2 , E X I respectively. The induction step for & and 1 i s obvious. Finally, if (TE, d l , d2) 1 ( y = U & x = Z & -+
[V(Y, x,
then (TE, d l , d2) t-
(X
0,
0)
+
x , 0 ) = $(K z,0 , x , 4 3
=Z&
0)
+ [(3y) cp(y, X,
0,
7
X,
= ( W ,w)(u E W & $(U, 2, 0 , x, o)] .
0)
1303. METATHEOREM. TC is an etheory. Demonstration. T C is an extension of (TE, d l , d2). Axiom (Al) is a closed formula normal in (TE, d l , d2) and is therefore fundamental in this theory. (d3)-(d6) are again definitions. The axioms (Bl) to (B7) are closed formulas of the form (VX, 0 ) ( 3 2 ) p(X, 0 , Z ) where cp is normal in (TE, d l , d2, Al, d3); for, by 1126 the formula ( z , , z 2 ) E Y is normal in this theory. All free variables of are class-variables and no constants occur in q ;therefore axioms (Bl) to (B7) are fundamental formulas in the latter theory. Thus T C is an E-theory since all of the remaining axioms are definitions. Remark. All extensions of TC considered in the sequel are €-theories; this fact will be evident in each particular case and will not require explicit verification. Godelian terms will play an important role in our considerations. For the moment we shall only prove a lemma concerning certain ‘‘c~mutativities~’ between godelian operations and the operation of intersection.
1304. LEMMA(TC). The following holds for arbitrary classes X, Y, 2: E(X)nZ =E(XnZ); ( X - Y ) n Z = ( X n Z ) - ( Y n Z ) ; ( X r Y ) n Z = ( X n Z ) r ( Y n D ( Z ) ) ; Cnv(X)nZ =Cnv(XnCnv(Z)); Cnv, ( X ) n Z = Cnv, ( X n Cnv, (Cnv, (2))).
The proofs of these equations are elementary and can be left to the reader. Note that we do not state anything about the operation D. 56
CHAP. I SEC. 31
THEORY OF CLASSES CONTINUED
1307
b) The normal F-definition 1305. METADEFINITION. The pair of formulas (1)
(2)
(VX) [(3~*)(x* = X ) (VX*, Y*)(X* E* Y*
= Sat, (x)]& sec ,
= ( 3 z E Y*) ( X *
= Exta ( z ) ) ) &sec
is called the normal F-definition (R being a constant) and is denoted by %(R). Thus classes in the sense of 8 (R) are saturated parts of C (R).
1306. METATHEOREM. % (R) with the specification Elk (R) determines a model of TC in TC. Moreover, IDir (i?TC/a%(R)) is a faithful model of TC in (TC, Elk (R), aTC/a%(R)). Demonstration. We demonstrated already in Section 1 that IDir (aTC/d% (R)) is a model of T C in (TC, Elk (R), dTC/a%(R));*) it remains to demonstrate that it is faithful. Observc that R = E is a specification for % (R) in T C stronger than Elk (R) and such that air (dTC/d% (R))asamodel in (TC, R = E,dTC/d% (R)) is equivalent to the identical model. Indeed, it is easy to see that the following formulas are provable in (TC, R = E. dTC/a% (R)):
(vx)(3x*)(x*= x), x*E* Y* = x* E Y* . Hence, by 1269 and 1270 the direct model of T C determined by 9t(R) with the specification R = E is faithful. By 1266(b) the dircct model of TC determined by % (R) with the specification Elk (R) is also faithful. We shall often denote Dir (dT/a% (R)) simply by Yl (R) if the theory T is clear from the context.
1307. METALEMMA. The formulas X E Y and U = X - Y are absolute in YI (R)as a model in TC, Elk (R), dTC/a% (R). Demonstration. We proceed in the theory indicated. For any X I , Y*, X * E * Y * = (VX*) (x* E* X* + x* E* Y*) (Vz E C (R))( z E X * -+ z E Y *) G = X * E Y* (recall that X*, Y* are saturated subclasses of C (R)!). Further, we want to prove U* = X * - * Y* = U* = X * - Y*. This formula is equivalent to (Vu E C (R))(u EX* - * Y* E u EX*- Y*). But u E X * - * *) The specification should be written in the form Elk(R) & sec, i.e. Elk (R) & ( 3 X ) (R = X); but we shall always omit the part sec in specifications.
1308
[CHAP. I SEC. 3
THEORY OF CLASSES CONTINUED
-Ext,(u)E*X* -*Y* -Ex~,(u)E*X*&EX~,(U)$*Y* =UEX* = 21 EX* - Y*. lJnfortunately the formula Re1 ( X ) is not absolute. However the following device enables us to handle relations in the sense of %(R).
-*}*
& u $Y*
2308. DEFINITION. (TC, Elk(R), i?TC/d%(R)). For any X * such that Rel* ( X * ) we denote by Dec (X*) the unique relation 2 defined as follows: (u, V }
E
Z
. U ,u E C (R) & (ExtR (u),EXtR (u))*
E*
X* .
Dec ( X * ) is called the decoded relation. 2309. LEMMA (TC, Elk (R), dTC/i?%(R)). If Rel* ( X * ) and 2 then
=
Dec (X*)
D*(X*) = D(Z) , W*(X*) = W(2) , C*(X*) = C ( 2 ) ;
(a)
Ext$ (Ed, (u)) = Ed, (u)
(b)
for any u E C (R) .
P ~ o o f(a) . If u E D* ( X * ) then Ext, (u) E* D* (X*) and there exists v such that ( E d R( u ) , ExtR(u))* E* X * ; hence ( u , u ) E Z , i.e. u E D (Z). The proof of the converse is analogous, and similarly for W and C . (b) u E Ext, (11) iff ( u , u ) E 2 iff (EdR( u ) , Ext, (u))* E* X * iff Ext, ( u ) E* E* E x$:* (EdR(u)) iff u E Ext,*, (Ext, (u)).
-
1310. I)I:I:INIIION (TC). A relation H is a iizorphisnt between two relations R , and R , if D ( H ) = C (Rl), W ( H ) = C (R,), and for all x, y E C ( R , ) and all u , u E C ( R , ) uHx& u H y implies x R , y uRzz;. If H is one-to-one we call it an isoinorplzisnz between R , and R,. In this case, if P , c C (Rl), P , 5 C ( R , ) and H " P , = P,, then H is also called an isomorphism between PI and P , with respect to R , and R,. 1311. LEMMA (TC). If H is a morphism between relations R , and R , then
(1) Cnv ( H ) is a morphism between R, and R , , ( 2 ) U H X implies EAR, (u) ( 3 ) Sat,,
,
( x )implies Sat,,
=
H" Ext,, (x),
(H"X),
(4) R is extensional iff R, is extensional. Proof. (1) Obvious. (2) Suppose first that v E Ext,, (u). We have ( u , u) E 58
CHAP. I SEC. 31
THEORY 01:CLASSES CONTINUED
1313
R , : if v H y then (y, x) E R , , i.e. y E Ext,, (x). Hence L: E H” Ext,, (x). Now suppose that u E H” ExtR, (x). We have v H y for some J’ such that (J’. x) E R , ; hence (21, u ) E R,, i.e. u E ExtRz(u). (3) Let X be a saturated part of C ( R , ) , let x E X and let u H x ; suppose that ExtR2( u ) = EX*,, ( u ) and that v H y . By (I) and (2) we have Ext,, (x) = (Cnv ( H ) ) ” (ExtR, (u)) = = (Cnv (H))” (ExtR2(u)) = Ext,, ( y ) ; hence y E X and t’ E H ” X . Thus H”X is a saturated part of C(R,). (4)Suppose that R , is extensional, i.e. that (Vx E C(R,)) (Sat,, (Ed,, ( x)). If u E C(R,) and u H x then Ext,, ( u ) = = H”(ExtR, (x)) and Sat,, (H”((Ext,,(x))). Hence R , is extensional. We sometimes encounter the following situation: we have a specification n,(R) for %(R) in a theory S stronger than TC and another specification n2(R) for %(R) and we can prove in S that for every relation X such that n2(X)there is a morphic relation Ysuch that n,(Y). We ask what can be said about the corresponding direct models. E
131-7. METATIIEOREM. Let S be an E-theory stronger than TC and let n,(R), n,(R) be specifications for %(R) i n S. Suppose that the following is provable in S: For every X such that n,(X) there is a Y such that rl(Y) and a morphism H between X and Y. Then any formula which holds in %(R) as a model (of the theory with the fundamental language and no special axioms) in S, nl(R), % (R) also holds in % (R) as a model in S, n2(R), % (R).
Demonstration. Consider the conservative extension S = (S,7r2(R), % (R)) of S. In 5 we have variables X * and the predicate E*. Let 6(R, X , 2) be a formula saying “(2 is a morphism of R and X ) and n,(X)”. By assumption, S I- ( 3 X , 2 ) 6(R, X, 2); fix $, H in 5 by d(R, S, H). Let %(S)be the F-definition obtained from %(R) by replacing R by S, X * and Y* by X u and Y Erespectively, and E* by E O ;thus % (S) is a “copy” of % (R). We have 3, d(R, S, H) k n,(S) and obviously %(S)is an F-definition in (S,6(R, S, H)); furthermore, (S,6(R, S, H), % ( S ) ) is stronger than (S, nI(S), ‘iR(S)).(See diagram 1313.)
1.313.
59
1314
[CHAP. I SEC. 3
THEORY OF CLASSES CONTINUED
From Lemma 1311 we easily deduce that the following i s provable in ( S , 6(R, S, H), %(S)):
(a) (VX”) (I! Xo)( X u (b) (X‘ = H”X* & YC
=
H’X*), (VXn) (El! X * ) (Xu = H”X*),
=
H’Y*)
--$
[X*
E*
y*
X‘;
yc]
Consequently, if cp(X, . .., Y ) is a fundamental formula then (c) S, 6(R, S, H), % (S) I- (Xu = H”X* & 0) -+ [cpo(Xu, 0 ) = cp*(X*, o ) ] is provable. If cp is a FF holding in %(S)as a model in (S, nI(S), Yt(S)) (arrow 2) then cp holds in 91(S) as a model in ( S , 6(R, S,H), 91 (S)) (arrows ;? * 4), i.e. cp“ is provable in the latter theory. By (a) and (c), cp* is also provable in this theory. But cp* is an S-formula and ( S , 6(R, S, H), 91 (S)) is a conservative extension of 5 (arrow 3), hence S t. cp* i.e. cp holds in 1. c) The stnndarti F-definition We shall often be concerned with relations which are sub-relations of the “membership” relation E. Thus we now state a theorem which gives sufficient conditions for such a relation to be E-like. 1314.
DEFINITION
properties:
(TC). X is a V-like class (Vlk ( X ) ) if it has the following
(i) X is not empty, (ii)
11,
t’
EX
= ( u , 1.1 E X ,
(iii) ( u , 1) E S & u
+ c) -+
u nX
+ u nX .
1315. LEMMA (TC). (a) If X is V-like then E A X is E-like (and, moreover, strongly extensional) and X z = X n V2. (b) V is V-like. PmoJ Suppose X is V-like. From (ii) we have X 2 = X n V2. Consequently, E n X = E n X 2 and k:xtEnX(u) = u n X for all u E X . Thus E A X is strongly extensional by (iii). By (i) X is non-empty; if u E X then by (ii) ( u } E X and so ( u , ( u ] ) E E n X . Hence E n X is non-empty. If u. c E’X then { t i , u } E X and Ext,,,((u, u } ) = {u, u } so that E n sY is a pairing relation. Hence E n X is E-like. (b) is obvious. 1316. MEIADEFINITION. The pair of formulas
3) 60
(VX) [(IX*) ( X * = X )
=X
E M] & sec ,
CHAP. 1 SEC. 31
THEORY OF CLASSES CONTINUED
(VX*, Y * ) ( X * E* Y*
(4)
= ( 3 z E Y*) ( X *
= z n M)) &
1317
sec
is called the standard F-deJinition and denoted by Gt (M). Thus classes in the sense of 6t (M) are subclasses of M . 131 7. METATHEOREM. The F-definition 6 t (M) with the specification Vlk (M) determines a model of T C in TC. Moreover, the model Dir (dTC/dGt (Dl))is a faithful model of T C in TC, Vlk (M), Gt (M).
Demonstration. (1) We observe that the standard F-definition is closely related to the normal F-definition; it is in a sense a particular case. Consider Diagram 1317a: 13170
I
2I
Arrow t is air (LJTC/d%(R)); which we know to be a model. Arrow 2 is air (?TC/dGt (M)); we want to demonstrate that it is a model. Arrow 4 is cvidently a faithful identity model. Arrow 3 is the direct translation which translates the symbols of (TC, Elk (R)) identically and the symbols defined by dTC/d% (R) onto the corresponding symbols defined by dTC/dGt(M). The diagram obviously commutes because the image of a TCformula by 2 (or, equivalently, by 2 * 4) coincides with the image by 1 * 3. We demonstrate that arrow 3 is a model. All axioms of T C hold in 3 since they are translated identically. By Lemma 1315 Elk (E n M) is provable in the terminal theory of 3. Now consider the F-definition % (R) (cf. the formulas (l), (2) above). To prove the images of these formulas in 3 it suffices to show that X c M = . X c C (R)& Sat, ( X ) , (3z E Y*) ( X * = = Ext, ( 2 ) ) = ( 3 z E Y*) ( X * = z n M). But this follows (in the terminal theory of 3) from the fact that R = M n E, sExtl (R)and Ext, ( z ) = z n M. Hence both (L) and (2) hold in 3. Then all members of 8TC/d% (R)hold in 3 because the definitions following % (R) in 8TC/d% (R)are translated into precisely the definitions following Gt (M) in dTC/dGt (M). Thus 3 is 61
1318
THLORY O r CLASSES CONTINUED
[CIIAP. I S E C . 3
a model. Therefore all axioms of TC hold in 2 * 4 (which is equivalent to 1 * 3); but thc image of an arbitrary axiom cp of T C in 2 * 4 does not contain the constant H and, since 4 is faithful, cp holds i n 2. This demonstrates that 2 is a model of TC. (2) There i s a specification for Gt(M) in TC stronger than Vlk (M), namely M = V. By 1269, 1270 IDir(G:TC/dGt(M)) is equivalent to the identity model as a model of T C in TC, M = V, Gt (M). Hence by 1266(b) IDir (dTC/dEt (M)) is a faithful modcl of T C i n TC, Vlk(M), ciTC/ciGt (M). The next Metathcorcm tells more on the relation of the standard F-definition to the normal F-definition. 2318. MCTATHCOKEM. Let T be a theory stronger than TC, let n(R) be (R)and Ict $(M) be a spcciIication for Gt (M) i n T stronger thanVlk (M). Suppose T, tI/(M)t n(E n M). Then for any TC-formula q~ we have: a pxification for Y2(R) in T stronger than Elk
(a) If cp holds in a i r (dTC/C% (R))asa model inT, n(K), (:TC/ZIl (It) then cp holds i n Dir (iTC/CSt (M)) as a model in T, $( ?TC/dGt % (M). I), (b) If cp is absolute from above (from below) in a i r (G:TC/d%(R)) as a model in T, n(R). CTC/d% (R)) then cp is absolute from abovc (from below)
air (dTC/?Gt (M)) as a model in T, $(M),i)TC/dGt (M).
jii
Dernotzst~ation.Consider the following diagram: 1318a
Hcrc the arrows have the same meaning as i n Diagram t317a. We may suppose w.1.o.g. that the language of the derivation dTC/d% (R)is the same as the language of dTC/dGt (M) and we denote images of TC-formulas in the corresponding direct model by asterisk. All arrows in Diagram 1318a are models (the assumption T, $(M) t n(E n M) ensurcs that 3 is a model). Neither the image cp* of cp nor the formula cp(X*, 0 ) --* (p*(X*, 0 ) expres62.
CHAP. I SEC. 31
1321
THEORY OF CLASSES CONTINUED
sing the absoluteness from above contains the constant R. Hence if either formula is provable in the terminal theory of 1 then it is provable in the terminal theory of 3 (because it is translated identically by 3). Since 4 is faithful, this formula is provable in the terminal theory of 2 which completes the demonstration. According to Lemma 1304 the following equalities may be proved in TC: E(X)nZ =E(XnZ), (X-Y)nZ=(XnZ)-(YnZ). Assuming that Z is V-like we can prove more: 1319. LEMMA (TC).If Z is V-like then
( X r Y ) n Z = ( X n Z ) r ( Y n Z ) , Cnv(X)nZ=Cnv(XnZ), Cnv, ( x )n z = Cnv, ( X n Z ) . (We may summarize these equalities in the form F,(X, Y ) n Z = = Fi(X n 2, Y n 2 ) for i = 2, 3, 5, 6, 7; where Fi denotes the i-th godelian operation .) Proof. Suppose Z i s V-like. Since ( y , x) E Z + x E Z we have D(Z) E Z; hence, by 1304,
We have Cnv (Z) E 2, U c Z
-+ Cnv ( U )
c 2 ; hence
Cnv (X) n z = Cnv ( X n Cnv (z))c Cnv (X n Z ) E Cnv (x)n z Cnv, is treated similarly. 1320. THEOREM (TC). Let Z be V-like. If X, Y _C Z then F,(X, Y ) E Z for i = 2, ..., 7. Proof. For i 4 this follows from the preceding lemma; indeed, if X, Y c E Z then X n Z = X , and Y n 2 = Y so that F,(X, Y ) = F,(X, Y ) n 2, i.e. F,(X, Y ) E 2. Furthermore, X c Z -+ D ( X ) E Z follows from the fact that if ( u , u ) E Z then v E 2.
+
1321. LEMMA (TC). If Z,, Zz are V-like classes and if F is an isomorphism of Z,, 2, w.r.t. E, E then for arbitrary X , Y E 2, we have
F”(F, ( X , Y ) ) = Fi (F”X, F”Y) for
i
=
2,
. .., 7 . 63
1322
[CHAP, I SEC. 3
THEORY OF CLASSES CONTINUED
Proof. This is trivial for i = 3. For other cases note that if {u, u ) E 2, then {F’u, F’u} E 2, and F‘{u,u } = {F’u, F’u}. Hence if (u, u ) E Z1 then F‘(u, u ) = (F’u, F’u). The lemma follows by an easy examination of cases ( i = 2. 4, ..., 7). E.g. s E F”D ( X )
( 3 y , U) ( ( u , y ) E X & x = F’y) ( 3 y E D ( X ) ) (X = F’y) (3y, fl) ((F’u, F ’ y ) E F”X & x = F’y) = = (30) ( ( u , x) E F”X) = x E D ( F ” X ) . EE
1322. METATIIEOREM. Tn the model air (dTC/aGt (M)) as a model in TC, Vlk (M), dTC/dG;t(M) (a) the formulas X E Y, Re1 ( X ) are absolute, (b) the giidelian operations F,(X, Y ) (i = 2, ..., 7) are absolute (i.e., the formulas U = Fi(X, Y ) are absolute).
Demonstration. (a) The absoluteness of X E Y is trivial. Let us proceed i n the theory TC, Vlk (M), dTC/dGt (M). Recall that sets in the sense of Gt (M) are o f the form x n M where x E M. Further, if { u , u ) E M then 11, 1’ E M, ( u , u} n M = {u, u } and { u , u ] = { u n M, u n M}*. Indeed, u n M E * { u , u } , L’ n M E * ( u , v } and if w n M E * { u , u ) then either \v n M = u n M or w n M = u n M, hence, by (iii) in Definition 13 t4 w = u or w = u. Consequently ( u , u ) = ( u n M, u n M)* and Rel(X*) = Rel*(X*) fo I lows. (b) The absoluteness of U = F,(X, 0 ) means that Fi(X*, 0 ) = FT(X*,o) is provable. This is obvious for X - Y. For the other operations recall
(‘1
u, u E
M -+ [ u
(4
u, u E
M
(3)
ti,
uEM
Eu
3
u n M E* u n M] ,
= (u, u ) E M , + (u,
(4)
u ) = ( u n M, u n M)* ,
Fi(X*, 0 ) C M .
Let us consider E ( X ) arid D ( X ) for example. We suppose x
X E E( X * )
E M.
. = . ( ~ u , u ) ( x= ( U , U ) & U E U & X E X *=) . ( ~ u , u E M ) (=x ( u , u ) & u E u & x E X * ) =
= .(3u*, u*) (x E
64
= (u*, u*)* & u*
.x€*E*(X*).
€*
= .xEE*(X*);
u* & x
Ex
*)
=
CHAP. I SEC. 31
x E D (X*)
1325
THEORY OF CLASSES CONTINUED
= .( 3 y ) ( ( y , x) EX*) E
= . ( 3 y ~ M ) ( ( y , x ) ~ X *=)
= . (3y*) ((y*, x n M)* E* X*) = = . ( x n M ) € * D * ( X * ) .E . x E D * ( X * ) . d ) A stronger specijication f o r Gt (M); restricted formulas 1323. DEFINITION (TC). X is a complete class (Comp ( X ) ) if X contains with every element u all elements of u, i.e. (Vu, v) (u E v & v E X + u E X ) .
.
Note that T C I- Comp ( X ) E UX E X = X E P ( X ) . Evidently, T C t t Comp (V). Consequently, the formula Vlk (M) & Comp (M) is a specification for Gt(M) in T C which is stronger than the specification Vlk (M). First observe that in the theory TC, (Vlk (M) & Comp (M)), Gt (M)) (which we denote by TCComp in the present discussion) we can replace the definition of E* by the following equivalent but simpler definition: X * E* E* Y* = X* E Y*. Indeed, TC, Comp (M) t z E M + z n M = z. Consequently the sort of variables for sets in the sense of Gt (M) is subordinate I- M* ( X ” ) .+ M ( X * ) ; thus sets in to the sort of set variables, i.e. TC,,,, the sense of Gt (M) are simply elements of M. The formulas x E y , x E Yare absolute. Further recall that Godelian operations F, to F, are absolute (by 1322). We shall find a whole system of formulas (having a certain syntactic structure) which are absolute in Gt(M) with the specification Vlk (M)& Comp (M). 1324. METADEFINITION (restricted formulas). Let T be an €-theory stronger than (TE, d l , d2). I. (a) Every atomic normal T-formula is a restricted T-formula. (b) If cp, $ are restricted T-formulas and x, y are set variables then cp& $, 7 cp, (3x) (x E y & cp) are restricted T-formulas (provided they arc formulas). (c) Every restricted T-formula is obtained by a finite number of applications of the rule (b) beginning with formulas described sub (a). We say “RF” instcad of “restricted T-formula”. 11. A T-formula cp is said to be restricted in T if there is a restricted T-formula $ such that T t cp z $. 1325. METADEFINITION (set formulas). Set formulas are formulas of the language (x, E). “SF” means “set formula”. 65
1326
THEORY OF CLASSES CONTINUED
[CHAP. I SEC. 3
1326. Remarks. 1) We compare the notions FF, SF, NF and RF. The FF’s are the formulas of the language ( X , E), while the SF’s are the formulas of the language (x, E). A formula ‘p is N F iff the two following conditions hold: (1) each atomic subformula of cp has the form x E Y where x is a set variable and Y is a constant or a variable different from x (thus no equality sign occurs); (2) only set variables are quantified. If, in addition, all quantifiers are restricted to set variabes then cp is a RF. 2) By a normal set formula (NSF) we shall mean a formula which is both NF and SF. Similarly we shall speak of RSF’s. 1327. METATHEOREM. Every restricted TC-formula without constants is absolute in Dir(aTC/aG;t(M)) as a model of TC in TC, Vlk (M)& & Comp (M), aTC/dGt (M). Demonstration. The metatheorem is true for atomic RF’s and the induction step is obvious for &, 7 . Suppose that q ( y , x, 0 ) is absolute and consider (3y E x) cp (y, x, 0 ) . This formula is absolute from below (cf. 1268); hence it remains to demonstrate that it is absolute from above. We proceed in TC,,,,. Suppose x, 0 EM, X , 0 E M, x* = x, 0 , X * = X , 0 and ( 3 y E x) ~ ( yx,, 0 ) . Then we have ( 3 y E M) ( y E X & 9 ( y , x, 0 ) ) and hence (3y*) (y* E* x* & cp*(y*, x*, 0)). RF’s are very important. In the remainder of this Section we shall show that quite a number of formulas are restricted in TC (or in stronger theories). Evidently, x = y is restricted in TC because it is equivalent to (Vu E x) (u E y ) & (Vu E y ) (u E x). Further we have the following
1328. METALEMMA. Let us say - in the context of this metalemma only that a term t is a pairing term if it is built up from set variables and the operation { >.Then the following holds: (a) For any pairing term t, the formulas y
Et
and y = t are RSF’s in TC.
(b) Let t be a pairing term which is not a variable. If u l , .. ., u, are variables in t and if cp is a RSF in TC then (3ul, 0) (x = t & ‘p) and (Vu,, 0 ) (x = t + ‘p) are RSF’s in TC. Demonstrution. (a) If t is a set-variable then y E t and y = t are RSF in TC. Suppose that t is {tl, t,) and that the assertion holds for t, and t,. The formula y E t is equivalent to y = t, v y = t, which is a RSF in TC; the formula y = t is equivalent to the formula (Vu E y ) (u E t) & t, E y & t2 E y which is a RSF in TC, since t i E y is equivalent to a RSF in TC, namely (3z E y) (2 = ti). 66
CHAP. I SEC. 31
1330
THEORY OF CLASSES CONTINUED
(b) If t is a pairing term which is not a variable then it has the form
it,, t,} for some pairing terms t, and t,.
We distinguish three cases as follows: (i) if t , is u1 and t , is u, then the formula ( 3 u , , u 2 ) (x = {ul, u2}& cp) is equivalent to the formula ( 3 u , E x) (3u, E x) (x = { u l , u,} & cp) which is a RSF in TC; analogously for the formula (3u,) (x = { u l , u2}& cp); (ii) if t, is u , and t , is not a variable then the formula (3u,, o)(x = = { U l , t2} & 40) is equivalent to the formula (3u1 E x) ( 3 u , E x) (x = = { u , , u , ) & (3u,, 0 ) (ul = t 2 & cp)) which is a RSF in TC by induction hypothesis; the demonstration is analogous if t, is a variable and t , is not; (iii) if neither t, nor t , is a variable then the formula (3u1, 0) (x = t & cp) is equivalent to the formula (3,E x) (30, E x) (x = {o,, u 2 } & (3u,, ..., ui) ( v , = t, & ( 3 ~ ~ 0+) ~( u ,, = t, & cp))) which is a RSF in TC by induction hypothesis. The assertion for V can be derived by considering the formula 1cp. 1329. COROLLARY. If the formula cp(x, u l , u p )is a RSF in TC then the formula ( 3 u , , u2)(x = ( u , , u 2 ) & ~ ( xu,, , u,)) is a RSF in TC; similarly for ( u , , u2, u3>. 1330. METATHEOREM. The following formulas are RSF's in TC:
Demonstration. All of the equivalences below are provable in T C and their right-hand sides are RSF's in T C by the preceding metatheorem.
= . (3u,u) (y = ( u , U ) & u E u)& - x, = . y E X 1 & y & X z ,
y E E (x) YEXI
y
E
D (x)
yEX1
y
E
J E
E
= ( 3 E~X ) ( 3 ~()Z = (u,y)) , = (3u,u)(y = ( u , u ) & y E X l & u E X 2 ) ,
rx2
Cnv (x) z (314,u) (y = (u, u ) & (3z E x) (z Cnv, (x)
Y E U(x)
y
yEx ,
P (x)
= ( j u , u, w) ( y
=
=
(u, u))) ,
( u , u, w) & (3z E x) ( Z
=
(w,u, u ) ) ) ,
= (3u E x) (Y E ). ,
= (vu E y ) (u E X ) . 67
1331
[CHAP. I SEC. 3
THEORY OF CLASSES CONTINUED
1331. METAIHEOREM. The following formulas are RSF’s in TC:
Demonstrotior!. All of the equivalences below are provable in TC:
similarly for sI - x2 and x, J’ =
& (vu
1‘ x2,
D (s) E
= . (VU E y) ( U E D (s))&
x) (vs. t ) (24
= (5, t )
-+
t
E
x) ,
Cnv (x) 3 . (Vu E !.) ( u E Cnv (s))& s)(VS, f ) ( z = (s. t ) + ( 3 4 E y) ( u = ( f , s)) =
& (Vz
E
.
simiiarly for Cnv, (x),
(Note that nothing is asserted about
j’ =
P(x).)
1.333. MI‘TAIEMMA. (a) Thc formula Un ( , f )is a RSF in TC. (b) If q(s. y l , e) is a RSF in TC then Un ( f ) & ~ ( f ’ z ,y l , 0 ) is a RSF in TC. Dcnionst~Ntion. The following equivalences are provable in TC:
& (VU? r f f )
=
Un ( f ) = (Vu ~ f(h, ) Y ) ( u = (v, x>)& y , z ) (u = ( y , x) & 1’ = ( z , x) . y
(VX.
--f
Un (f)& cp(f’z, j v 1 . e) . 3 . un ( f ) & p u E j ) (3s)(11 = (N, z ) & q ( x , y , ,
= 2))
0))
.
,
CHAP. I SEC. 41
THEORY OF SEMISETS
1400
SECTION 4
The theory of semisets
Semisets will now be defined as those classes which are included in sets (as subclasses). We shall present some axioms for semisets and collect them together into the axiom system of the theory of semisets (TSS). We shall also present a theory stronger than T S S and called the theory of sets (TS). The theory of semisets is unusual in several aspects; however it plays a fundamental role throughout the book. The theory of sets is equivalent to the usual (Bernays-Godel) set theory but it will only be used exceptionally. From the point of view of the theory of semisets T S seems to be too restrictive, assuming simply that all semisets are sets. In this way some very interesting developments are excluded, especially the construction of certain models. On the other hand, the connection between TSS and T S is quite close. (1) These theories are equiconsistent, i.e. TSS is consistent iff T S is. (2) Moreover, upon the addition to T S S and T S of a very weak “regularity” axiom (Dl) whose consistency with TSS (and with TS) will be proved, then we obtain theories TSS‘ and TS’ in which the same statements concerning sets (set formulas) are provable. Thus nothing concerning sets is lost in TSS‘. Indeed, one can construct a model of TS’ in TSS‘ which, so to speak, only excludes certain proper classes. This construction which might be called “downward”, will be described in the present Section. (The demonstration of the consistency of (Dl) with TSS will be postponed to Chapt. I11 Sect. 1.) (3) There is, however, another important method for obtaining models of T S in certain extensions of TSS (consistent relative to TSS) which will be presented in Chapt. V Sect. 1; this is the “upward” construction. Here one constructs a (faithful) model of a certain extension of TS; this model has, roughly speaking, more sets than the original theory of semisets. (In particular, the independence of the continuum hypothesis 69
I401
[CHAP. I SEC. 4
THEORY OF SEMISETS
can be proved in this way.) It is this possibility for constructing models which gives the theory of semisets its importance.
u ) Semisets
1401. DEFINrTIoN (TC). A class X is a serniset (Sm ( X ) ) if it is a subclass of a set, i.e. ( 3 y ) ( X c j*).We use variables D , e (and others if necessary) for semisets.
1402. DEFINITION (TC). A class X is a real class (Real ( X ) ) if its intersection with every set is a set, i.e. (Vx) M ( X n x). 1403. LEMMA (TC). (a) V is a real class. (b) 0 is a semiset. (c) Every set is a semiset. (d) Every subclass of a semiset is a semiset. (e) The intersection of a claca with a set is a semiset. (f) If a class is a real class and a semiset then it is a set. Proof. (a) to (e) are obvious. (f) Suppose a semiset D is a real class. There is a set y such that u E y and so D n y = u. Since D is real, D n y is a set. We now introduce some notions concerning relations in the context of semisets. A function was defined in the usual way as a many-one relation (Un(R)). This definition can be generalized so that every relation can be considered as a “mapping which assigns to each y E D ( R ) the class ExtR(x)”. Such a generalized mapping might be called a functor. The following lemma shows that this really is a generalization of the definition of a function:
1404. LEMMA (TC). If F js a function then there is a unique relation R such that D(R) E D(F) and
(Vx E D ( F ) ) (F’x = E d R(x))
.
Proof. Let ( z , x) E R = z E F ’ x ; clearly ExtR (x) = F’x. If F ’ x = 0 then x 6 D ( R ) and Ext, (x) = 0 = F’x. Notice that the values of a functor may be proper classes. Functors whose values are semisets play an important role in our further investigations. We call such functors regular relations or senii.set-valued functors.
1405.
DEFINITION
Reg ( R ) 70
(TC).
= . Re1 ( R ) & (Vx E D ( R ) ) (Sm ( h f(x))) ~.
CHAP. I SEC. 41
I410
THEORY OF SEMISETS
Remark. The predicate Reg ( R ) is normal since it is equivalent in TC to the formula
Functors which are “one-to-one” will be called nowhere constant. 1406. DEFINITION (TC).
Ncon ( R ) = . Re1 ( R )& (Vx,y
E
D ( R ) )(x =I=y
-+
Ext, (x) =I= Ext, ( y ) ) .
1407. LEMMA (TC).(a) sExtl ( R ) --* Ncon ( R ) , (b) Un ( R ) + (Un, ( R ) = Ncon (R)).
Proof. (a) sExtl ( R ) means that any two different elements of C ( R ) have different extensions; Ncon ( R ) follows immediately. (b) Suppose that R is many-one; then Ext, (x) = {R’x}. If Un, ( R ) then for X, y E D (R), x =ly=we have R’x R’y and hence (R’x} =I={ R ’ y } ; if Ncon ( R ) then for x + y we have {R’x) 9 {R’y} and so R’x $. R’y.
+
1408. DEFINITION (TC).A relation R is an exact functor (Exct ( R ) ) if it is regular and nowhere constant. 1409. LEMMA (TC).Un, ( R ) + Exct ( R ) .
b) Exact functors 1410. We now present two axioms which will be denoted by (Cl) and (C2). (Cl) Axiom of Infinity
(C2) Axiom of Exact Functors Exct ( X ) + (Sm (D ( X ) ) = Sm (W ( X ) ) ) According to (Cl) there exists a nonempty set which contains the empty class and for each element y the oneelement set { y } . We shall see later that this set is - in a precisely defined sense - infinite. However this axiom will not be used for the present; when it is used we shall mention the fact 71
1411
THEORY OF SEMISETS
[CHAP. I SEC. 4
explicitly. Axiom (C2) is of basic importance. According to this axiom an exact functor has a small domain iff it has a small range, where by a small class we mean a semiset.
2411. THEOREM (TC, C2). For any semisets u, e the classes U o, a u e, P (0) and o x e are semisets. Proof. (a) Let o be a semiset; we prove that its sum is also a semiset. Since E is an exact functor, F, (T is also an exact functor and since D (E o) E o we have by (C2) that W (E o) is a semiset. This proves the assertion, since u o = w (E 0). (b) Let a c s,Q c y. Then CJ u g c U((.Y,y } ) , the latter class is a semiset. (c) Let o be a semiset; we prove that P(o) is a semiset. Set F,, = = ( ( u , u ) ; u E v& u c a]. The relation El is an exact functor and W (El) C G a, hence W (El) is a xmiset. If the empty class 0 is not a set then D (E,)= == P (a) and we are done. In the oppossite case D (El) = P (o) - (0) and P(a) = D(E,) u YO], hence P(a) is the union of two semisets and the assertion follows by (b).
r
r
r
(4
o x
Q
r
c P (P((7 u e))
1412. THEOREM (TC, C 2 ) . For any semisets cr, e the following classes are semisets:
E (a), o - g , D (a). a
; 1
e , Cnv (a), ~ n v (o) , .
re
Proof. Let o and e be arbitrary semisets. Since E (o), o - g and a are included in o it follows (without using (C2)) that these classes are semisets. To prove that D (o), Cnv (0) and Cnv, (0) are semisets we use the preceding theorem; indeed, D(o) E U Ua, Cnv(o) E PPUUo and Cnv, (a) c PP (ao u PPo,) where a. = UUo u UUUUo.
1423. COROLLARY (TC, C?). For any semisets o,Q the following classes are semisets: a n 9. W (0). C (a), a”@. (Recall that a n e = o - (o - e), W (0) = D (Cnv (o)), C (0) = W (o) u D (a), d’e = W (o i‘g).) Axiom (C2) also has the following consequences:
1414. LEMMA (TC, C2).
1.i
.
Ncon (X) & Sm (W ( X ) ) -+ Sm (D ( X ) ) ,
CHAP. 1 SEC. 41
THEORY OF SEMISETS
(b)
Reg ( X ) & Sm (D (X)). 4 Sm (W ( X ) ) ,
(4 (4
Reg ( X )& Sm (D ( X ) ) . --t Sm ( X ) ,
1416
.
Ncon ( X )& Sm (W ( X ) ) -+ Sm ( X ) .
Proof. (a) If X is a relation and W ( X ) is a semiset then X is trivially regular; hence if X is nowhere constant then it is exact and the assertion follows by (C2). (b) Let X be a regular relation with a semiset domain. Define a = {((y, x), x); ( y , x) E R } . Then D ( R ) = D (a) and so D (a) is a semiset. For x E D ( R ) , ExtK(x) = Ext, (x) x (x}, hence ExtK(x) is a semiset by 1411. If x =yi= then ExtK(x)n Extx ( y ) = 0 and so a is nowhere constant. By (C2), W (R) is a semiset and W ( R ) = W (W (a))is also a semiset by 1413.(c) If X is regular and D ( X ) is a semiset then, by (b), W(X) is a semiset; further X E W (X)x D (R) and so X is a semiset by 1411. (d) is proved similarly. There is a specification for the F-definition %(R) such that %(R) with this specification determines a model of TC, C2 in TC, C2. Define in TC: 1415. DEFINITION (TC). A relation X is almost universal (AUn(X)) if for every semiset (T E C ( X ) there exists z E C (X)such that (T E Extx ( z ) . Remarks (TC). 1) Note that E is regular and almost universal. 2) The formula AUn ( X ) is normal in TC because it is equivalent to (Vx) (3y E C ( X ) ) (x n C ( X ) c Extx (y)). 3) If C ( X ) is real then X is almost universal iff (Vx c C (X))(33. E C ( X ) ) (x C Extx (y)). 1416 . METATHEOREM. Consider the formula
sExtl (R) & Prg (R)& Reg (R) & AUn (R) which in the context of the present Metatheorem will be denoted by Mrel (R)(“R is a model relation”). Mrel (R)is a spccification for 8 (R)in TC, C2 such that 8 (R)with this specification determines a model of TC, C2 in TC, C2. Moreover, air (dTC/ll% (R)) is a faithful model of TC, C2 in TC, C2, Mrel (R), dTC/d% (R) and the formula Sm ( X ) is absolute in this model.
Demonstration. Mrel (R) is a specification because TC, C2 I( 3 X ) Mrel ( X ) ; indeed, TC I- Mrel (E). Moreover, this specification is stron13
I41 7
[CHAP. I SEC. 4
THEORY OF SEMISETS
ger than Elk (R)since T C k AUn ( X ) -+ X $: 0; indeed, 0 is a semiset,O c E C ( X ) and there is a y E C ( X ) such that 0 c E d x (y). Hence !Jl (R)with Mrel (R)certainly determines a model of T C in TC, C2 by Metatheorem 1266. It remains to prove (C2)* in the theory TC, C2, Mrel (R), 8TC/8% (R). First we prove Sm ( X * ) = Sm* (X*). If X * is a semiset then by almostuniversality there exists z E C (R) such that X* c Extn ( z ) ; we have M*(ExtR( z ) ) and X * G * Ext, ( z ) , since c is absolute; hence Sm* (X*). Conversely, if Sm* ( X * ) then X * C * z* and z* = Extn ( 2 ) for some z; by regularity, z* is a semiset and hence X * is also a semiset. Suppose now that Z* is some relation in the model and consider the decoded relation W = = Dec (2"). First we show that Exct* (Z*) implies Exct ( W ) . If Reg* (2") then (all extensions are semisets)*; for u E D(W) we have Extn (u)E* E* D* (Z*), Ext, ( u ) = Ext,*, (Ext, (u)) and Sm* ( E d w (u)); hence Sm (Ext, (u))and so Wis regular. If Ncon* (Z*), we let u and u be distinct elements of D ( W ) ;since R is strongly extensional we have u* = Ext, (u) =+ Ext, ( u ) = u* and Ex tE (u*) ExtZ* ( u * ) ; hence Ext, (u)$: Ext, ( u ) and Wis nowhere constant. Thus Exct* (2") implies Exct ( W ) . Further we have D* (2") = D ( W ) and W* (Z) = W ( W ) ; hence Sm* (D* (2')) = _= Sm (D ( W ) ) = Sm (W ( W ) ) 3 Sm* (W* (Z*)). We have proved that %(R) with the specification Mrel(R) determines a model of TC, C2 in TC, C2. The faithfulness of the corresponding direct model is proved in exactly the samy way as in the demonstration of Metatheorem 1306.
=+
+
1417. I ~E F I N IIION (TC). A class X is called an alniost uniuersal class
( AUncl ( X ) ) if every subsemiset of X is a subsemiset of an element of X,i.e. (va
c x)( 3 y E X ) .(
E y)
.
Ekidently, X is an almost universal class iff E n X z is an almost universal relation; V is an almost universal class. 1418. METATHEOREM. The F-definition Gt (M) with the specification Vlk (M) & AUncl (M) determines a model of TC, C2 in TC, C2. The model Dir (?TC/dGt(M)) is a faithful model of TC, C2 in TC, C2, Vlk (M)& & AUncl (M), dTC/dG?t(M) and the formula Sm (X) is absolute in this model. The demonstration is analogous to the demonstration of the preceding Metatheorem but is much simpler because Re1 ( X ) is absolute. Thus it can be left to the reader. 14
1421
THEORY OF SEMISETS
CHAP. 1 SEC. 41
c ) Comprehensiveness; the theory of semisets
1419. METADEFINITION. Let T be a theory stronger than TC and let cp be a normal T-formula. Let X , 0 be all free variables of cp which are not set variables and let xl, 0 be a sequence of distinct set variables which is free for cp (cf. 1115). If T is any term representing p in T w.r.t. sl,0 then cp is said to be comprehensive in T w.r.t. x,, 0 in case
(*)
T I- (Real ( X ) &
0) 3
Real (T)
(If cp has no free non-set variables replace (*) by T I- Real (T).) The formula cp is fully comprehensive in T if it is comprehensive w.r.t. any sequence of distinct variables free for cp. 14410. Note that if (*) holds then
(**I
T I- (Real ( X ) & 0 ) 3 (Va) (3b) (Vu) ( u E b
= .uEa&(3x,,.)(u
=
= (x,,o)&cp)):
for. given a we may take b = T n a. Thus, given real classes X , 0 and a set a, there is a subset b of a which consists of all tuples ( u l , 0 ) E a satisfying cp. O n the other hand, if (**) holds and T is a term representing cp in T w.r.t. xl, 0 then we can proceed as follows in T: If X , 0 are real classes, a is an arbitrary set and b is the set satisfying the condition of (**) then necessarily b = T n a; hence T is a real class. Thus we see that the conditions (*) and (**) are equivalent and, in particular, the notion of comprehensiveness w.r.t. xl,0 docs not depend on the particular choice of the term T. 1421. We now add to TC, C2 a group of axioms which will ensure that
in the extended theory every RSF is fully comprehensive. The situation here is analogous to the introduction of the axioms B1 - B7 to ensure the “class comprehension scheme” ( 3 A ) (VX)
(X
EA
( 3 ~ 1 O, ) (X
= (u1, 0 ) & cp))
(Metatheorem 1120 and Metadefinition 1115). There it was sufficient t o take as axioms seven particular cases of the scheme. For the restricted comprehension scheme it suffices to take six
1422
[CHAP. 1 SEC. 4
THE ORY 0 1 : Sf MISI- IS
particular cases; since they are analogous to the axioms B2 to B7 we shall denote them by A2 to A7. We already know that the formulas y E Fi ( i = = 2, . .., 7), where Fi are the gijdelian terms E (x), x 1 - x2, D (x), x1 x2, Cnv (x) acd Cnv, (x), are restricted in TC; further we know that in TC + C2, the classes corresponding to these terms are semisets. Thus if the formulas y E Fi are comprehensive in some extension T of TC + C2 then one can prove in T that the Fi are sets (see 1330, 1412, 1403(f)). This is precisely the content of the following axioms:
r
1422. AXIOMS. (h2) (V.Y) (32) (VU)
(A3)
(U E
(Vx,1.) (k)(Vu) ( t i
(A4) (Vx) (32) (Vu)
(A5) (VX,J) (3:)
(11 E
( 3 ~W) , (U
z E
z
(VX)(3:) ( V U )(11
= ( 3 ~( ()u , 21)
(A7)
(VX) ( 3 ~(VU) )
E IG
( I / E 2 EZ
(u, \v) & 21
E
x & 'L E
w)) ,
= u E x & u 6y ) ,
:
("I{) ( I I E z
(Ah)
=
EX))
( 3 u , W)(I/
(30,11')
=:
,
(G, W)
(// = (0, W )
(3~, W , t ) (U
=
& u E x & \Y
& (w,
C)
E
x))
( v , \\', t ) & ( w , f,
E 11)) ,
,
V) E
x))
.
1423. The theory of semisets T S S is the extension of TC by (A2) to (X7), (Cl) and (C2). Axioms (A2) to (A7) differ from (B2) to (B7) only by having (Vx) (3:) ... and (Vx, y ) ( 3 z ) . . . instead of (VX) (32) . .. and (VX, Y ) ( 3 Z ) ...; hence they assert nothing more nor less than the fact that for a n y set x (or any sets N, the class given by the axiom (Hi) IS . a set. 19)
1424. We shall now showthat in T S S every R F is comprehensive.We shall not need the axiom (Cl) of infinity; to stress this fact we denote by TSS-, the theory T S S with the axiom (CI) removed. First some lemmas.
1425. LEMMA(TSS-,). For any sets s. y : E(x), x - y, D(x), s r y , Cnv (x), Cnv, (x) are sets. Immediate from the axioms A2 to A7.
1426. LIMMA(TSS-,). (a) Every set is a real class. (b) If classes X , l-arc real then the following classes are also real:
E(X). X
-
Y , X r Y , Cnv(X), Cnv,(X).
(Note that nothing is a w r t e d about D ( X ) . ) 76
1430
THEORY OF SEMISETS
CHAP. 1 SEC. 41
Proof. (a) If x and y are sets then x n y = x - (x - y) is a set. (b) The assertion follows from the equalities 1304 writing 2 instead of Z ) . 1427. LEMMA (TSS-,) (a) 0 is a set. (b) For any sets x, y the following classes are sets: x n y , W (x), x”y, Ux,x u y , C (x), x x y.
Proof. (a) 0 = x - x. (b) The assertion concerning x n y , W (x), x”y follows immediately from Lemma 1425. For Ux recall that in the proof of Lemma I41t we showed that U x = W (E x). Now, E x is a real class and. in addition, a semiset (by Lemma 1414). It follows that E x is a set and hence W (E x) E Ux is a set. Consequently, x u y = U({x, y ) ) is also a sct. Finally, C (x) = D (x) u W (x) and x x y is a real class and a semiset . We are now going to prove that every R F without constants is comprehensive in TSS-,. However it is possible to prove the full comprehensiveness of all formulas of a family of NF’s substantially larger than the family of RSF’s. (Cf. Metatheorem 1438.)
r
r
r
r
1428. METALLMMA, Every atomic N F without constants is fully comprehensive.
Der17onstration. We follow the demonstration of Mctalemma t 1 t9. (a) For the term T = D ((Y n {x}) x V“) we have TSS-, t T = 0 v T = = \‘”and so TSS-, k Real (T) trivially. (b) The operation D is not used in the construction of the term T and so T is comprehcnsive by Lemma 1426. (c) It suffices to prove in TSS-, that (y; x E y } is real. We proceed in TSS-,. Denote ( y ; x E y} by Y ; Ict n be a set. Evidently a n Y = {y E a ; x E J.) = D (E n ({x) x a)), while E n ((x} x a ) is a real semiset and therefore a set. It follows that a n Y i s a set because it is the domain of a set. Hence Y is real. (d) The operation D is not used in the construction of the term T.
1429. METALEMMA. If y , $ are fully comprehensive then ip & $, 1ip are fully comprehensive. This follows trivially from the demonstration of Metalemmas 1116, 11 17.
2430. METALLMMA. If cp is fully comprehensive and x, y are set variables not bound in cp then (3y E x) cp is fully comprehensive. Dentonstration. Let u I , 0 be free for ( 3 y E x) ip. We want to prove in TSS-, that a n {(u,, 0 ) ; ( 3 y E x) (p> is a set for any a. Obviously y , u l , 77
1431
THEORY OF SEMISETS
[CHAP. I SEC. 4
is free for cp and also for y E x & cp. Let us proceed in TSS- ., Suppose (Real (X)& 0 ) . Let a be any set. Then a n((u,.
0)
; ( 3 ~E' X ) cp) = a n D ({(p, u l , 0 ) ;
J' E
x & cp}) .
By the full comprehensiveness of y E x& cp the class A = ( ( y , u l , 0 ) ; y E E x & cp) is real; we want to prove that D ( A ) is real, i.e. that a n D ( A )
is a set. (a) If x does not occur in the sequence u l , 0 then we reason as follows: If ( t i 1 . 0 ) E a n D ( A ) then there exists y E x such that (y, u , ,
0) E
A n (x x u ) , i.e. a n D(A)
= a n D(A n (x x
LI))
.
But x x a is a set by 1427, hence A n (x x u ) is a set and u n D ( A ) is a set. (b) Suppose now that x occurs among u,,0 so that x is u1 and { u l , 0 ) is ( u l , u , ) , say. If { u , , u , ) E u n D ( A ) then u l E U U ( u ) and if y E u , then {y, t i 1 , uz> E U U U ( u ) x a. But U U U ( a ) x u is a set (cf. 1427), so that u U U ( a ) x a = c, say. Hence a n D(A) = a n D ( A n c) and a n D(A) is a set. Consequently, we have the following 1431. MLTATHEOREM. Every RF without constants i s fully comprehensive i n TSS-,. 1432. THEOREM (TSS-,).
For every set x, the class P (x) IS . a set.
Proof. By 1411 P (x) is a semiset. If a is a set such that P (x) a then P (x) = a n {p; J' c x). The formula y E x is a RF without constants and so it follows by the preceding Metathcorem that P (x) is a set.
We shall now establish a generalization of Metatheorem 1431; the rather technical demonstration may be omitted on first reading. 1433. METALEMMA. If cp is fully comprehensive and x, y, z are not bound in cp then ( 3 y E (x, z ) ) cp, ( 3 y E U ( x ) ) cp and (3y E P (x)) cp are fully comprehensive. Demonstration. (1) If the sequence u l , 0 is free for ( 3 y E (x,z}) cp(y) then it is free for cp(x) v cp(z), the latter formula is fully comprehensive and TSS-, I- ( 3 y E {x, z } ) cp _= cp(x) v cp(z).
.
(2) If ul, 0 is free for ( 3 y E U(x)) cp and v is a new set variable not bound in cp then u l , 0 is free €or (3v EX) ( 3 y E v) cp, the latter formula is fully comprehensive and TSS-, t- (3y E U ( x ) ) cp = ( 3 u E x) (3y E v) cp. 78
CHAP. I SEC. 41
THEORY OF SEMISETS
1436
(3) Let ul,0 be free for (3y EP(x))cp. We prove in TSS-, that (Real ( X ) & 0 ) + Real ( { ( u , , 0 > ; ( 3 y E P(x)) cp}). Suppose Real ( X ) & 0 , let a be an arbitrary set. We have: a n ( ( u , , 0); ( 3 y E P (x)) cp} = a n n D ( ( ( y , u , , o ) ; ) ; ~ P ( x ) & c p } ) . S e t A= ( ( y , u l , * ) ; y ~ P ( x ) & c p } ;A is real, since the formula y E P (x) & cp is fully comprehensive. We show that D ( A ) is real, i.e. that a n D ( A ) is a set. (a) If x does not occur in u l r 0 then as in the proof of Metalemma 1430 w e h a v e a n D ( A ) = a n D ( A n ( P ( x ) x a)), a n D ( A ) i s a s e t . (b) If x occurs in ul, 0 , x = ui, say, then we reason as follows: If (u,, 0 ) E a n D ( A ) then there exists y E P (ui)such that ( y , u,, 0 ) E A ; but then y E P ( U ... U(a)), hence (y, u,, 0) E P ( U ... U(a)) x a and a n D ( A ) = a n D ( A n (P . .. U(a)) x a)); this last is a set.
(u
1434. METALEMMA. y = P (x) is fully comprehensive. Demonstration. TSS-, t- y = P (x) = (Vu E y ) (u E P (x)) & (Vu EP(x))( u E y). The right hand side is fully comprehensive by the preceding Metalemma. 1435. METALEMMA. Let cp be fully comprehensive and suppose that x. y , z are not bound in cp. Then ( 3 y = (x,z } ) cp, ( 3 y = U(x)) cp, ( 3 y = P (x)) cp are fully comprehensive. Demonstration. Let T be {x, z}, U(x), P(x) respectively, and let u l , 0 be free for ( 3 y = T) cp. Put A = ( ( y , u,, 0); y = T & cp}. We want to prove (Real ( X ) & 0 ) -+ Real (D ( A ) ) in TSS-,. (Recall that X , 0 are the class variables free in cp.) If x (and z ) do not occur in ul, 0 then TSS-, tI- a n D ( A ) = a n D ( A n ({T} x a)) hence TSS-, t- (Real ( X ) & o) -+ -+ Real (D (A)). If x (and z ) occur in ul, 0 then TSS-, t- a n D ( A ) = = a n D ( A n ( S (a) x a) for an appropriate term S ( a ) built up from a, P; hence TSS-, t- (Real (X)& 0 ) -+ Real (D (A)). (In case where T is ( x , z> and x occurs in u,, 0 but z does not we find a term S (a, z ) such that TSS-, t- a n D ( A ) = a n D ( A n ( S (a, z ) x a)).)
u,
1436. METALEMMA. Let cp be fully comprehensive, and let T be a term built up from set variables not bound in cp and from the operations ( }, P. Then (a) ( 3 y E T) cp is fully comprehensive, (b) (3y = T) q is fully comprehensive. Demonstration by simultaneous induction. Let T,, T, be terms for which (a), (b) hold. Then
u,
TSS-,
I- ( 3 y E {T,, T,}) cp
E
( 3= ~ TI) ( 3 = ~ T,) (3y E (u,u)) cp, 19
THEORY OF SEMISETS
1437
[CHAP. I SEC.
4
the right hand side is fully comprehensive. Similarly for =. If F is either U or P then TSS-, TSS-
~
t ( 3 . ~E F(Tl)) ip
( 3 = ~ T,) ( 3 y E F(u)) cp ,
t (31'
( 3= ~ T I )(31' = F(u)) cp
=
F(Tl)) ip
.
We are now able to formulate the generali~ationof Metatheorem 1431. 1437. METADEFINITION. Define PUP-fournulas inductively: (a) Every atomic N F IS a PUP-formula. (b) If cp, $ are PUP-formulas then cp & $, l c p are. (c) If cp is a PUP-formula, T is a term built up from set variables ul, 0 free for cp and from the operations { }, P and if y is a variable not bound in cp and different from u l , 0 then (3y E T) cp is a PUP-formula.
u,
1438. McrATIIEOREM. Every PUP-formula without constants is fully comprehensive in TSS53.
143Y. METALEMMA. Let T be a theory stronger than TSS-,, let cp be a PUP-formula, and let x be a variable free in cp; let x l , 0 be a sequence free for y and let G be a term constructed from the variables xi, and the operations { }, E, -, D, Cnv, Cnv,, P. Then the result of replacing all occurrences of x in ip by G is equivalent to a PUP-formula in T. Denionstrution. Writc q(x) instead of ip and q ( G ) instead of the formula described above. q(G) is equivalent in T to (3x) (x = G & cp(x)). Using the technique of the demonstration of Metatheorem 1438 we see that it wffices to show that (3x) (x = F, & cp(x)) is a PUP-formula in T where Fi is one of the terms (u, r } , E (u), . .., U(u),P (u). This can be left to the reader ac an exercise. The following Lemma provides a useful test for the realness of classes:
r,
u,
1440. LI:MMA (TSS-,). Let X be a class which is not a semiset and suppose that for all x, y E X either x E y or y E x. Then u ( X ) is a real class. Pronf. Let a be a set; we prove that a n U(X) IS a set. Define (x, u> E R E = ( I L E a n U(X) & x E X & u $ x). R is regular: if u E a n u ( X ) then there cxists x, E X such that u E x,; hence for (x, u ) E R we have x E xo and h x e f o r e Ext, (u) E P (xo). Further, D ( R ) E a n u ( X ) , so that D ( R ) is a cemiset. It follows from (C2) that W ( R ) is a semiset. But W ( R ) G X and X is not a semiset; consequently, there exists x1 E X - W (R). We have (a n U(X)) -C x,; hence ( I n U ( X ) = a n x I arid CI n x1 is a set.
80
CHAP. I SEC. 41
I445
THEORY OF SEMISETS
d ) Model-classes 1441. DEFINITION (TSS-,). A class X is closed (Clos(X)) if, for all x , y ~ X ,F i ( x , y ) ~ X( i = 1, ..., 7), i.e. Clos(X) _= ( V x , y e X ) ( { x , y } , ~ ( x ) x, - Y , D (x), x Y , Cnv (x), Cnv, (x) EX). Note that TSS,, t- Clos (V).
r
1442. DEFINITION (TSS-,). A class X is a model-class (Mcl (X)) if it is = . Comp ( X )&
a complete, closed and almost universal class, i.e. Mcl (X) & Clos ( X )& AUncl ( X ) .
1443. LEMMA (TSS)-,. (a) V is a model-class. (b) Every model-class is V-like. Proof. (a) is obvious. (b) Suppose X is a model-class; then X is nonempty because it is an almost universal class. Further, x, y E X 5 (x, y } E X because X is closed (+) and complete (t). It follows by completeness that x n X = x for EX; hence x, y
EX
&x
+y.
.+ x
n X =+ Y n X
.
1444. THEOREM (TSS-,). Every model-class is a real class. Proof. Let P be a model-class and let x be a set. Then x n P is a semiset, (x n P ) s P ; since P is almost universal class there is an y E P such that x n P E y . It follows, using the completeness of P , that x n P = x n P n n y = x n y and so x n P is a set. 1445. METATHEOREM. Gt (M) with the specification Mcl (M) determines a model of TSS-, in TSS-,. Dir (dTSS/dGt(M)) is a faithful model of TSS- ,in TSS- m, Mcl (M), dTSS/dGt (M); the formulas M ( X ) , Real ( X ) are absolute from below and the formula Sm (X) is absolute. By Metatheorems 1317, 1318, 1416 Gt(M) with Mcl(M) determines a model of TC, C2 in TSS-,, and Sm(X) is absolute. Hence we must prove (A2)* to (A7)* in TSS-,, Mcl (M), dTSS/dGt (M). Denote the last theory by TSS,,, (in the context of the present metatheorem only). We know (Metatheorem 1322) that godelian operations are absolute, i.e. TSS,,, tF?(X*, Y*) = F, (X*, Y*). In particular, TSS,,, I- F: (x*, y*) = = Fi (x*, y*) hence it remains to show that TSS,,, I- M* (F, (x*, y*)). But sets in the sense of Gt(M) are elements of M; since M is closed it follows that TSS,,, k Fi (x*, y * ) E M and so TSS,,, t- M* (F: (x*, y*)). This proves (A2)* to (A7)*. 81
1446
THEORY OF SEMJSETS
[CHAP. I SEC. 4
Specifying by M = V we obtain the result concerning faithfulness. The fact that M(X) is absolute from below is trivial. Finally we prove TSSM,, k t Real* ( X * ) -+ Real ( X * ) . We proceed in TSSM,,. Suppose Real* ( X * ) and let x be a set; since x n X * E M there exists y E M such that x n X * E c y . Thus x n X * = x n X * n y ; X * is a real class in the sense of Gt (M) and y is a set in the sense of Gt (M); hence there is a u E M such that u = = X * n y and we obtain x n X * = x n u which is a set. Consequently, X * is a real class.
e) The theory of sets
1446. The theory of sets T S is the extension of TC, C1, C2 by the axiom
(C3)
(VO)
(3x) ( 0 = x)
.
Thus (C3) says that every semiset is a set. We denote by TS-, T S with the axiom (Cl) removed.
the theory
1447. METATHEOREM. (TS-,) is stronger than TSS-,. Demonstration. It suffices to verify that (A2) to (A7) are provable in TS-,. But in TC, C2 we proved that E (x), x - y , ..., Cnv, (x) are semisets; hence by (C3) these classes are sets. Thus we have the following result in TS-,: 1448. THEOREM (TS-,). For any x, u x and P (x) are sets; if F is a function then F”x is a set for any x. 1449. Remark. Our theory TC is equivalent to the theory with Godel’s axiom system A, B. The formulas M (U(x)), M (P (x)), Un ( F ) +. M (F”x) are in fact Godel’s axioms C2, C3 and C4; we denote them by GC2 to GC4. Godel’s C1 has the same meaning as ours. We have seen that GC2 to GC4 are provable in TS-,. We now show that C2 and C3 are provable in TC, GC2, GC3, GC4. For any X we define a relation I by (x, y ) E I = y E X& x = y ; Z is a function and if X is a semiset then X 5 z for some z ; since X = I”z, it foflows that X is a set. Thus we have proved C3. Let R be an exact functor; we prove that D ( R ) is a set iff W ( R ) i s a set. This will prove (C2), since by (C3), Sm(X) 5 M(X). R is nowhere constant and (Vx)(M(Ext,(x)). We define a function F by (x, y ) E F E x E D ( R ) & y = ExtR(x). F is 82
CHAP. I SEC. 41
THEORY OF SEMISETS
14S4
one-to-one, D ( F ) = D ( R ) and U(W ( F ) ) = W ( R ) . If D ( R ) is a set then D ( F ) is a set, W ( F ) = F”(D ( F ) ) is a set and hence W ( R ) = U(W ( F ) ) is a set. Conversely, if W ( R ) is a set then P (W (R)),W ( F ) c P (W ( R ) ) and D ( F ) = (Cnv (F))” (W ( F ) ) are sets; hence D ( R ) is a set. Thus we have shown that TS is equivalent to Godel’s axiom system A, B, C.
f) Thefirst axiom of regularity; the real F-definition 1450. We shall now extend TSS (or TS) by the addition of the following axiom (the first axiom of regularity):
The theoiy TSS,D l is denoted by TSS’;similarly, TS, D1 is denoted by TS’. TSSL, and TSL, denote TSS‘ and TS’ with the axiom of infinity removed. 1451. THEOREM (TSSL,).
The domain of a real class is also a real class. Prooj. If X is a real class and if x is a set then by DL there is a set y such that D ( X ) n x = D ( X n y ) , hence D ( X ) n x is a set by (A4). 1452. THEOREM (TSS-,). The axiom (Dl) is equivalent to the following assertion: For every relation X whose domain is a semiset there is a subrelation Y E X which is a semiset and has the same domain. Proof. Suppose (Dl). If X is a relation and D ( X ) C x then D ( X ) n x = = D(X) and by (Dl) there is a y such that D(X) = D(X n y). Hence X n y is the required semiset. Conversely, assume the assertion of the theorem. Given X and x, consider X’ = X x. The domain of X’ is a semiset and so there is a semiset Y such that Y E: X’ and D ( Y ) = D (X’). If Y c y then D ( X ) n x = D (X’) = D ( X n ( y x)). We have proved (Dl).
r r
1453. METATHEOREM (Comprehension scheme). Every normal formula without constants is fully comprehensive in TSSL,. This is equivalent to the following assertion which may be proved by induction using Theorems 1426, 1451. 1454. METATHEOREM: For any godelian term T (XI, 0 ) without constants we have TSSL, k (Real (XI)& 0 ) 4 Real (T(X,. 0 ) ) . 83
I455
[CHAP. I SEC. 4
THEORY OF SEMISETS
1455. METADEFINITION. The pair of formulas (5)
(VX) [(3X*)(X* = X )
(6)
(VX*, Y*) ( X * E* Y*
= Real ( X ) ] & sec, = X* E
Y * ) & sec
is called the real F-deJnition and denoted by seal. Thus classes in the sense of s e a l are real classes. 1456. METATHEOREM. The F-definition s e a l determines a model of TSL, in TSSL, and of TS‘ in TSS’. The model rDir (dTS/ZReaI) is a faithful model of T S I , in TSSL,, dTS/d%eaI and of T S ’ in TSS’, dTS/d%eaI.*) All normal TS-formulas without constants are absolute; in particular, if 9 is a closed set formula then
TSSL,,
dTS/8%caI t- q
= cp* .
Denionstration. By (6), the formula X E Y is absolute. Since TSSY, It (V.Y) Real (x), (FL) holds in the model. Moreover, denoting the theory
TSS’,, aTS/d%eaI by TSS,,,, (in the context of the present Metatheorem) we have TSS,,,, t- M(X) 3 M*(X), i.e. sets of the model are exactly all sets. This implies immediately that all NF’s without constants are absolute. Axiom F2 holds in seal, since TSS,,,, t (Vx*) (x* E* X * = x* E* Y*) = = (Vx)(x E X * = x E Y * ) . By absoluteness of NF’s, (Al) holds in the model and if C1 is assumed in the theory we are dealing with then it holds in the model. The axioms of group B havc thc form ( V X , 0 ) ( 3 2 ) v(X,0 , Z) where cp is normal in TE. I n TSS,,,,, we have for arbitraryX*, 0 :
Real ( Fi ( X * , o)), i.e. (3Z*) ( Z * = Fi (X*, 0 ) ) ;by the absoluteness of NF’s we obtain(32*) cp* (X*, 0 , Z*). This proves (B1)* to (B7)*. Axiom C2 has the form (VX) cp(X) where q is normal in TC, hence (C2) is absolute from above and wc have TSS,,,, I- (C2)*. The same can be said of (Dt). Finally, since semisets of the model are real subclasses of sets, they are sets and therefore sets in the sense of the model. We havc TSS,,,, t (C3)*. s e a l is a model of TS’_ , in TSS,,,,.
*) This can be also formulated as follows: The model %eat makes TSS’ to a conservative extension of TS’ by identifying class variables in TS’ with variables for real classes in TSS’.
84
CHAP. I SEC. 41
THEORY OF SEMISETS
1459
To demonstrate the faithfulness consider the diagram:
The composition of these two arrows is evidently equivalent to the identity model of the first theory in the third one and therefore, by 1233(2), Seal is faithful. 1457. METACOROLLARY. (The equiprouability principle.) If cp isa SF then r is a sequence of closed SF’s,
TSSL t cp iff TSL I- q ;more generally, if then TSSL,, t- cp iff TSL,, t cp.
r
r
Demonstration. In Diagram 1458 the arrow 1 is a faithful model; since it is direct it is essentially faithful; thus 2 is a faithful model. Since NF’s and therefore also SF’s are absolute, we have TSS,,,, t II/ = for in the terminal theory of 2 can be replaced every SF.)I This means that rWea‘ by r. By faithfulness, TSL, r t cp iff TSS,,,,, r I- qRea’; however TSS,,,,, r t cpwear means exactly the same as TSS,,,,, I‘ t cp which is equivalent to TSSL,, r t q because the former theory is a conservative extension of the latter.
1459. METATHEOREM. % (R)with the specification sExtl (R)& Prg (R)& & Reg (R)& AUn (R)(cf. Metatheorem 1416) determines a model of TC, C2, Dl. in TC, C2, D1. Demonstration. We proceed in TC, C2, D1, Mrel (R), JTC/8% (R)(where 85
I460
[CHAP. I SEC. 4
THEORY OF SEMISETS
Mrel (R)is an abbrcviation for the specification above). Let X * be such that Rel* ( X * ) and Sm*(D*(X*)). We denote by Z the decoded relation Dec ( X * ) ; Z is a relation whose domain is a semiset and hence there exists S E Z such that Sm (S) and D(S) = D ( Z ) . If we define U E Y *= = (3s,y ) ( ( s , j~)~S&(Ext,(x),Ext,(y))* = Ext,(u)),thenS = Dec(Y*), Y* E * X* and D* (Y*)= D* (X*). It remains only to verify that Sm* (Y*); to do this it suffices to show that Y* is a semiset. Wc define a relation N as follows: ( I ( ,V ) E
If
= (3x, r) (U = (Ext,
= (x, 1) E S & Ext,
(u) =
(x), E d , (Y))*) ;
H is onc-to-oile and hence an exact functor; sincc D ( H ) = Y*, Y* is a semiset.
=
S and W ( H ) =
g ) The notion of support; the support F-definition
To end this Section we generalize the construction of the real model. The notion of a support which we shall define here will play an extremely important role in Chapter IV et seq.
1460. DEFINITION. A class X is dependent on a class Z (Dep (X, Z)) if there is a set I‘ such that X = r”Z. 7462. LEMMA (TSS-,). (1)
Z =/= 0 -+
(Vx) Dep (x, Z ) ,
(2)
Dep ( X , Y )& Dep (Y, Z ) . 3 Dcp ( X , 2) ,
(3)
Dep ( X , Z ) -+ Sm (X) ,
(4)
Dep(X, Z)& Dcp(Y, Z ) .
+
Dep(X u Y, Z ) ,
(5)
DeP (0, ).
(6)
M ( Z ) & Dcp ( X , Z ) . 3 M(X)
7
.
Proof. (1) We choose sonic 1.E Z and let r = x x ( y } ; then x = r”Z. ( 2 ) If X = r”Y and Y = s”Z then wc define (x, y ) E t = (3u) ((x, u } E E r Rr ( 2 1 , I.) E s) so that X = t”Z. ( 3 ) If X = r”Z then X G W (r). (4) If X = r”Z and Y = s”Z then X u Y = ( r u s)” Z . (5) If 0 c a and r = 86
CHAP. I SEC. 41
r
1464
THEORY OF SEMISETS
r
= I a (where I = {(x, x); x EV}) then cr = r”cr. (6) If 2 r ” ~= r’’z = w ( r z) is a set.
=z
then
1462. DEFINITION (TSS-,). Supp ( 2 ) = . 2 4 0Lk (Vo, +
e) (Dep (a,2)& Dep ( Q , 2 ) . +
Dep (0 - PI z )),
i.e. 2 is a support if 2 is a non-empty class such that cr - Q is dependent on 2 whenever both v and Q are dependent on 2. Remark. There exists at least one support; e.g. (0) is a support and X is dependent on (0) iff X is a set. 1463. METADEFINITION. The pair of formulas
(8)
(vx*,Y*) (x*E* Y* = X* E Y*)& sec
is called the support F-definition and denoted by Gupp (Z). Thus classes in the sense of Gupp (2) are classes which intersect each set in some semiset dependent on Z. 1464. METATHEOREM. The F-definition Gupp (Z) with the specification Supp (Z) determines a model of TSSL, in TSSL, (and of TSS’ in TSS‘). Demonstration. We proceed in (TSSL , Supp (Z), dTSS/dGupy (Z)). Consider the support Z; the classes of the model are all classes which satisfy the condition (Vx) Dep ( X n x, Z). By Lemma 1461, every set is a class of the model. It suffices to show that Fi ( X , Y ) , i = 2, ..., 7, is a class of the model whenever X and Yare classes of the model. In this case we may follow a procedure analogous to that used in proving that the real model is a model. It suffices to prove Dep ( X , Z) & Dep (Y, Z) . + Dep (F i ( X , Y ) , Z) ( i = 2, . .., 7) ; (cf. 1304 and Axiom (Dl)). The case i = 3 follows from the definition of support. In case i = 2 , 4 , 6 , 7 we show that F,(X) is dependent on X . Since X is dependent on Z, X is a semiset and hence X c u for some u; we have E ( X ) = PYX where r2 = I E ( u ) , D ( X ) = rTX where ( x , ( y , x)) E rs E ( y , x) E u and Cnv CX) = rk’;u where ( ( y , x ) , (x, y ) ) E E r6 = (x, y ) E u ; similarly for i = 7. Finally in case i = 5, let X and Y be
r
87
1465
THEORY OF SEMISETS
[CHAP. I SEC. 4
r
two classes dependent on Z; we prove that X Y is dependent. Since X and Y are semisets, X E u and Y E u for some u and v. If we let z=W(u)thenXrY=Xn(z xY)andifwedefine((y,x),x)~u~ = y E z & x E v then we have z x Y = a”Y. Hence z x Y = u”Y is dependent, and X Y = X n ( z x Y ) = X - (X - (z x Y)) is also dependent. This completes the proof.
.
r
1465. Remark. a i r (aTSS/aSupp (Z)) as a model in TSSL,, Z = {0}, aTSS/dGupp (Z) is equivalent to air (aTSS/a%eaI) as a model in this theory. (Exercise.) 1466. LEMMA (TSS’_,).-A class Z =# 0 is a support iff it has the following property: (Yo) (Dep (0, 2 ) + ( 3 a ) (o c u & Dep ( a
- o,2)).
Proof. Clearly every support has this property. Conversely, suppose that a class Z has the property. If crl, o2 are two semisets both dependent on Z then there are a,, a , such that crl E a,, cr2 c a, and Dep (al - ol,2) and Dep(a, - 02,2). Set a3 = a , u a,; we have Dep (a3 - ol,2) and by 1461 we obtain Dep ( ( a 3 - ol) LJ cr2, 2). Put ug = (a3 - ol)u (T, and observe that a3 2 0 3 .We show Dep (a3 - 03,2).Indeed, there is an u4 2 2 cr3 such that Dep (a4 - cr3, Z). Furthermore, u3 - o3 = ((a4 - 03)u u ( a 3 - a4)) - (a4 - u 3 ) because u3 2 0 3 . But a3 - o3 = u3 - ((u3 - ol) u 0 2 ) = crl n (a3 - 02)= crl - cr2. This proves that Z is a support. 1467. For the sake of those readers who prefer axiom-schemes, we shall now collect the axioms of TSS and TS into the following system:
TSS
TS
(vii) Restricted I (viii) Axiom “Semisets Comprehension are sets” (C3) Scheme (A2 - A7)
All models to be considered in this book are determined by one of the F-definitions 112 (R), G;t(M), Gupp (Z) with suitable specification or by the F-definition Seal or are compositions of such models. 88
CHAP. I1 SEC. 11
ORDINAL NUMBERS
2101
CHAPTER II
In Chapter I we introduced the fundamental Godelian theory of semisets
TSS and the fundamental Godelian theory of sets TS. Wc also considered
some general models of these theories. Since we have not proved very many theorems in these theories, we have only been able to establish general properties of the models. In this Chapter we shall therefore define a numbcr of set-theoretical concepts and prove a number of theorems in TSS; the models will be dealt with in later chapters. We investigate ordinal numbers and well-orderings in Sect. 1, and equivalence and subvalence of sets in Sect. 2. Sect. 3 is devoted to a weak form of the axiom of choice and its consequences; in particular we consider the consequences for powers of sets. Sect. 4 is devoted to Boolean algebras and Sect. 5 to ordered sets.
SECTION 1
Ordinal numbers Throughout this and following sections we shall work within the theory of semisets TSS or one of its extensions. We first consider the theory of ordinal numbers due to von Neumann. Let us recall here the following remark of Godel:
2101. “The ordinal ct will be the class of all ordinals less than a. For instance, 0 = the null (empty) set, 1 = (O}, 2 = (0, 11, o = the set of all 89
2102
[CHAP. I1 SEC. 1
ORDINAL NUMBERS
integers, etc. In this way, the class of ordinals will be well ordered by the relation, so that a < p corresponds to a E p. Any ordinal will itself be well ordered by the erelation since an ordind i s a class of ordinals. Moreover, any element of an ordinal must be identical with the segment generated by itself, since this segment is the class of all smaller ordinals.”
2102. Reniarli. If R is a relation then x R y means the same as (x, y ) E R . 2103. DEFINITION (TSS). a) Arelation R is an ordering (Ordg(R)) if it has the following properties: (1) (Vx, y, z ) ( x R y & y R z
.
--f
xRz)
(2) (Vx) (1x R x )
(transitivity), (irreflexivity).
b) A relation R is a linear ordering (LOrdg ( R ) ) , if it has the following properties:
( I ) R is an ordering, (2) (VX,
~3
E
C ( R ) ) ( x R y v ~ R vx x
= y)
(trichofom~?)
2104. DEFINII‘ION (TSS). a) X is ordered (linearly ordered) by a relation R if the relation R n X 2 is an ordering (a linear ordering) and X c C ( R ) . b) A one-to-onc mapping F of X onto Y is called an isomorphism between X and Y w.r.t. the relations R and S if (Vx, y
E
X ) (xRy
= (F’x)S(F’y)).
2105. D E r l N l I ION (TSS). X is a n ordinal (Ord ( X ) ) if it has the following
properties:
a) x is complete, b) (VX, J E X ) (X E J’ v y E x v x = I)), c) (Vu c X ) ( u =I= 0 -,( 3 z E u ) ( z n u = 0)). 2106. Remurk. The predicate Ord ( X ) is obviously normal: it is easy to find a PUP-formula equivalent to Ord ( X ) .
2107. DEFINITION (TSS). x E On = Ord (x). On is the class of all ordinal numbers; ordinal numbers are the ordinals which are sets; we use lower-case letters from thc beginning of the Greek alphabet to denote ordinal numbers. 2108. LI:MMA (TSS).(a) Ord (0). 90
CHAP. I1 SEC. I ]
2112
ORDINAL NUMBERS
(b) If Ord ( X ) , Ord (Y), Ord (2) then
x $6 x ;
l ( X E Y& Y E x) ; l ( X E Y& Y E Z &
zEX) .
(c) If Ord ( X ) and y E X then Ord ( y ) . (d) If X , Yare ordinals, then X n Y is an ordinal. Proof. (a) is obvious. (b) If X is an ordinal and X E X then (X} c X ; putting u = {X} we obtain a contradiction with (c) in the definition of an ordinal. Similarly put u = ( X , Y}, u = {X, Z } in the remaining cases. (c) Suppose Ord (X) and y E X. Then y is complete. For suppose u E u E y , then, by (b) in the definition of an ordinal, u E y v u = y v y E u ; the last two cases contradict (b) of the present lemma and so u E y. If u, u E y then u, D E X since X is complete; hence u E u v u = D v u E u. Finally, if 0 ==! u -C y then u E X and therefore by (c) in the definition there is a z E u such that z n ti = 0. (d) is obvious. ,1109. LEMMA (TSS). Let Y be a non-empty real subclass of On. Then there exists z E Y such that z n Y = 0. Proof. Choose y E Yand set u = y n Y . If u = 0 we are finished If u 0 then u E y and thus there exists z E u such that z n ti = 0, i.e. z n y n Y = = 0. Since z E y we have z E y , so L n Y = 0.
+
2110. LEMMA (TSS). If X and Y are real ordinals and if Y is a proper subclass of X then Y E X. Proof. Since X - Y is non-empty there exists z E X - Y such that z n n ( X - Y) = 0; hence z E Y. It suffices to prove z = Y. Suppose z c Y. Then there exists z1 E Y - z such that z1 n (Y - z ) = 0; hence z1 c z . If rl = z then z E Y , a contradiction; thus z1 c z. Since z,z1 E X and X i s an ordinal we have either z1 E z or z E z , . But z E z1 implies z E zl, which contradicts z1 c z,and so z1 E z. Since z1 E Y we have z1 $6 Y - z , a contradict ion. 2111. LEMMA (TSS). If X and Y are different real ordinals then either X E Y o r YEX. Proof. Set Z = X n Y. Z is a real ordinal and 2 E X , Z E Y. If Z = X then X c Y; hence X c Y and so X E Y by Lemma 2110. Similarly if 2 = Y then Y E X . The remaining possibility Z c X & 2 c Y can be excluded; for by Lemma 21.10 it implies Z E X& 2 E Y, i.e. 2 E X n Y = 2 which contradicts Lemma 2108. 2112. LEMMA (TSS). Let Y be a non-empty real subclass of On. Then there is exactly one element z of Ysuch that z n Y = 0. 91
2113
ORDINAL NUMBERS
[CHAP. I1 SEC. I
Prooj”. By 2109 there is at least one such z . Now we prove the uniqueness. Let zl, z2 EY, z1 z 2 . z1 n Y = 0, z2 n Y = 0. Then by the preceding Lemma e.g. z1 E z2 so that z1 E z 2 n Y, a contradiction. Thus the lemma is proved.
+
2113. DEFINITION (TSS). Let Y be a non-empty real subclass of On. An element z of Y is called minimal ( z = Min (Y)) if z n Y = 0. 2114. LEMMA (TSS). The class On is linearly ordered by the erelation E. Proof. The irreflexivify of E on On (a4 a) follows from Lemma 2t08. For the transitivity suppose a E p and p E y ; then fl E y and hence a E y. The trichotomy follows from Lemma 2111. 211 5. LEMMA (TSS). (a) On is a n ordinal; it is a real class and is not a set (consequently On is not a semiset).
(b) If X is an ordinal and X is not a semiset then X
=
On.
Proof. (a) On is an ordinal by Lemma 2108(c), Lemma 2LlL and Lemma 2109. On is a real class by Metatheorem 1438. (For Ord ( X ) is a PUPformula.) On is not a set by Lemma 2108(b). (b) Let X be a n ordinal and suppose X is not a semiset. Then X E On and, for every ordinal number a, X $ a, i.e., for any a there is a p such that a E p E X . By the completeness of X every a is an element of X and so X = = On. 2116. Reinark We have seen that there are ordinals which are sets (the ordinal numbers) and that there is just one ordinal (the class On) which is not a semiset. Nothing is asserted about ordinals which are semisets but not sets; it can be shown by the ultraproduct model that the existence of such ordinals is consistent but we shall not be interested in them in the present book. 21 17. LEMMA (TSS). Let a be a n ordinal number. Then
(a) cx u (a} is a n ordinal number, (b) there is no
such that z
E
p& p E ( a u ( a } ) .
Proof. (a) Obvious. (b) Let z E p, p E cx u ( a } . Then either j? E a or p E (a}, i.e. a E fi it follows in either case that a E a - a contradiction. 92
j? = a. Since
CHAP. I1
SEC. I f
2124
ORDINAL NUMBERS
2118. DEFINITION (TSS). a
+ 1 = CL u (a}. (the successor of a).
2119. METADEFINITION. We define in TSS: l = O + l ,
2 = t + l ,
3=2+1
etc.
(TSS). If X is a class of ordinal numbers then its sum U ( X ) 2120. LEMMA is an ordinal. (Obvious.)
2121. DEFINITION (TSS).For X c On define Sup ( X ) = U ( X ) (the supremum of X )
If Sup (X) E X , we write Max ( X ) instead of Sup ( X ) (the maximum of X). 2122. LEMMA (TSS). The supremum of a set of ordinal numbers is an ordinal number. 2123. THEOREM (TSS). (The principle of the Transfinite Induction.) Let X be a real class satisfying the following conditions: (a) 0 EX, (b) (Va) (a E X
-+
(c) ( v x ) (x E x
(a
-+
+ 1) EX),
s u p (x) EX).
Then X contains all ordinal numbers. Proof. Suppose that not all ordinal numbers are in X. Then On - X is a non-void real class of ordinal numbers; let z = Min (On - X ) . We will prove z E X , a contradiction. Obviously z E On n X. Set z1 = Sup (z). By (c) z1 E X . If z = z1 we are finished. If z E z1 then z G z1 and z1 2 z ; hence z E z , a contradiction. If z1 E z set z 2 = z1 1. By (b) z 2 E X . We claim that z 2 = z. Otherwise we would have z 2 E z. Then z1 E z2 E z and hence z1 E Sup ( z ) = zl, a contradiction.
+
2124. THEOREM (TSS). (Construction by Transfinite Recursion.) If G is a real function (i.e., a function and a real class) then there i s exactly one real function F on the class On with the following property: (va) (F’a = G‘(F”a)) .
Proof. Lct K be the class of all set functionsf such that D (f)is an ordinal number andf’a = G’(f”a) for each U E D(f). It is easy to check that the 93
2125
ORDINAL NUMBERS
[CHAP. 11 SEC. 1
condition defining K can be written as a PUP-formula, hence K is a real class. Now we prove that, for an] f , g E K , a E D ( f ) n D ( 9 ) implies f ’ a = = g’a. Assume f , g E K and f ’ a += g’a for some a E D ( f ) n D (9). Let Y = { a E D ( f ) n D ( g ) ; f ’ a =k g’a}. Y is real by Metatheorem 1438. Hence, there is a least a E D ( f ) n D (9) with f ’ a =Ig’a. = Then f ” a = g”a and so f ’ a = G ( f ” a ) = G’(g”a) = g’a, a contradiction. Set F = u ( K ) ; obviously, F is a function and F’a = G’F”a whenever O L ED (F). We prove that F is not a set. If F is a set then D ( F ) is a set by Axiom (A4) and therefore D(F) is an ordinal number. Set f = F, y = D ( j ) and g = f u ((G’(f”y), y ) ) . Then f , g E K , D (9) = y + 1, hence y E D ( j ) = y which contradicts 2108. Hence F is not a set and consequently K is not a set by 1427. We now prove that D(F) is not a semiset. S e t X = { ( f , a ) ; f ~ K & & D(f) = a). Obviously X is a one-to-one mapping, D(X) = D(F) and W ( X ) = K . If D ( F ) is a semiset then K is a semiset and therefore a set since K is real. But we proved that K isnot a set and so D (F) is not a semiset. Furthermore, F is not a semiset since Sm ( F ) -+ Sm (D (F)). It follows by 1440 that F is real and by 2115 that D (F) = On. The proof of uniqueness is straightforward. Remark. I n the course of the above proof we gave the definition of F in terms of G. Hence we are justified stating the following matemathical version of Construction by Transfinite Recursion:
2125. METATHEOREM. There is a godelian term F ( G ) with one class variable G such that TSS
t Real (G) & Un (G) =
. + . Real (F (G)) & Un (F ( G ) )& D (F ( G ) )=
On& (VLY) ( F (F)’ a = G’(F (G)” a)) .
Demonstration. The formula Un ( f ) & D (f)E On & (Va) ( f ’ a = G’(f”a)) is normal in TSS, i.e. there is a N F q(f,G) equivalent to the former one in TSS. Hence, by Corollary 1122 there is a godelian term K (G) such that T S S I- K (G) = ( f ; Un (f)& D ( f ) E On& (Va E D ( f ) ) (f’a= G’(f”a))}. The formula x E U(K (G)) is equivalent to (If) (q(f,G ) & x ~ f in) T S S and is therefore normal; hence there is a godelian term F (G) such that TSS t F (G) = U(K (G)). In the same way as Theorem 2124 was proved we can prove in TSS: If G is a real function then F(G) is a real function, D ( F (G)) = On and F (G)’ a = G’(F (G)” a) for every a.
2126. DEFINITION (TSS). An ordinal number a is called isolated if + 1 = a.An ordinal number which is not
a = 0 or if there is a p such that
94
CHAP. I1 SEC. 11
3129
ORDINAL NUMBERS
isolated is called a Zimif number. The classes of all isolated and all Iimit numbers are denoted by On, and On,, respectively. 2127. THEOREM (TSS). Both On, and Onll are real classes and proper classes, i.e. they are not sets. Proof. For any x we have x E On, iff Ord (x)& (x = 0 v (3y E x) (x = = y u ( y } ) ) . The last formula is evidently equivalent to a PUP-formula, hence, by Metatheorem 1438 On, is real. Since On,, = On - On,, OnlI is also real. Suppose that the isolated numbers form a set and let y = Sup(OnI). Then y + 1 is again isolated, a contradiction. Suppose that the limit numbers form a set and let y = Sup (On,,). Then for each nonempty set x of ordinals, x E On - y implies Sup (x) E x and consequently x has a maximal element. Let a be a set satisfying the axiom C l of infinity, i.e. suppose that 0 E a and that (x} E a whenever x E a. Let X be the class of all functions f such that D (f)= B - y for some fi and such that f’y = 0 and f ’(a 1) = { f ’ a } for each a such that a + 1 E D ( j ) . It is easy to write the condition defining X as a PUP-formula; hence X is real. We first prove that every f E X is one-one and has its values in a. Let a be the least element of D (f)such that f ’a = f ’p for some P E a.There exist a, and Po such that a = a, + 1 and j? = Po 1. Then f ’a, = f’P,, a contradiction. Hence f is one-one. Suppose that W (f) is not included in a and let a be the least element of D (f)which is not in a. Then f ’ a = {f ’ao}, a = a, + 1, f ’ a , E a and hence f ’ a E a, a contradiction. We can prove in a similar manner that if D(f) c D (9) then f E 9, whenver f and g are in X. Set F = U(X). We prove that F is not a set. Suppose that F is a set and let a, be the maximal element of the domain of F. Set a = a, + 1, g = = F u (({F’a,], a>}. Then we have g E X and D (9) = a 1, a contradiction. Hence F is not a set and consequently X is not a set. Since X is real X is not a semiset. Analogously to the proof of 2124 we prove that neither F nor D (F) are semisets. But then F is real by 1440. F is one-to-one and its range is a set. Hence F is itself a set, contradiction.
+
+
+
2128. DEFINITION (TSS). The least limit number is denoted by o (or
0,).The elements of
m, n etc.
o are natural numbers and are denoted by the variables
2129. THEOREM (TSS). (The induction principle.)
If a real class X contains 0 and with each natural number n also its successor n + 1, then X contains all natural numbers. 95
2130
[CHAP. I1 SEC. 1
ORDINAL NUMBERS
Proof. Suppose that o - X 9 0 and let n be the least natural number not in X . The number n is a successor of a number m E X and hence is itself in X , a contradiction. The following theorem is a variant of Theorem 2124: 2130. THEOREM (TSS). If GI, G , are real mappings and a is a set then there is exactly one real mapping F such that
t 1)
F’O = a ,
(2)
F’(a
(3)
F’M = G;(F”a)
+ 1) = G;(F‘a), for M limit
,
The proof is analogous to the proof of Theorem 2124. We define K as the class of all functions f such that D ( f ) E On and f fulfils (L), (2), (3) for any a for which these equations make sense. K is real (because it is defined by a PUP-formula), u ( K ) is real by Lemma 1440 and F = u ( K )fulfils (I) -(3). 2131. COROLLARY (TSS). (Construction by Recursion.) Let a be a set and let G be a real mapping. Then there is exactly one mappingfsuch that D (f)= o , f ’ O = a andf’(n + 1) = G’(f’n)for each n. (Set G I = G , and let G , be an arbitrary real mapping. (Take f = F o where F fulfils (1)-(3). f is a set because F is a real mapping.) The reader may formulate Metatheorems analogous to Metatheorem 2125 which express the fact that we may “describe” F in terms GI, G 2 ,a (fin terms of G and a) by means of godelian operations. In particular define:
r
2132. DEFINITION (TSS).
Roughly speaking Unv (x) is the “infinite union” x u u(x) u uU(x) u ... It follows from the definition of Unv(x) that T S S t (Unv(x) is a set). (Unv(x) is the union of the domain of values of some set function f x . ) The following definition is a generalization of Definition 2132: 2133. DEFINITION (TSS). If R is a real regular relation then
Unv, (x) = u ( W ( f ) ) where f ’ 0 96
=
EXtR (x), f ’ ( n
+ 1.) = R”(f’n) .
CHAP. I1 SEC. 11
ORDINAL NUMBERS
2137
The definition is justified by the following easy 2134. LEMMA(TSS). If R is a real regular relation then, for every x, ExtR (x) is a set. Proof. Clearly Ext, ( x ) = W (R n (V x {x})); R n (V x {x}) is real and it is a semiset (indeed, ExtR(x) is a semiset and if Ext, ( x ) c a then R n (V x {XI) c a x { x } ) , hence R n (V x {x}) is a set and therefore W ( R n (V x {x})) is a set. Note that T S S k Unv (x) = Unv, (x). 2135. Remark. The first place in this Section where axiom C1. was used is the proof of 2127. Hence if we denote by CLbiSthe assumption On,, 0 ("limit numbers exist") and by TSSbi"the theory T S S with C1 replaced by CIbis,then we can define o in TSSbi"and prove Theorem 2131. Hence in TSSbiSdefine f'0 = 0, f ' ( n + 1) = { f ' n } , a = W (f);it follows in TSSbis that 0 E a and that ( y } E a for any y E a . This means that C1 is provable in TSSbiS.Consequently, C1 can be equivalently replaced by Ctbi" in the axioms of TSS.
+
In the rest of this Section, we shall consider the theory TSS' (i.e. TSS + + Dl). Recall Metatheorem 1453 which enables to prove many classes to
be real. We shall now investigate well orderings.
2136. DEFINITION (TSS'). A relation R is a well-ordering (WOrdg(R)) if it has the following properties:
(0) R is a real class, (L) R is a linear ordering, (2) each nonempty real subclass of C ( R ) has a least element, i.e. (VX) [0
+X
c C (R) & Real ( X ) . -+
(3x E X ) [ X n Ext, (x) = 0 ] ] .
2137. DEFINITION (TSS'). A relation R is called a regular well-ordering (RWOrdg (R)) if it has the following properties: (0) (1) (2) (3)
R is a real class, R is regular, R is a linear ordering, each nonempty subset of C ( R ) has a least element.
A real class P is well-ordered (regularly well-ordered) by a real relation R if P G C (R) and R n P z is a well-ordering (a regular well-ordering). 97
2138
ORDINAL NUMBERS
[CHAP. I1 SEC 1
2138. LEMMA (TSS’). A real relation R is a regular well ordering if and only if it is regular and a well ordering. Proof. A real regular relation which is a well ordering is obviously a regular well ordering; conversely, let R be a regular well ordering. Let X be a nonempty real subclass of its field. To find the least element of X , take an arbitrary z E X and set u = X n Ext, ( 2 ) . If u = 0 we are finished; otherwise, take the least element x of u. We assert that X n Ext, ( x ) = 0. Assume the contrary and take y E X n Ext, ( x ) . We have y R x , x R z whence y R z . Since y E u we have u n Ext, ( x ) =k 0, which contradicts to the minimalify of x in u and our assertion is proved.
2139. Remark. The formula RWOrdg(R) is normal in TSS’. 2140. THEOREM (TSS‘). The class of all ordinal numbers is regularly well ordered by E (the erelation). Proof. The theorem is an immediate consequence of preceding lemmas. 2141. THEOREM (TSS‘). Let P , R be real and suppose that P is regularly well ordered by R . Then there exists a unique real ordinal X and a unique real isomorphism F between P and X w.r.t. R and E. Proof. Let P be regularly well ordered by R and suppose, for simplicity, that 0 4 P. First we shall prove the existence of an ordinal X and a n isomorphism F. Define the function G as follows: G’x = y if y is the least element of P - x, and G’x = 0 if P - x = 0. By Theorem 2124 there is a real function F such that F’a = G’(F”a) for each a. Hence F’a is the least element of P - (F’a). Set X = { a ; F’a + 01. Then F X is the required isomorphism. To prove the uniqueness suppose first that there are two real isomorpbisms F , and F , between P and an ordinal X . Let a be the least ordinal in X such that F ; a Fia. Then F;P = FiP whenever P E E . By trichotomy we have e.g. ( F ; a , F i a ) E R . Then there exists y E X such that F;a = F i y . Obviously y E a and hence F ; y = F i y . It follows that F;a = F;y, a contradiction. Now suppose that there is a real isomorphism F , between P and XI and a real isomorphism F , between P and X , where X , and X , are different ordinals. We have e.g. X i EX^. Define the function G on X , as f(>llcws: G’a = B = F;/I = F i a . G is obviously an isomorphism between X , and X , (w.r.t. E). Since G’X, EX^, we may denote by y the least elen-ent cf X , such that G’y E y. We have G’(G’7)E G’y and so by the minirrality of y, either y E G’y or y = G’y, contradicting G’y E y.
r
=+
98
CHAP. I1 SEC. 11
2145
ORDINAL NUMBERS
We conclude our treatment of ordinal numbers with some metarnathematical results concerning the possibility of definitions by induction in TSS’ even in the case where the arguments are proper classes. Although we cannot deal with ordinal functions which assume proper classes as values, we can employ relations such that the extensions of ordinal numbers (e.g.) are proper classes with certain properties. 2242. METADEFINITION. Let T be a theory stronger than TC. Let X be a variable. A term T(X) is local in T if TWX)=UT(y) YCX
(i.e. if we can prove that u E T ( X ) iff there is a subset y of X such that u E E
T (Y)>*
Note that if T ( X ) is local in T then T t X and Yare variables of the same sort).
c Y -+ T ( X ) c T ( Y ) (where X
2143. METALEMMA. A term T ( X ) is local in T iff the following is provable in T: T(X) =
u T(Y)
YEF
whenever F E P ( X ) contains for each x c X a superset y called a S-coJinaZ system of subsets of X ) .
zx ( F r a y
be
2244. METADEFINITION. Let T be a theory stronger than TSS’. A godelian term T (X) of T is iterable in T if (a) T ( X ) has the set images property (i.e. T t T (x) is a set), (b) T (X) is local. 2145. METATHEOREM. Let T be a stronger than TSS’. Let T(X) be a godelian term of T iterable in T. Then there is a godelian term S ( X ) (called the class iteration of T(X)) such that the following is provable in T: If X =# 0 is real and H = S ( X ) then H is a real relation, D ( H ) is an ordinal, H”{O} = X and H”{a} = T (H”a) for every a > 0.
Demonstration. PutK(X) = { r ; Re1 ( r ) & D(r) E On&(3z c X ) ( r ” { O } = = z) & (Va > O)(a E D(r) --f r”{a} = T(r”a)). The formula defining K(X) is normal (see 1126), hence the godelian term K(X) can be constructed following 1122. Furthermore, by Metatheorem 1454 T !- Real (X) 3 99
2146
[CHAP. I1 SEC. 1
ORDINAL NUMBERS
-+ Real (K ( X ) ) . Put S (X) = U ( K (X)); we show that S ( X ) has the desired properties. We proceed in T. Denote S ( X ) by H . Then H is real relation and D ( H ) is an ordinal. Evidently H”{O) = X. Further, H”(a} = U r”{a}
and H”a
T(ur”a)
=
u
r”a.
rsK(X)
=
We
have
H”{a} =
u
rcK( X)
=
U T (r”a) r
reK(X)
and
T(H”cr). If we prove UT(r”a) = T ( u r ” a ) we have H”{a) r
r
r”{a}
=
r
T(H”a). It suffices to prove that ( F a ; r E K(X)} is a n G-cofinal system in Ur”cr. Let q 5 u r ” a ; we find ro E K(X) such that q c rza. Define =
r
r
( r , u ) E Q = . u E q & r E EC (X) & u E P a . Q is a relation and D ( Q ) is a set; by (Dl), there is a semiset relation e E Q such that D(Q) = D(Q). Put W (e) = u; u is a subsemiset of K(X). Let u s a ; we can suppose that a C E K ( X ) (since K ( X ) is real). U ( a ) is a relation and U(a)” ( 0 ) is a subset xo of X. There exists ro E K ( X ) such that ro”{O} = xo and 0 < /3 5 a -+ r y { p } = = T ( r Y P ) ; it follows that r”a c rycc for any r E a. (Prove r, s E K(X)& & cr E D ( r ) n D ( s ) & r”{O} C s”(O} . -+ r”{a} C s ” { a } by transfinite induction.) Therefore q c ro”a. 2146. METATHEOREM. Let T (X) be iterable in T where T is stronger than TSS‘ and let O(X) be the range of the class iteration of T(X). Then the following is provable in T: For every real X, Z = 0 ( X ) is the least real
Z and T(Z) c Z . class with the properties X Demonstration. We use the denotation from the proof of the preceding Metatheorem. In T we have Z = W(H), T(W(H))
=
U T(r”cr) = U
reK(X)
rsK(X)
r”(a} E W ( H ) ;
0 < acOo
aeon
furthermore, it is obvious that X E 2. If W is real, X C Wand T ( W ) c W, then by induction H ” { M } E W for each a; hence Z -C W. 2147. Example. Define H as follows (for X real):
H”fO}
H”{n and let
=
x,
+ I > = U(H”(n + 1))
for n E o
Unv ( X ) = W ( H ) .
Verify that the term Unv(X) is definable in TSS’ by a normal formula, that in case X is a set this term coincides with the previously defined term Unv (X) (cf. Definition 2132), and that TSS‘ k Comp (Unv ( X ) ) .
CHAP. I1 SEC. 21
EQUIVALENCE AND SUBVALENCE
2201
SECTION 2
Equivalence and Subvalence of Sets. Cardinal Numbers
Throughout the present Section we shall work within the theory TSS’. We shall define equivalence of sets (having the same power) and subvalence (having smaller than or equal power). We will not define “the power of x” for arbitrary sets x, because without assuming the axiom of choice or some stronger axiom of regularity we cannot define an operation assigning to each set its “power”. However we shall prove in TSS‘ a number of useful statements concerning cardinalities which can be applied in those theories where the axiom of choice is not assumed (or even does not hold), namely a number of statements on finite sets. On the other hand, in TSS’ we can define cardinal numbers as powers of well-orderable sets and prove a number of statements about them. The concept of equivalence will be defined for classes in general (not only for sets). Equivalence of two objects (sets or classes) usually means the existence of a one-one mapping of one object onto the other. But observe that - at least as far as sets are concerned - we have two possible ways to define equivalence. Given two sets x and y we may ask whether there is a set f which is a one-one mapping of x onto y or whether there is a class with this property. (It must necessarily be a semiset.) More generally, for real classes X and X we may ask whether there is a real class which is a one-one mapping of X onto Y or whether there is simply a class with this property. This gives the following definitions: 2201. DEFINITION (TSS’). (1) Let X , Y be real classes. X is equivalent to Y (or X has the same power as Y, X E Y ) if there is a real class which is a oneone mapping of X onto Y. X is subualent to Y ( X Y) if there is a real class which is a one-one mapping of X into Y. X is strictly subualent to Y ( X < Y) if X Yand not X w Y.
<
101
2202
EQUIVALENCE AND SUBVALENCE
[CHAP. I1 SEC. 2
(2) Let X , Y be arbitrary classes. X is absolutely equivalent to Y ( X & Y ) if there is a class which is a one-one mapping of X onto Y. X is absolutely suhoalent to Y ( X Y ) if there is a class which is a one-one mapping of X into Y. X is absolutely strictly subvalent to YifX Yand not X & Y.
3
3
2202. Retilurk. Sets x and y are equivalent iff there is a set which is a oneone mapping of x onto J’; x is absolutely equivalent to y iff there is a semiset which i s a one-one mapping of .x onto JJ. Similarly for subvalence. Obviously, X 2 Y -+ X 2 Y and X Y+X Y are provable in TSS‘; further, the reflexivity, symmetry and transitivity of = are provable in TSS’; i.e. X M X , X = Y - + Y = X . X M Y & Y w Z . + X M Z for real X , Y, 2 ; similarly for & instead of = (and arbitrary X , Y , Z ) . Also X X and X Y & Y Z . -+ X Z are provable in TSS‘ and similarly for 4 instead of
<
4
<
<
<
< <.
2203. DIIFIKITION (TSS’). A relation R is an equivalence if the following hold for all x. J’. z E C ( R ) : (s,s) E R ,
(1)
(2)
(I, J) E
R
-+ (y, x) E
(x. J,) E R & (J’. z )
(3)
E
R
.
R
. --+
(x, z }
E
R.
Evidently. the relation Eq defined by (x, y) E Eq E x w y is an equivalence. (Note that the formula x w J’ is normal i n TSS‘.) In the present section we shall be interested in the notions of equivalence and wbvalence (=, 4 ) ;we shall deal with absolute equivalence and absolute wbvalence later. For the moment we shall only show that the Cantor-Bernstein theorem holds for both notions. 2204. TIII-OKEM (TSS‘). (a) X 1‘.
c1a\\e\
(b)
x,
x 3 Y&
Y
3 x . -+ x
< Y& Y < X .
4
X w Y for any real
& Y for any classes X , Y.
Proof. Given one-one mappings F of X into Yand G of Yinto X we construct a one-one mapping H of X onto Y; then we show that H is real provided that F and G are. Obviously it is sufficient to construct a one-one mapping o f X onto G”Y which is real provided that F and G are. Let K bo the class I02
EQUIVALENCE AND SUBVALENCE
CHAP. I1 SEC. 21
2208
of all one-one mappings f such that D ( f ) is a natural number
(1)
(2) (3)
1‘0E X (VH)
(PI
+ 1 E D(f)
.
- G”Y,
-i f’(i1
+ I.)
= G’(F’(f’n)).
For any f , g E K , n + t E D ( f ) n D(g) implies f ’ ( n + 1 ) = g’(n + 1 ) = = f ’ n = g’n. Define X E W = ( 3 f ~ K ) ( 3 n ) ( =x j ’ n ) ; S = X - W ( = = G”Y - W). Note that if F and G are real then K , S and Ware also real. Now define H’x = G’(F’x) for x E W, H’x = x for x E S. We claim that H is the required mapping. First, H is real if F , G are. Further, H f S is a oneone mapping of S onto S and H f W is a one-one mapping of W onto (G”Y - S); hence H = (H W ) u ( H S) is a one-one mapping of X onto G”Y, q.e.d.
r
r
2205. LEMMA (TSS’). Every set x is strictly subvalent to P ( x ) . Proof. The mapping f on x which assigns to each y E X the set {J’} is one-one and into P ( x ) . Hence x P ( x ) . Suppose now that x M P ( x ) . Let f be a one-one mapping of x onto P ( x ) . Consider the set a of all J’ E x which do not belong to their images, i.e. y 4 f ’ y . Since a E P (x) there is a y, E x such that a = f’y,. If y o E a , then yo $f’y,, a contradiction. If y o 6 a , then yo ~ f ’ y ,= a , a contradiction. Hence x x P ( x ) .
<
2206. DEI-INITION (TSS’). X is a cardinal (Card (X)) if it is a real ordinal and is not equivalent to any of its elements. Ordinal numbers which are cardinals are called cardinal numbers. The class of all cardinal numbers is denoted by Cn. 2207. LEMMA (TSS’). The class On is a cardinal. Proof. On cannot be equivalent to any ordinal number a,since it is a proper class. 2208. LEMMA (TSS’). The class Cn of all cardinal numbers is a proper class. Proof. Suppose the contrary and set t = Sup (Cn). Clearly ( is equivalent to each a > 5. Let the function F assign to each a > ( the set of ordering5 of 5 which are isomorphic to the natural ordering of a (by E).The function F is defined for every a > ( and assumes different values at different points, 103
220Y
[CHAP. I1 SEC. 2
EQUIVALENCE AND SUBVALENCE
i.e. is one-one. Since all of its values F’a belong to P (P (c x c)), the range of F is a set. Consequently the domain of F is a set too, a contradiction. 2209. LEMMA(TSS’).If X is a real class which can be regularly wellordered then X is equivalent to a cardinal, and this cardinal is unique. Proof. By Theorem 2141 every real class X which can be regularly wellordered js equivalent to an ordinal. Each ordinal is equivalent to a cardinal. The uniqueness is obvious. 2210. LEMMA(TSS’). The supremum of any real class X of cardinal numbers is a cardinal. Proqf. The assertion holds whenever X is a proper class since then Sup X = = On. Let X be a set and let SupX $ X . Suppose that there is a B E SupX equivalent to Sup X . Then there is a y E X such that p E y E Sup X . We have then p z y by 2204, since y is a cardinal number, this is a contradiction. 2211. LEMMA(TSS’). x c FI + x < n . Proof. By induction. Let n be the least natural number equivalent to a proper subset, let x be such a subset and let f be a one-one mapping of n onto x. Since n 0 let n = no + 1. The mapping f no is one-one and onto x - { ~ ‘ F I , } If . f ’ n , = no or if no is not in the range o f f , then x {f’n,} c no, which contradicts the induction hypothesis. If no = f’i and i E ti,, define f on no as follows: f y =f ’ n , if j = i and f? = f ’ j otherwise. The function f is then a one-one mapping of no onto a proper subset, a contradiction.
+
2212.
THEOREM
r
(TSS’). a) Every natural number is a cardinal number.
b) The class of all infinite cardinal numbers is isomorphic to On. Proof. a) follows from 2211. because m E n 4 m c n and hence m i.e. each n is a cardinal number. b) follows from 2208, 2141, 2115.
< n,
221 3. DEFINITION (TSS’). The unique enumeration of infinite cardinal numbers by ordinal numbers is denoted by N. The a-th infinite cardinal number is denoted by Nu or 0,. 2214. Remark. Both o, and H, denote the same operation on ordinal numbers. The former notation is used if the ordinal number o, is involved, whereas the latter one is used of the cardinality of the number o,is under consideration; however, both symbols may be used interchangeably. 2215. LEMMA (TSS’).Each o,is a limit ordinal number. 104
CHAP. I1 SEC. 21
EQUIVALENCE AND SUBVALENCE
2219
Proof. Suppose w, = p + 1. Define f as follows: f’y = 0 if y = p , f’y = y + 1 if y E w and f’y = y otherwise. Then f is a one-one mapping of o,onto 8, so that w, x j?,a contradiction. Tn what follows we investigate the properties of finite sets. 2216. DEFINITION (TSS’). A set x i s j n i t e (in the sense of Tarski), if every non-void subset of the power-class of x has a maximal element with respect to inclusion, i.e.
A set x is countable
if
x x
No.
2217. LEMMA (TSS’). (a) A singleton is finite. (b) The sum of two finite sets is finite. Proof. (a) Obvious. (b) Let y be a non-void subset of P ( x , u x,). Set u E y, = ( 3 v ~ y(u ) = u n xl). If y, = (0) then y contains only subsets of x, and hence has a maximal element. Tf y, =l= (0) then there exists z1 E y, such that (Vu E y,) ( l z , c u). Set u ~ y -, ( + ~ y ) ( u = u n x 2 & z l = u n x , ) .
If y, = { 0 } ,then z1 is maximal in y, i.e. (Vu E y) (1 z, c u) . If y, =I= (0) take a maximal z2 in y, and set z = z, u z,; then z is maximal in y. (TSS’). (a) If x1 is subvalent to x2 and x, is finite then x, 2218. LEMMA is also finite. (b) A set which is equivalent to a finite set is itself finite. Proof. (a) Letf be a one-one mapping of x1 into x2 and let y be a non-void set of subsets of x,. Set 7 = {f”z; z E y}. Obviously 7 E P (x,) and 7 =k 0 and so J has a maximal element z. Then the inverse image of z under f is maximal in y. (b) follows from (a).
2219. LEMMA (TSS’). (a) Each natural number is finite. (b) No is infinite. Proof. (a) Denote by F the class of all finite sets. F contains the void set and, by 2217, if x E F then x + 1E F . Hence, by the Induction Principle, F contains all natural numbers. 105
2220
EQUIVALENCE A N D SUBVALENCE
[CHAP. 11 SEC. 2
(b) We have ooE P (ao) and oohas no maximal element with respect to inclusion.
2220. THEOREM(TSS’). (a) A set is finite if and only if it is equivalent to some natural number. (b) If x contains a countable subset ( i t . if KO ,Y) then x is infinite. Proof. (a) If s = n then x is finite by preceding lemmas. If x is finite q q x ) s e that Y i \ not equivalent to any natural number II and set u E y = = u c .Y & (31)( 1 1 x n ) . We have y =+ 0 and y G P (x). Let z be a maximal element of y and denote by the natural number equivalent to 2 . Since x -2 0, tahe q E X - z and set z,= z u ( 4 ) . We have z 1 =I z and z,w ,v tio 1- 1. a contradiction. (b) fo!lows from 2219(b) and 2218(a).
<
-+
3321. COROLLARY (TSS‘). x is finite if and only if x
< coo.
2222. Remurk. The converse of 2220(b) is not provable in TSS‘, see
62 17. 2223. LEMMA (TSS’). The power set of a finite set is finite. Proof. It suffices to prove that Y ( n ) is finite for each 11. Let ti be the least number with an infinite power set. Clearly n =+ 0 and thus n = no + I. for some no. We have P ( n ) = P ( n o ) u (P ( n ) - P (no)). By the induction hypothesis, P ( t i , , ) is finite. Definef’by settingJ’x = x u { H ~ }for x E no. ‘The function.f’is a one-one mapping of P ( n o ) onto P ( n ) - P ( n o ) and hence P ( 1 1 ) is a union of two finite sets; thus P ( n ) is finite, a contradiction.
2223. LEMMA (TSS’). If .Y is a non-void finite set of ordinal numbers then Sup (x) E x (i.e. Sup (x) = Max (x)). Proof. Suppose that 11 0 is the least natural number for which there is a n n-element set x E On which has no maximum. Obviously n =k 1. Set ti = no + I . I f /’ is a one-one mapping of n onto x then by the induction hypothesis the image of under f has a maximum u. The number y = = Max ( x , f ” i i o ) is obviously the maximum of x, a contradiction.
+
2225.
DEI:INITION (TSS’).
Pfi,,( X ) is the class of all finite subsets of X .
2226. DEFINII-ION (TSS’). The relation Sd on the class of all finite sets of ordinals is defined as follows: ( s . .I.)E Sd 106
= .x
=k y & Max ((x
- 4’) u ( y - x)) E y
CHAP. I1 SEC. 21
2230
EQUIVALENCE AND SUBVALENCE
2227. LEMMA (TSS'). The relation Sd is regular and real. Proof. Let y E Pfin(On).If y = 0 then Sd"(y) = 0. If y 0 let a = = Max(y). Since ( x , y ) E Sd implies x E a + 1, we have Sd" { y } E E P ( a + 1) and so Sd is regular. Sd is real by Metatheorem 1453.
+
2228. LEMMA (TSS'). Pfin(On) is linearly ordered by Sd. Proof. The only non-trivial part of the proof is the proof of transitivity. Suppose that (x, y) E Sd and (y, z ) E Sd. Set a1 = Max ( ( x - J') u u (p - x)), u2 = Max ((y - z ) u ( z - 4')). We have a l E y - x , a2 E z - y and a I z2. Suppose that a, E cil. Then y E y = y E z wheneber a2 E y and hence a l E z - x. Let ctl E y and y E X - 2 for some y. Then y E X - y, contradicting the definition of a l . Similarly, if a , E Y and Y E z - x then y E y - x, contradicting the definition of a l . Hence al = Max ((x - z ) u u ( 2 - x)), and since a1 E z we have ( x , z ) E Sd. The case a l E a2 is treated analogously.
+
2 2 3 . THEOREM (TSS'). Pfin(On) is regularly well ordered by Sd. Proof. It suffices to prove that each non-void subset of P,,,(On) has an Sd-least element. Each u E Pfin(On) is a subset of Pfln( a ) for some a ; we take the least CI such that there is a non-void u c P,,, ( a ) having no Sdleast element. Set 5, = Min ((Max (x) ; x E u } ) ; obviously [j E a. Denote by uo the set of all x E u such that b = Max (x). The set u,, has no least element, for otherwise the least element of u o would also be least in u. We have uo c Pfln(/3 + 1) and hence a = b 1 . Let x E u , = x c j?& & x u ( f i ) E u,,. Since u o is non-void, the set u1 is also non-void. Hence, u 1 has a least element (by the induction hypothesis). Denete by z 1 the least element of u 1 and set z = z l u ( p } ; z is the Sd-least elemelit of uo, a contradiction.
+
2230. THLORFM (TSS'). (a) Pfin(On)z On, (b) P t i n (om) x m . Proof. (a) Since Pfin(On) is a proper class regularly well-ordered by Sd, there is a real isomorphism F between On and Pfin(On) with respect to E and Sd. Hence P,,, (On) and On are equivalent. (b) Let F be the above-mentioned isomorphism. We shall prove that F " o , = Pfin(0,) for each 0,. Suppose the contrary and consider the least a, such that F " o , ==! Pfin(oa).Since p E y implies ( ( p } , {y}) E Sd, Pfin(om) contains a subset isomorphic to o,,and Sd"(x} c Pfin(0,)whenever x E Pfin(om). Hence P o , c Pfin(om) and there exists a finite y E o, 107
2231
EQUIVALENCE A N D SUBVALENCE
[CHAP, 11 SEC. 2
such that y $ Po,.Let fl be the maximum of y. Then F ” o , c Prin(/?+ I). Since fl 1 < K,, we have Plin( p + 1) < K,, a contradiction.
+
2231. THEOREM (TSS’). If X is a cardinal such that X /> S o then X x X 2 . Proof. Since X 2 c Prin(Pfi,,( X ) ) , we have X 2 X by the preceding theorem; obviously X X z and so X x X 2 .
<
<
2232. DEFINITION (TSS’). o,is coJinul with op(Conf (o,,op)) if o, 5 - og and if there is a function f such that D(f) = o,,W ( f ) c og I and U(W(f))
=op.
2233. LEMMA (TSS’). If o,is cofinal with wI, and o, 5 o66 ogthen o6 is cofinal with wp. (Obvious.) 2234. DEFINITION (TSS‘). cf(og) is the least cardinal number o, which is cofinal with op. 2235. LEMMA (TSS’). If o, = cf (og)then there is a functionf such that D (,f) = a,,W ( f ) c ob,U(W ( f ) ) = opand (Vl,
K E 0,) (1
<
ti
= f”1
Proof. Let g be a function from o, into opsuch that U(W (9)) = wg. Define the set d by the following induction: O E and, ~ for 0 < 1 < o,, 2 E d iff g ’ l > g’y for all y E (1 n 3.. Then d is a subset of oar g d is a 1 - 1 function and L < K = g’r < g’ti for all 1, ti E d. Clearly d is isomorphic with a n ordinal 6 5 0,; but 6 cannot be less than o, because otherwise some cardinal less than o,would be cofinal with op.Hence there is an isomorphism h of o,onto d ; we setf’i = g’(h’i) for L E 0,.
I‘
2236. LEMMA (TSS‘). cf (cf (cob)) = cf (ap). Proof. Let o, = cf(op), o6= cf(o,). Let f be a monotone mapping of o,into opsuch that U(W (1))= opand let g be a monotone mapping of o6into o, such that U(W ( 9 ) ) = 0,. Define h’i = f ’ ( g ’ i ) for L E 0,; then h is a (monotone) mapping of oainto wp such that U(W (11)) = og;hence o, 5 ma. Evidently w6 I o,,so that o, = oh. 2237. Rerizark (TSS’). There exist infinite cardinal numbers such that cf ( a ) = a (e.g. KO)and infinite cardinal numbers with cf (a) E a (e.g. Nmo). 2238. DEFINITION (TSS’). (a) A cardinal number a is regulur if it is infinite and cf (z) = a. 108
2242
EQUIVALENCE AND SUBVALENCE
CHAP. I1 SEC 21
(b) A cardinal number cz is singular if it is infinite and cf ( a ) E a. 2239. LEMMA (TSS’). The class of all singular cardinal numbers is a proper class. Proof. Suppose that the class of all singular cardinal numbers is a set. Since it is non-void, this set has a supremum y ; we have oo c y. Put f ’ 0 = y, f ’ ( n + 1) = the first cardinal number greater than f ’ n , x = f”o. We have Sup ( x ) # x and y E Sup ( x ) ; furthermore Sup ( x ) is a cardinal number. Sup ( x ) is singular, since it is cofinal with a countable subset; this is a contradiction. 2240. THEOREM (TSS’). For every limit number a, cf(a) is a regular cardinal. To close this Section we define a regular-ordering of On2 which will be useful later. 2241. DEFINITION (TSS’). (a,
p ) Maxlex ( y , S) = . Max (a, p) < Max ( y , 6)
v [(Max (a, p) = Max ( y , S) & ct v [(Max(a,
/?) = Max(y, S)&
v
< y)] v
cz = y &
fi < S)] .
(the maximo-lexicographical ordering). 2242. LEMMA (TSS’). Maxlex is a regular well-ordering of On’; there is an isomorphism Is of On2, On w.r.t. Maxlex, E. Proof. Exercise.
230 I
FIRST AXIOM OF CHOICE
[CHAP. I1 SEC. 3
SECTION 3
The First Axiom of Choice
We shall study three forms of the axiom of choice E l , E2 and E3. E l is weaker than E2 and E2 in turn is weaker than E3. The present section deals with the first of these which is a “set form” of the axiom of choice; it ensures the existence of a selector (choice-function) on each set. In addition, we define the exponentiation of cardinal numbers and prove the recurrence formulas for cardinal exponentiation. It turns out that problems concerning cardinal exponentiation can be reduced to the investigation of a certain unary operaticn 1 on cardinal numbers. 2301. DE121NITION (TSS’). A function f defined on x is called a selector f o r x, if f ’ v E y whenever 0 =I=2’ E x. Sel (x) denotes the set of all selectors for x. 23013. AXIOM ( E l ) . Every set has a selector. In the sequel we shall work in the theory (TSS’, El). 2303. THEOREM (TSS’, El.). Every set can be well ordered. Proof. Let f be a selector for P (x). Define the function F by transfinite recursion: F’O = f’x, F’a = f ’ ( x - F”a) for a < 0. Note that F is a real class by 2124. Since x is not a proper class, there exists a E On such that x = = F”a. If a is the least number with this property, then F ci is a one-one mapping of a onto x. F induces a well ordering T on x, namely (F(/3), F ( y ) ) E ~r = P ~ y f o r f iy, ~ r .
r
2304. LEMMA(TSS’, El). For every set x there is a unique cardinal number equ‘valent to x. ._
2305. DEFINITION (TSS’, E l ) . X is the unique cardinal number equivalent to x. (The cardinality of x.) 110
CHAP. I1 SEC. 31
FIRST AXIOM OF CHOlCE
2310
2306. LEMMA (TSS’, El). Let 5 K, and suppose that y (= K, whenever _. y E x. Then u ( x ) 5 K,. Moreover, if x = K, or if = Kz for some y E x, then UG) = Nu. Proof. For each z E u(x) choose (by EL) some y E x such that z E y and denote it by k’z. For each y E x choose a one-one mapping f,, of y into o, and let g be a one-one mapping of x into 0,. Let 11 be the following function on u(x) : k’z = (g’(k’z),fkPz’z). The function h is obviously one-one and into o, x 0,. This completes the proof by Theorem 223 I . 2307. LEMMA (TSS’, El). For every a, Nu+ is a regular cardinal number. Proof. Take the least a such that Nu+,i s singular. Then there is a subset x of ma+, such that 3 S K, and u(x) = ma+ By 2306, Sup (x) = u(x) is strictly subvalent to X u + , . This is a contradiction. 2308. COROLLARY (TSS‘, El). The class of all regular cardinals js a proper class.
2309. DEFINITION (TSS’, El). The set of all sets which are mappings of the set y into the set x is denoted by exp (x, y). From now on we shall be interested in cardinal exponentiation, i.e. in the cardinalities of the sets exp (x, y ) where x, y are cardinal numbers at least one of which is infinite. From Theorem 2231. we immediately deduce that __
eG(h’,Gj = K, _ for every a and every n + 0. Also, _- - -
as we shall see later, . -exp ( n , K,) = exp (2, K,) for every n 2 2. Hence, we are interested in cardinalit’es of sets exp (K,,K f l ) and exp (2, Nu).These cardinalities have traditionally been denoted by K p , 2’“ respectively. This notation is very common but it i s defective in that the operation symbol for exponentiation is absent. (Let us mention that for this reason it is necessary to write (e.g. n * * m instead of nm in programming languages.) We preserve this traditional notation in the present section for the sake of intelligibility; but we introduce simultaneously another notation which will turn out to be useful in the sequel; ra.nely, we take the Hebrew letter 1for the sign of exponentiation. The reader may rewrite the recurrence formulas given in the sequel in the 1 symbolism.
2310. DEFINITION (TSS‘, El). If x, y are cardinal numbers then x y = - .~ = x 1 y = exp (x, y). We ncw define some important suprema. In each case we give two alternat.ve nntations; one of these is traditional and the other uses the 1 symbolism. ~~
111
231 1
[CHAP. II SEC. 3
FIRST AXIOM OF CHOICE
2311. DEFINITION (TSS', EL).
kp 'PI
2 u
=
E,
=
2 1 gg= sup (2'7; y
1 K, = sup ( K F ;y E
2312. LEMMA (TSS', El). If x, y , s, t E Cn and x
E P}
,
P> . 5 y, s 5 t then xs 5 y'.
2313. LEMMA (TSS', El). ( K ~ ) ' Y = K ~ ( ' p * ' y ) .
Proof. Let f be a mapping of my into exp (a,,a,), i.e. f't is a mapping of al, into a, for each 5 E a,.Define a mapping f of a, x my into o, by f ' ( q , 5 ) = (f't)'q. The function which assigns to eachf E exp (exp (ma, o,), a,)the corresponding f~ exp (ma, m, x my) is one-one and onto. Hence exp(exp(m,, 0,).my) and exp(o,, mPI x my) are equivalent. By 2231. the latter set has cardinality K ~ ( ' ~ * ' y ) . 2314. LEMMA (TSS', El). 2'1 = q w , , > K, for each a. If x E Cn and 2 5 x 5 K, then xu, = 2".. Proof. If f~ exp (2, 0,) (i.e. iff is a function on a, assuming the values 0. I ) we set Flf' = {y, y E ma&f'y = I}; F is obviously a one-one mapping of exp (2, 0,) onto P (ma). This proves the first . -. - assertion; now we prove the . second. Let x 5 Xu. We claim that exp (x, 0,) 2"-. I f f € exp (x, am) then f E x x ma s o that f~ P (x x 0,). Since x x ma = K, the assertion is proved. ~
2315. DEFINITION (TSS', El). P,. (x) is the set of all subsets of x having power at most Ka. 2316. LEMMA (TSS', El). If a 5 P then K F = PN,&,) Proof. Let f be a mapping of a, into mg. Clearly f is a subset of o, x mu .~and has cardinality K,, so that f E P,. (a,x ma). Since a, x mu = N, we have K p 5 PK,<mg). Conversely, if y is a subset of 0, having cardinality at most K,, then there is a functionf, of ma into 0, whose range is exactly y. We choose such a function f , for each y G 0, with F E Nu.Obviously if y, y , then ___ f Y l =+ f.Y 2 . Thus the mapping associating f , E exp (as, a,) with each y E P x (0,) is one-one and hence P,.(Os) 5 Kp. ~
+
CHAP. I1 SEC. 31
2320
FIRST AXIOM OF CHOICE
<= K, < K,. Then N,Urn - (‘Nabcf(Kb) ) .
231 7. LEMMA (TSS’, EL). Let cf (N,)
Proof. Denote cf (a,) by oBo. There is an increasing function g from obo into a cofinal subset of 0,. We claim that PUrn(o,)is subvalent to exp ( PUrn(g’6), oB0). If x E o, is of power at most Nu,set g:y = g’y n x
u
OF PO
for each y E ope. The function associating gx with each x E PKrn (oflo) maps the set P,. (0,) into exp ([ U PRa(g’h)], ope) and is one-one. Since K, <
< K,, there is a 6,
,=#O
E a,,,such
x exp (9’6, a,) whenever 6,
exp ( U PUa(g’d), a,,,)is *SU#O
that o, E 9’6,. By 2316 we have PNa(g’6) x
< 6 < wp0. Hence the cardinality of the set
(G?)’#o
and we have proved K p
(e?)
<= (G?)’#O.
As for the converse inequality is at most K p (since 51 < p) and hence the right-hand side is at most (K?)’,o. Since 3/, 5 tl we have (K?)K,o = = N? by 2313. 2318. LEMMA (TSS’, El). 2’P = (2 ?)d(U#). Proof. We again set oso= cf (0,).Let g be an increasing function with domain wD0and range cofinal with a,. As in the proof of the preceding lemma it can be shown that P (0,) is subvalent to exp ([ U P (g’s)], os0) *-%O
and hence 2Nf’5 (2 ?)‘#o. The converse inequality is obvious; indeed we have (2 = 2Np.
2319. DEFINITION (TSS’, EL). X(K,)
=
% U)~,O
=< (2HB)K,~ =
K$(KB).
2320. THEOREM (TSS’, El). (1) If K, =< K, then K F = 2’.. (2) If K, 5 K, then N> = Max (N,+ Np) (Hausdorff’s recurrence formula). (3) If p is a limit number and N, < cf(K,) then Hp = (Tarski’s recurrence formula). (4) Let K, be singular and let cf(Ks) K, < K,. Denote by K, the cardinal number We distinguish two cases as follows:
Ep
(ep).
(a) if K, = K? for some y < p then
113
2321
FIRST AXIOM OF CHOICE
(b) if K, >
[CHAP. I1 SEC. 3
K F for every y < jl then
Kp
=a
(Gp)= a (K,).
Proof. (1) See Lemma 2314.
RE1,
Gp
(2) and (3): Since K, 5 whenever B is limit, and K p = both cases are included in the formula K p = Max (K,,62) whenever K, < < cf (Kfi).The left-hand side is obviously at least as great as the right-hand side and hence it suffices to prove K p 5 Max (K,,Ep).Here we use 2306 and the fact that exp (a,,a,) = exp (y, a,); for if f E exp (afi,a,),
u
YE-#
then there is a y E assuch that W7(f)c y since K, < cf (Nfi). The cardinality of exp ( y , a,) is just the right-hand side of the equality.
u
YEW#
(4a) Suppose that K, = Rp is equal to K p for some 7 < 8. By 2317 we have = (gp)d(H#) (S\rF)cf(Kg) = KMax(%cf(H#)) = N N U =
~p
~
I
7
$ip.
(4b) Suppose that there is no maximal number among the cardinals
K p (y < B). We claim that cf (a6)= cf (a,). Since cf (a6) cf (ag)i s obvious, it suffices to prove cf(o,) 2 cf(a,). Suppose that cf(a6) <
< cf (a,)and let f be an increasing function of cf (a6) onto a cofinal subset of a,. For every 5 E D ( f ) let g'c be the first y < B such that K F 2 f 'c; set y o = u ( W (9)). Then K,, < K, and Ny; 2 f '5 for all 5, i.e. Ky; = 06. Consequently, for each y between yo and B we have K F = Ky; which contradicts the assumption. Now, we have K p = Kf('#) by 2317 and hence K p = X(K,). 2321. COROLLARY (TSS', El). If /3 is a limit number and 27' every y < B then
< K, for
(i) K> < K, whenener a,y < jl , (ii)
K?
= 3,
(K,) .
Proof. (i) Let 6 (ii) By 2318
=
K;P
Max (a,y ) . Then K> 5 K p = 2" < N,. =
20'
=
H#
(2 ~)c'('~)). Using the assumption we have
# '
2 u = K, and hence K\'a""= 1(K,). 2322. THEOREM (TSS', El.). (1) If K, is regular, then
114
CHAP. I1 SEC. 31
FIRST AXIOM OF CHOICE
2323
(2) If K, is singular, then we again distinguish two cases: H.
(a) if 2 u = 27' for some y < a, then
U.
(b) if 2 u > 2Nyfor every y < a, then N,
2"" = 3, (2 u) . Proof. (1) follows from Theorem 2320 Case (1). Nu
(2a) Let 2 u = 2Hvfor some y < x ; we may suppose that cf (K,) 5 h",. By . N Nu 2318 we have 2" = (2")Cf(Nd = (2%)d(Hd = 2MadN7, Cf(u=)) = 2% = 2 u N.
(2b) It suffices to prove cf (2 u) = cf (8=) and then to use Lemma 2318. The proof is analogous to the proof of Theorem 2320 Case (4b).
Remark. Theorems 2320 and 2322 give the promised recurrence formulas for cardinal exponentiation. 2323. THEOREM (TSS', El). Properties of the function 1.
Prooj. (L) If K, is regular then 3, (K,) = 2'B and the assertion follows from Lemma 2314. Suppose that A", is singular. Let g be a function on cf(a,) whose range is cofinal with a,. Let a be the set of all pairs (6, y ) such that y < cf (0,) and 6 E g'y. Obviously, = K,. We have, of course, K, <= 6 3, (K,), so suppose N, = 3, (K,). Then there is a one-one mapping of a onto exp (a,,cf (ag)). Denote byfdy the image of (6, y ) under this mapping and define a mapping h of cf(a,) into o, as follows: h'y = Min(a, - ($,(y); 6 E g(y)}). The function h differs from all f&;s; for, if h = fcp where (c, y ) E a, then h'y 4 f&(y) whenever 6 E g'y and hence Ii'y =k += f&(y), a contradiction. But the functions f a y were supposed to include all functions in exp (a,,cf (a,));thus we have reached a contradiction. 115
2324
[CHAP. I1 SEC. 3
FIRST AXIOM OF CHOICE
(2) Suppose that cf (1 (His))
=
cf (Kis). Then
a contradiction. 2315. The Continuur~A x i o m :
(Cont) (va) (
F a
=
Nu+
.
23-76. THEOKI 51 (TSS', El). The Continuum Axiom is equivalent to the statement (Vu) (1(xz)= xu+
~3
roof. Since 3 (sJ 5 = 2'~, the Continuum axiom implies (VZ) (1(Xz) = N u , ,). Conversely, suppose that (Vz) (1 ( H a ) = Nu+ and consider the least r with 2'" > N u , If H a is regular then 1( N u ) = 2'I by2322, Ka
Case L. If Xz is singular then by the induction hypothesis 2 u EC,
we have 2'= = 1 ( 2 u)
1 I6
= 1(Xz) =
Xn+ by 2322, Case (2b).
=
K, and
CHAP. I1 SEC. 41
2402
COMPLETE BOOLEAN ALGEBRAS
SECTION 4
Complete Boolean Algebras
This section deals with the basic properties of complete Boolean algebras, which play an important role throughout the book. A complete Boolean algebra is a class which contains the complement of each of its elements and the meet of each of its subsets and which satisfies certain additional conditions. Thus a complete Boolean algebra is determined by a class B, a function C mapping B into B and a function F of P ( B ) into B . It is convenient to consider only real B, C, F. Thus, a Boolean algebra is formed by three classes fulfilling certain conditions. If these classes are sets we speak of a set Boolean algebra. In this case it is more convenient to consider the ordered triple of these sets rather than the three sets separately. To be able to speak of an ordered pair (triple) of classes which are not necessarily sets we introduce the notion of generalized pair (triple) as follows: 2401. DEF~NITION (TSS’).
[x,y, z ] = (x x
(0)) u (Y x (1)) u
(zx
(2)).
2402. DEFINITION (TSS’). B = [ B , C , F ] is a complete Boolean algebra (Cba ( B ) )if B, C , F are real classes, C is a mapping of B into B, F is a mapping of P ( B ) into B and the following conditions hold: (1) F’{x, C’x} = C’F’O for each x E B ,
(2) F’(x, C ’ y } = C’F’O = F’(x, y } = x for each x, y
E
B,
(3) F’z = F‘F’q provided q -C P ( B ) and z = U(q). 117
2403
COMPLETE BOOLEAN ALGEBRAS
[CHAP. I1 SEC. 4
2403. Remark. We introduce the following notation: (the z i t i i t element) , (the zero element), (the meet of x and y) , (the complement of x) .
F’O = 1, c ’ 1 B = OB F’[.Y,y> = .Y C’.V = - B X
If z
5
1’
A
B we write
(the meet or Boolean product of z ) .
F’r = AB3
Iff is a function with values in B we write
The subscript will be omitted if there is no danger of misunderstanding. The conditions (1)-(3) may now be rewritten as follows: 0, ,
(I.)
X A
(I)
S A -J‘=O,=XAj’=.U.
(3)
if z
-
E
N =
B and
z = U(q), then /\z =
A (At). t=l
(This is in fact the associative law.) 2404.
LEMhlA
(TSS’). Tf X. y, z E B then =
X A J’
.Y A
(1. A
Z)
V A X ,
= (X A J’) A Z .
-3405. DEFINITION (TSS’). (1) R is a reflexive ordering of A (Ordg,(R, A)) if for all x, y , z E A we have
(a) x R x , (b) sRy& y R z .
-+ x R z ,
(c) x R y & ~ 7 R x-+. x
= 1’ .
( 3 ) A = [ A , R ] is an ordered class if R is a reflexive ordering of A . If A = [ A , R ] is an ordered class then A is said to be the field of A and the reflexive ordering R is often denoted by
s.
CHAP. I1 SEC. 41
2408
COMPLETE BOOLEAN ALGEBRAS
2406. Remark. If f = {(x, x); x E V } and R is a reflexive ordering of A then R - I is a n (irreflexive) ordering of A. 2407. DEFINITION (TSS‘). Let 5 be a reflexive ordering of A and let z G A , x E A . The element x is the infimum of z (inf.(x, - z)) if
s
(a) (VY E =)(x Y ) (b) (Vw) [(Vy E z ) [w 5 y] 7
-+ w
5
x]
.
2408. THEOREM (TSS’). Let 5 be a complete Boolean algebra. For x, y E B define x y by s A - y = 0, (i.e. x 5 y iff x A y = x). The relation 5 is then a (real) reflexive ordering of B and for each z E B, Az is the infimum of z. If x 5 v then - y 5 - x; - - x = x; x =l= - x unless B contains the single element x. Proof. Clearly x 5 x7 since x A - x = 0, by (1). Suppose x 5 y z, i.e. x A y = x and y A z = y . Then x A z = (x A y ) A z = x A A ( y A z) = s A y = x, i.e. x 5 z. If x 5 y 5 x then x = x A y = y . Hence S is a reflexive ordering. Since x A - x = 0, we have x A x = x. Suppose z E B and let x = A z = F’z; we shall prove that x 5 y for every y E z. We have X A
y = AZ A y = F’{F’Z, y } = F)F”{z, { y } } =
= F’(z u {y}) = F’z = AZ = x
.
Suppose that w 5 y for all y E z; we shall prove w 5 s. IfyEzthenw A y = w ; h e n c e f o r 2 = z u { w } w e h a v e F ’ 1 = ~ z ~ w = = x A w. On the other hand,
Hence w A x = w , i.e. w 5 x, and x = Az is the infimum of z. By (l), - x A - - x = 0,; i.e. - - x 5 x. If x 5 y , then - - x 5 y ; i.e. - - x A - y = 0, and so - y 5 - x. Since - - x x we deduce that - x 6 - - - x; on the other hand, substituting - x for x in - - x 5 I x, we get - - - x - x and hence - - - x = - x. Using x A A - x =0 , we get x A - - - x = O,, i.e. x 5 - - x. Hence x = = - - x. If x = - x then O , = X A - X = X A X = X and 0,=-0,=1,. Since 0, y 5 1, for each y E I?, we have 0, = y = 1, for each y E B.
2409
COMPLETE BOOLEAN ALGEBRAS
[CHAP. I1 SEC. 4
2409. DEFINITION (TSS’). (1) Let B be a complete Boolean algebra. The canonical ordering of the algebra B is the (reflexive) ordering defined by x 5 y X A JJ = X. (2) For x, y E B define X
E X
s y&x * y ,
y = - (-X
X V
A
-y)
(the joirt or the Boolean s u m of x, y), for z G B define
V Z= C’(hC”z) (the j o i n of
2).
2410. DEFINITION (TSS’).Let be a (reflexive) ordering of A and let z E A, x E A. The element x is the supremum of z (sups(x, z)) if
(a) ( V Y E Z ) ( Y 5 x) (b) (Vw) [(Vy E Z ) [ y 5 w ] 9
-,x 5 w ] .
2411. LEMMA(TSS‘). Let B be a complete Boolean algebra and let Vz is the supremum
z E B. Let 5 be the canonical ordering of B. Then of z and we have
C’(/\u) = V(C”u)
and
C’(Vu) = A(C”u)
(De Morgan rules); in particular, for x, y -(X
A V)
= -X
V
-y,
E
-(X
for each u E B
B, V
y)
= -X
A
-y
,
2412. LEMMA (TSS’). Let B be a complete Boolean algebra. For any elements X, y , z E B we have (1)
120
x A y = y A X
x v y = y v x
COMPLETE BOOLEAN ALGEBRAS
CHAP. 11 SEC. 41
X V
(8)
(y
A Z)
= (X
V
y)
A
2415
(X V Z)
Proof. We shall only prove (6) and (7); the proofs of (1)-(5) are easy, while (8) is analogous to (7). - y , i.e. x A y = 0,. Similarly we have (6): If y = -x we have x 5 y, i.e. -x A -y = 4;hence x v y = 1,. On the other hand, assume that x A y = 0, and x v y = 1,. Then we have x A - - y = = Os, i.e. x 5 -y, and -x A - y = O,, i.e. -y 5 x; hence x = -y. -x
(7): We have x A y x and x A z i_ x; hence (x A y) v (x A z ) 5 x. Similarly we have x A y y v z and x A z 5 y v z; i.e. (x A y ) v v (x A z) 5 y v z. Hence (x A y ) v (x A z) 5 x A ( y v z ) . It remains toprove theconverseinequality. Set b = y v z and a = (x A y) v (XAZ). Since x A y 5 a , we have x A y A - a = 0,. Similarly x A z A - a = = 0,. Hence x A - a 5 -y and x A - a 5 - z. Consequently x A - a 5 5 -y A -z = -(y v z) = -b. Thus we have x A - a A b = O , , i s . x A b 5 a ; hence x A ( y v z ) 5 (x A y) v (x A z). 2413. DEFINITION (TSS’). Let B = [ B , C , F ] be a complete Boolean algebra and let x E €3. The partial algebra determined by x is the algebra B I x formed by the following classes:
5
(~1x)’y = -y A x for y ~ ~ ( x , ( F I x ) ’ z = A z ~ x for z ~ B l x .
B l x = {y;y
x),
2414. LEMMA (TSS’). If B is a complete Boolean algebra and if x E B then B I x is a complete Boolean algebra. Proof. Condition (3) is satisfied trivially. We have OBI, = 0,. Since y A -y = 0, we have y A -y A x = 0, hence (1) is satisfied. Finally, if u, u E B 1 x, i.e. u, u 5 x, then u A - u A x = 0, means the same as u A --2) = 0,, i.e. u A u = u. Hence (2) is satisfied. 2415. DEFINITION (TSS’). Let B be a complete Boolean algebra and let Z E B. (a) 2 is aJilter on B if (1)
2
* 0,
121
2416
COMPLETE BOOLEAN ALGEBRAS
(2)
(vx, Y E Z ) ( X
(3)
(vx E z)(Vy E B ) (x 5 y
A
YE
[CHAP. I1 SEC. 4
Z ) , +
y
E
z).
(b) Z is a properjilter if it satisfies (l), (2), (3) above and if (4)
oB#z.
(c) Z is a rornpletejilter if it satisfies (3) above and if
(5)
(vq
c Z ) (A q E Z ) .
(d) Z is a n ultrafilter if it satisfies (l),(2). (4) above and if (6)
( V X E B ) ( X E Zv - X E Z ) .
2416. LEMMA (TSS’). Every ultrafilter is a proper filter; every complete filter is a filter. Proof. Let Z be an ultrafilter; we shall prove (3). Suppose that for some x , y we have X E Z , y # Z and x 6 y . Then - ~ E Zand so x A - - y e Z ; i.e. 0, E Z , a contradiction. Let Z be a complete filter; we shall prove (l),(2). We have 0 E Z and so A 0 = 1 , E Z . ( 2 ) is a particular case of (5). 2417. Rernurks. 1) A complete ultrafilter is an ultrafilter which is a complete filter. Obviously a class Z 5 B is a complete ultrafilter iff it satisfies (4), (5) and (6) above. 2) A proper filter Z is an ultrafilter iff (6’)
(Vx, y
E
B ) (.Y v y E Z
-+
x
E
Z v y
E
Z)
(obvious).
2418. THEOREM (TSS’). Let B be a complete Boolean algebra and let Z be a proper filter on B. Then Z is an ultrafilter iff there is no proper filter Z , on B such that Z c Z,. PROOF.Let 2 be an ultrafilter. If u E Z , - Z then - u E Z and so - u E A - u E Z , , a contradiction. Now suppose that Z is not an ultrafilter. Then there exists u E B such that neither u E Z nor - u E Z . We have z A u =I= 0 , for each z E Z; for, if z A u = 0, for some z E Z then z - t i , so that - u E Z , a contradiction. Now define x E Z , = (32 E Z) (x 2 z A t i ) . Z , is a proper filter and Z , z 2; since u E Z , - Z , we have Z , =IZ= . E
Z1; thus u
122
CHAP. ll SEC. 41
2424
COMPLETE BOOLEAN ALGEBRAS
2419. THEOREM (TSS’). Let 2, be a real proper filter on B and let H be a real one-one mapping of an ordinal onto B. Then there exists a real ultrafilter Z on B such that Z, c 2. Proof. By induction we define H‘a E 2 = (VU) (Fin (u)& u E 2, u (H”a n 2 ) + Au A H’a 0,) . We have 0, # Z and 2 , c Z. If u is a finite subset of Z then Au E 2; hence 2 Is a proper filter. Suppose that there exists an x E B such that neithcr x E Z nor - x E 2. There exist finite subsets u, u of Z such that x A Au = - x A = 0,. Set w = u v U ; then x A AW = --x A AW = 0, SO that A (x v -x) A A w = 0,, i.e. A w = 0,. This contradicts w c 2, since w is finite. Hence Z is an ultrafilter.
+
2420. DEFINITION (TSS‘). Let B be a complete Boolean algebra. An clement u of B is an atom if it is non-zero and if therc is no u E B such that 0, < v < u. An algebra B is atomic if for each non-zero element x E B there exists a n atom y such that y x. B is atomless if it has no atoms. 2421. LEMMA(TSS’). If u is an atom then {x; u 5 x} is a complete ul t rafilter. 2422. DEFINITION (TSS’). Let B be a complete Boolean algebra. A subset
u of B is a partition of B if
Part (B) (briefly Part) is the class of all partitions. If u and u arc partitions, we define u Q ti h
= (vx E u ) ( 3 y E u ) (x 5 = (x
h
y) ,
y, x E u & y E u & x
A
y $. OB} .
2423. LEMMA (TSS’). The relation Q is a reflexive ordering of the class Part; u A u is the infimum of { u , v } in this ordering.
2424. DEFINITION (TSS’). A non-void real class P titionjilter (on B) if (1)
(vx, y E P ) (x h y
(2)
(Vx
E
E
c Part is called a par-
P) ,
P ) (Vy E Part) (x
Q
y
+
y E P) 123
2425
COMPLETE BOOLEAN ALGEBRAS
[CHAP. 11 SEC. 4
2425. DEFINITION (TSS’). Let 2 be a filter on B. Set Pt (2) = {u E Part, 11 n z $:
01
(the partition jilter induced b y a jilter).
2426. LEMMA (TSS’).Pt (2) is a partition filter. ,7427. DEFINITION (TSS’) Let B and B , be coniplete Boolean algebras. (a) B , is a subalgebra of B if
B , c B ,
F , c F ;
C,cC,
(b) A mapping H of B onto B , is a horiiomur.phis/n of B onto B, if H is real and (1)
(vx E B ) ( C i H ’ s = H’C’X) ,
(2)
(vz
c B)(F;H”z
=
H’F’z)
If H is one-one it is called an isomorphism. An isomorphism of B onto itself is called an autornorphism.
-3428. LEMMA(TSS’). Let B , be a subalgebra of B. Then 0, = O,,, 1, = I,, and for z G B , we have V,z = Vz, where V1 is the symbol for the Boolean sum in B , . If x, y E B , then x 5 y = x 5 y , where 5 is the canonical ordering of B , . (Obvious.)
,
,
2429. LEMMA (TSS’). Let B be a complete Boolean algebra; let B , and suppose (VX E B , ) ( - x E B , ) , (VZ
r
c B , ) (F’z
E
cB
B,) .
Then [ B , , C B , , F P ( B , ) ] is a complete Boolean algebra which is a subalgebra of B. (Obvious.) 2430. LEMMA (TSS‘). Let If be a homomorphism of B onto B , . Then (VY, y E B ) (x
5y
-+
H ’ x -I H’y) ,
H’O, = o,, , H’1, = I,, , H’ V u = V H ” u for ZI c B . 124
CHAP. I1 SEC. 41
2436
COMPLETE BOOLEAN ALGEBRAS
The class {x E B, H ’ x = lB,} is a complete filter on B. If H is an isomorphism, then its inverse is an isomorphism of B , onto B. (Obvious.)
2431. THEOREM (TSS‘). Let B be a complete Boolean algebra and let X be a real class such that X E B. Then there exists an algebra B , which is the least subalgebra of B (with respect to inclusion) such that X E B1. Proof. By recursion we define H ” { O ) = X and H “ { a } = {X;(3yEH”sr)(X = - y ) v (3z E H”z)
(X =
F’z))
r
for ct > 0. Set B , = W ( H ) . By 2429, [ B , , C [‘ B , , F P ( B , ) ] is a complete Boolean subalgebra of B; from 2146 it follows that B , is the least subalgebra containing X .
2432. DEFINITION (TSS’). Let B be a complcte Boolean algebra and let X be a real class such that X E B. The class B , of Theorem 2431 will be denoted by Gen, ( X ) (or simply Gen ( X ) ) . If B = Gen, ( X ) then we say that X generates B. 2433. DEFINITION (TSS’). Let B be a complete Boolean algebra; a real class X E B is a base (for B ) if 0, 4 X and if for each non-zero x E B there x. exists JTE X such that y 2434. In the remainder of this section, we restrict our attention to those complete Boolean algebras which are sets. We say that a triple ( b , c,f) is a complete Boolean algebra, if [b, c,f] is a complete Boolean algebra. The set 6 is then called the field of the complete Boolean algebra (6, c,f) etc. 2435. LEMMA (TSS‘). Let b = ( b , c,f) be a complete Boolcan algebra, and let z 5 b be a complete filter. Then there exists u E b such that z = = {x, u 5 x}; z is an ultrafilter iff this u is an atom. Proof. Take u = Az E z. If u is an atom then z is an ultrafilter (see above); if not, then there is some w such that 0, < w < u. Suppose z is an ultrafilter; since w q! z we have - w E z ; hence w 5 - w , a contradiction since w =k 0.
2436. LEMMA (TSS’). If z is a base for a complete Boolean algebra b = = ( b , c,f) then for each u E b there exists some t E z such that u = Vt. Proof. Set t = { u E z , v 5 u } . We have Vt 5 u ; suppose Vt < u. Then u A -Vt =!= 0, and there exists u E z such that v 5 u A -Vt; i.e. v E t and u 5 Vt. Since u 5 -Vt we have v 5 Vt A -Vt = O,, a contradiction.
125
2437
[CHAP. I1 SEC. 4
COMPLETE BOOLEAN ALGEBRAS
2437. COROLLARY. If z is a base for a complete Boolean algebra b = ( b , c, f ) then z generates b.
=
2438. THEOREM (TSS’). Let q be a base for a complete Boolean algebra b = ( b , c, f > and let q , be a base for a complete Boolean algebra 6 , = = ( b , , cl, f , ) . Let Ii be a t - 1 mapping of q onto q , such that
(vu, 2, E q ) (u 5 v = h’u 5 1 h’u)
( s ,is the ordering of ( b , , c , , f , ) ) . Then there is a unique isomorphism g
r
of b onto b, such that g
q = h.
s
Proof. Define g’x = V , { h ’ u , u x} for x E b. Since this must be satisfied by each isomorphism, the uniqueness will follow from the existence.
,
(t) x 5 y = g’x 5 g’y for x, y E b. If x 5 y then we have g’x 5 g‘y immediately from the definition. To prove thc other implication suppose g’x 5 g’y, i.e. V,(h’u, zi 5 x} 5 V l { h ’ v , v 5 v } . It follows that u x implies u 5 y for each u E q . Hence V { u , u 5 x] 5 V { v , v 5 y } . i.e. x 5 J’. This cornplctes the proof of (1); moreover, we have x = y = = g’x = g’y for x, y E b and hence g is one-to-one.
s
(2) g maps b onto b,. For y E b , set x = V { u , h’u 5 y } . Then g’x = y . We also have g’0, = o b , and g’l, = l,,. (3) g ’ h z = A , g ” z for z c b. To prove the inequality S1, consider some w E z;then Az 5 w implies g ’ A z 5 g’w. Hence g’hz 5 A, {g’w, w E z } = A1g”z. To prove the converse, set A,g”z = y. Then w E z implies y & g’w; if y = g’x then x 5 w, and hence x 5 Az. It follows that g’x = y = Alg”z g ’ h z . It can be proved analogously that g ’ v z = = Vlg”z for z c b.
,
s1
(4) g’( -x) = 8’0, = 0,,
= -
, g’x for x E b. We have g’x , g’( -x) A
and g’x v g’(-x) = g’(x v -x) g’( -x) = - (g’x) by Lemma 2412 (6).
= g’(x A -x) = = 9’1, = lb,. Hence
This completes the proof of the Theorem.
2439. DEFINITION (TSS’). We say that an ordering if for each x, y E A (vz
implies x 126
=< y.
5
x)(3t)(t
5 z& t 5
y)
5 on A is separative
2442
COMPLETE BOOLEAN ALGEBRAS
CHAP. I1 SEC. 41
2440. Remurk. The condition can be restated as follows:
2441. LEMMA (TSS’). Let q be a base for a complete Boolean algebra b = ( b , c , f ) . Then the ordering p of b is separative on q.
z
Proof. If x, y E q and x F y, then x A - y < 0;, hence there is some with z 5 x A - y . Now z satisfies the condition
Eq
sl
2442. THEOREM (TSS‘). Let be a reflexive separative ordering of a set q. Then there exist a complete Boolean algebra b = ( b , c,f) and a 1-1 function g which maps q onto some base for b and which satisfies the condition ( v u , u E q) (u
Proof. (1) We say that u hold :
cq
5* u
5
g’u
5 g’u) .
is saturated if the following conditions
5 y and y E u then x E u, (b) if (Vy 5 x) (3z 5 y ) ( z E u) then x E u. (a) if x
Denote by b the set of all saturated subsets of q . (2) If z is a non-void subset of b then n z is saturated. (Exercise.) Define F’O = q , F’z = nz for 0 z E b .
+
(3) For u E b define C’u = {x E q ; 1 ( 3 z 5 x) ( z
E u)]
.
Evidently
u n C‘u = 0.
(4) We prove that C’u is saturated for each u E b. If x 5 y E C’u then 1 (32 5 y) ( z E u); hence 1(3z 5 x) ( z E u ) and so x E C’u. Further, if we
suppose
(vs
5 x) (3t
s s) ( t
E
C’U)
then we may deduce successively the formulas (VS
s x) (3t s s) 1(32 2 t ) ( z
E u)
,
1 (3s 5 x) (vt 5 s) (3z 5 f) ( z E u ) , 127
2443
COMPLETE BOOLEAN ALGEBRAS
l(3s
5 x)(Vt 5
[CHAP. 11 SEC. 4
S)(tEU),
1 (3s ix) (s E u ) , J E
C’U
.
( 5 ) Define b = ( b , C , F ) ; we have Ob = 0 and u A u = u n u for u , u E b. Obviously u n C’U = 0. If z E. b and z = u(g) then = (fit). (Exercise.) tsg
nz n
(6) We prove that u n C’u = 0 iff u n II = u (i.e. u E u). If u c u then u n C’u = 0, since u n C’u = 0 (see (3)). If u $ u, choose x E u such that x $ c. It follows (from saturatedness) that 1 (VJ 5 x) (3: 5 y ) ( z E u), i.e. (3y 5 x) (Vz 5 y ) (z ef u). For any such y we have y E u and y E C’u; hence zi n C‘u =j=0. Thus b is a complete Boolean algebra.
(7) For w E q set g’w = ( x E 4 ; x
5
w] ;
g’w is saturated by the separativity. Evidently, w I
Finally, g”q is a base, which completes the proof.
ii
5 w 2 = g’w, E g’w, .
2443. LEMMA (TSS’ + El). If q is a base for a c. B. a. 6 and if u E b ar?d 0, then there exists u E q such that
+
I) v u = u ,
y = 0, whenever x nad y are distinct elements of a. Proof. Let f be a selector for the power-set of q and let G be defined as follows: 2) x
A
G’O = f ’ { u E q ; u G’ir = f ’ { u
E
q; u 5
U>, zi
- VG”CI) for
CI
> 0.
We may now let u = G”1 where 1 is the least ordinal number such that G’A = 0, i.e. such that u - VG”1 = 0,. ,7444. THEOREM (TSS’, El). If p is a partition of b and if q is a base for b then there exists a partition j Q p such that c q . Proof. Straightforward.
CHAP. I1 SEC. 51
2502
SEPARATIVELY ORDERED SETS
SECTION 5
Ordered and separatively ordered sets
The significance of separatively ordered sets lies in the fact, established in the preceding section, that they are precisely the bases for complete Boolean algebras which are sets. In the present Section we shall study some characteristics of separatively ordered sets and certain methods for constructing such sets. We shall use the axiom of choice (El) in several places. Some statements are formulated not only for separatively ordered sets but for ordered sets in general. 2501. DEFINITION (TSS’). If a = ( a , I , is) an ordered set then a segment of a determined by x is a set of the form Seg,(x) = { y E a ; y S a x } where x
E a.
If it is clear which ordered set is involved we may write 2 instead of Seg, (x). A set z G a is called an exclusive system (in a) (Ex,(=)), if (vx, y
E
z ) (x
*y
+
2 n 9) = 0
An ordered set is K,-multiplicative (v(Ec,, a)) if for any subset u of a which is linearly ordered by 5 and has cardinality at most K, there is an x E a such that (Vy E 2 1 ) (x 5 y). We say that a is K,-partitionable (p,(K,, a)) if there is an exclusive system u E a which has cardinality at least Ec,. 2502. LEMMA(TSS‘). If Ec, 2 K, then for any ordered set a we have
129
2503
SEPARATIVELY ORDERED SETS
[CHAP. I1 SEC. 5
2503. THEOREM (TSS’). If K; is a singular cardinal and if an ordered set a is Xp-multiplicative for every fl < 5 then a is Ks-multiplicative. Proof. Let u E a be linearly ordered by 5 , and suppose u x K,; let f be a one-one mapping of m, onto u. We let g’0 = f ’0 and for any CI > 0 we let E be the least /? such that (Vy < cc) (g’y >,f’p) and g’cc = f’E. Clearly D(g) 5 0,. If D(g) < mc we let S = g; if D(g) = aslet { T , ) , < ~ ( ~ ~be ) a sequence whose supremum is m5and let g’o! = g’z,. It follows that D (8)< < o, and that W (8) is cofinal with u in 2,; i.e., (Vz E u ) (3w E W (y)) (2 2, w). Let Xp be the cardinality of W (9); clearly, a is Kp-multiplicative and hence there is an x E u such that (Vy E W (x 5 , y). By cofinality ($7 E u ) (x 5 , Y ) .
(a))
2504. THEOREM (TSS’, El). Let a be a separatively ordered set and suppose that a has no minimal elements, i.e. that (Vx E a ) (3y E a ) ( y <,, x). If a is K,-multiplicative then it is K,, ,-partitionable. Proof. Using a selector for P ( u ) we construct a function f from ma+ into a such that j ’ f l >,f’y for any p < y < a,+ By separativity we have
y
+ (32
<,
x) (2 n 9 = 0 ) ,
and hence (32 5 , x) (Vt 5 , z ) (t $, y). If follows that for any j? E m,+, we can define a function g’p such that g’j? 5,f’fl and Seg, (g’fl) n Seg, (g’(j? + 1)) = 0. We have g”ma.+ M o,+ and Ex,, (g”ma+ ,).
+
2505. LEMMA (TSS’). Let u be an exclusive system in an ordered set a. If for a11 x E u, ux E 12 is an exclusive system in a then U u, is an exclusive XEU system in a. (Exercise.)
2506. D1:FINITION (TSS’). If u is an ordered set then a n element x of a is called p-saturated (psat, (x)) if for every Naand every y <, x the existence of an exclusive system u E R of cardinality K, implies the existence of an exclusive system of cardinality N, included in 9.
2507. LEMMA (TSS’). If x is p-saturated and y 5 , x then y is p-saturated. ,2508. LEMMA(TSS’). If x is an element of an ordered set a then there is a p-saturated y 5 , x. Proof. For any y 5 , x we let M ( y ) be the least ct such that there is no exclusive system of power K, of elements of 9. We let y o 5 , x be such that M ( y o ) 5 M ( y ) for every y 5 , x. Clearly y o is p-saturated. 130
CHAP. 11 SEC. 51
SEPARATIVELY ORDERED SETS
2510
2509. LEMMA (TSS’, El). If a is an ordered set then there is an exclusive system u of elements of a which contains only p-saturated elemems of a and such that (Vx E a) (3y E u ) (R n 9) =k 0. Proof. Let (xu},<1 be a sequence consisting of all p-saturated elements and let f’0
= xg,
f’u
=
{
x, if (VP < a) (a, n (f’fi) ’) = O , xo otherwise,
for any a > 0. We also let u = f”A. Clearly Ex,(u). If x E a then there is an a such that x, Sax. If xu E u then we let y = xu. If x u # u then thereexists b < a such that x, E u and 12, n R, 0; we then let y = x,. Clearly, y is p-saturated.
+
2510. THEOREM (TSS’, El). Let a be an ordered set, let Kcbe a singular cardinal and suppose that for every b < 5 there is an exclusive system u E a of cardinality K,. Then there is an exclusive system of cardinality K,. Proof. Let u be an exclusive system in a which satisfies the conditions of Lemma 2509. If this system has cardinality ZK, then we are finished; we therefore suppose its cardinality to be
u
XEY,
by = 5 then
u1 is a n exclusive system on 9; otherwise u1 is a subset of u.) It remains to prove that there is no j? < 5 such that, for any x, there is no exclusive system u, c R of cardinailty K,. Suppose there is such a /3 and let 6 < 5 be such that K, > K, and K, > K,,where K, is the cardinality of u. Let u be an exclusive system of cardinality K,. For every X E U we choose a y, E R such that y , & z for some z E u. If we denote by ij the set of all y, such that x E u then for every z E u , 5 n 2 E 2 is an exclusive system and has cardinality less than K,. Hence the cardinality of ij, and therefore also of u, is at most Max (N,, K,) < K,; a contradiction. This completes the proof. 131
2511
[CHAP. I1 SEC. 5
SEPARATIVELY ORDERED SETS
We now define two important indices for complete Boolean algebras. 2 j f l . D I F I N I T(TSS’). I ~ N If b is a complete Boolean algebra then the crrlrbw of b ( p ( b ) ) , i\ the least cardinal K, such that there is no q E b o f cardinality Nzsuch that (VS,
J’ E
9) (X
$. y + X A
y
=
0,)
2512. LEMMA(TSS’. El). If a n ordered set a is a base for a complete Boolean algebra b then p ( b ) is the least cardinal N, such that there is no exclusive system u E CI of cardinality K,. (Exercise.) 2.513. DEFINITION (TSS’). If b is a complete Boolean algebra then the dr-off of b ( v ( h ) ) .is the least cardinal Xg such that there is no K,-multiplicative base for 6. We h a l l now consider- products of ordered sets. 2514. DI~IINITION (TSS’). (I) If s is a nonempty set then 1 C P (s) is called a cut i n s if it has the following properties:
I ( 2 ) If
112
*0,
(vx E
I ) (Vy
G x) ( y E 1 ) .
is a cardinal number then a cut I is nz-additive if (vz E
<
I ) (.
I12 +
uz E I )
( 3 ) A cut is an itleu[ on F if it is ,-additive, i.e. if. (Vx,
(4) The that
I I O U J ~of
)* E
I ) (x u ?‘ E l ) .
a cut I (Norm ( I ) ) is the least cardinal number K, such
1 (3x
E
r) (x /> KJ .
2515. DEf 1 ~ 1 1 1 (TSS’). 0 ~ Let I be a cut in s and let b be a function on s such that, for any x E 5 , b’s is an ordered set; for convenience we write b, = = ( b x . 5,) imtead of b’x. The product of ordered sets b, over 1 is the ordered sct 6 , = ( u , 5 ) where
n‘
I E F
~1
=
{ f’; LJn (f)& D ( f )
f5g =
E
1 & (Vx
E
D (f))( f ’ x
D (f) 2 D ( g ) & (Vx E D (9)) ( f ’ x
E
b,)) ,
5 , g’x) .
2523
SEPARATlVELY ORDERED SETS
CHAP. 11 SEC. 51
2516. Remark. The reader may easily verify that n ' b K is an ordered
set.
XXES
n'
2517. LEMMA(TSS'). 1 ) If all b, are separatively ordered sets without greatest element then 6, is also separatively ordered.
2) Denote
n'bx
XES
by ( a , 5 ) . Iff and g are in a and iff
=g
XES
r D(f) then
g 2 f;if D (f) n D ( 9 ) =: 0 then f u g is the infimum off and g. 3) The empty function 0 is the greatest element of b,.
nz XES
2518. DEFINITION (TSS'). A cut 1 is said to have the singleton property on s if z E 1 and x E s imply z u (x} E 1. 251Y. Remark (TSS'). 1) Clearly if an ideal on s contains all singletons
then it has the singleton property. 2) Let s be infinite and suppose that 1 has the singleton property and s 4 1 (e.g. s = coo and 1 = Pfin(s)). If, for any x E s, the set b, has at least one element, then there are no minimal elements in b,. (Exercise.) Thus
nz XES
the algebra whose base is n Z b xdoes not contain any atoms.
r
XES
2520. DEFINITION (TSS'). For any set x, at (x) is the ordered set (x, I s). (It follows that every two distinct elements of at (x) are incomparable.) 2521. DEFINITION (TSS'). If b,, bl are ordered sets then b, = (b, x b,,
5 ) where
0 6, =
for any u,, L'E,b, and cil, u , E b,. We may prove (in TSS) that if b,, b , are separatively ordered sets (not necessarily without greatest elements) then b, 0 6, is a separatively ordered set. Further, we have the following 2522. LEMMA (TSS'). Let b,, b , be separatively ordered sets with greatest elements lo, 1 , respectively. If a, = ( b i - (li},S i ) (i = 0, 1) and 1 = = P ((0, 1)) then 6, 0 6, and ai are isomorphic.
ni
i=O,l
Proof. Exercise.
n'
2523. THEOREM (TSS', El). If bx is Nu-multiplicative for each x E s and if 1 is N,-additive then bx is K,-multiplicative. XES
133
2524
[CHAP. 11 SEC. 5
SEPARATIVELY ORDERED SETS
Proof. Let ( a , 5 )
=
n*b, and let
u be a linearly ordered subset of a
XES
of power 5 Nu.For every x E s the set { f ' x ; f E u } is linearly ordered by sx. By Nu-additivity we have U D (f) = z E 1. Let f be a function which assof
ciates with each .Y E z a n element f ' x of bx such that f ' x 2 , g'x for any y E u. The element f of u satisfies the condition of N,-multiplicativity. 2524. THEOREM (TSS', EL). If I is an ideal, if Norm ( I ) 5 K,+l and if no b, is K,+l partitionable then n ' b , is not K,-partitionable for any XES cardinal K, > 2". Proof. For any f~ a we shall call the elements of D ( f ) f i x i n g points off. The set of all fixing points o f f belongs to 1 and hence it is of power at most K,. For any f E a we choose some (not necessarily one-to-one) mapping of my onto the set of all fixing points o f f ; for each a < o, we may refer to the a-th fixing point to be denoted by x!. Suppose that u E a is an exclusive system in a ; we shall prove that u 4 2". Let A be a selector for P (u); i.e. we have A'u E u for any nonempty u E u . We shall prove that every element of u can be constructed by iterated application of A. For any f~ u and ci < os we let C(f, a) = = ( g E u; ({f'x:, g'x!})); G(f, a) is the set of all functions in u whose value at the a-th fixing point off determines a segment disjoint from the segment determined by the value o f f a t this point. We let D be the (real) class of all functions h such that D ( h ) E On and W ( h ) c m5. To each It E D we assign a sequence ft of elements of u as follows : f,"= A'u ,
f,"= A'
n C($,
h'p)
(0 <
C(
5 D (h)) .
B<S
If p < ci the values off," and$ at the (h'p)-th fixing off; are exclusive (in the ordered set corresponding to the (h'B)-th fixing point off;). We let Do be the class of all h E D such that G(f;, h'p) 0 for all a D (h); this
n
+
s
B
means that all f,"for a D(h) are defined as elcments of u. If a < p g 5 D (11) and if h E Do then f: f;, since they differ at the (h'cc)-fixing point off:. We shall prove the following:
+
1) For every h E Do we have D ( h ) < as+ 1. 2) For every f E u there exist /t E Do and a 5 D ( h ) such that f = f:. This will complete the proof since we shall have Do 2°F by 1)and u K, . 2" = 2' by 2 ) .
<
134
<
<
2526
SEPARATIVELY ORDERED SETS
CHAP. I1 SEC. 51
Sub 1): Suppose there exists h E Do such that D (h) = my+ and let f = For each a < my+, we let p’a be the (h’a)-th fixing point of f:. Each p’a is also a fixing point o f f , since f~ G(f,h, h’a). Hence we have p”wt+, < K,+ and so there exists x E s such that the set q = { a < wt+ p’a = x} has power Kc+l. The set w = (f:’x; a E q} is a n exclusive system of power K,+ I in 6,’ a contradiction. This proves 1). Sub 2): Iff
E
u then we construct h
E
Do as follows.
Iff = A’u we are finished sineeft = A’u for all h E Do. Otherwise we let h’O = y, where y is the least number such that f ~ G ( f t ,7); such a number exists since f += and bothf and f; are elements of the exclusive system u. Suppose that we have already constructed h’b for all < a ; hence f j for all j 5 CI. Tff = f: then we are finished. Otherwise we havef E n G($, h’j)
ft
and f $:
ft; we let h’a = y where y is the least number such that f E G(f2, y). B
We have f E
n G(f;,
BSa
h’p) and the induction may be continued. In this
way we construct a function h
E
Do; hence we have
ft = f for a = D (h).
The following theorem can be proved similarly: 2525. THEOREM (TSS’,El). If Norm(Z) 5 wo and if no b, is No-partib, is at most countable. tionable then every exclusive system in UES
Proof. We proceed as above; we construct the fixing points xf forf E
n’6, XES
and n E w, the function G(f, n), the class D of all functions with D (h) E On and W (h) E w and the class Do. For any h E Do the set D (h) is finite; for any f~ u there is some h E Do and some n such that f = f;; hence Do is countable which proved the Theorem.
2526. THEOREM (TSS’,El). Let Norm ( 1 ) 5 Ka where a is a limit number. If NB < K, and if no b, is KB-partitionablethen every exclusive system in b, is of power at most Sup 2”.
n’
b
XES
Proof. We let l ( y ) = {z E 1; z < K Y }for any y < a ; Z(y) is an ideal and 1). We let a(y) = (a(y), 5 ( y ) ) = b, so that a = U a(y
I = U Z(y
+
Y
+
XES
y
+
1). If u S a is an exclusive system in a then u n a(y) is an exclusive 1). system in a(y); we have u = U u n a(y B
+
By the preceding Theorem we have u n a(y follows immediately.
+ 1) < 2’7;
the result now
135
2527
[CHAP. 11 SEC. 5
SEPARATIVELY ORDERED SETS
2527. DEFINITION (TSS’). If t is a nonempty subset of s and if 1 is a cut in s then the restriction of the cut 1 t o t is the set l / t = 1 n P ( t ) . 2528. Reniark (TSS’). 1 ) Thc restriction of 2 to t is a cut in t . 2) If 1 is a cut then 1 = Z/u(l). 2529. THT;oREM(TSS’). Let I be an ideal on s and suppose that, for any x E s, b, is an ordered set. If t _C s, t 0 and s - t =+ 0 then n ’ b , is isomorphic to h, 0 b
n”‘
XE I
l-I2’(‘-‘’
+
XES
X.
XES-t
Proof. Excrcise. We shall now study products of separatively ordered sets as bases for complcte Boolean algebras. l f b is a complete Boolean algebra then the set of all non-zero elemcn~sendowed with the canonical ordering is a base for 6. This lead\ us to the following 2530. DEFINlrroN (TSS’). If b is a complete Boolean algebra with more than two elements then
is called the cnnonicrrl base for b.
6
=
( b - {Ob}. 5 )
is called the canonical base for b w i t h the unit.
As a conseqitence of Lemma 2522 wc obtain the following 2532. LEMMA (TSS’). Let b,, b , be complete Boolean algebras and let 1 be the ideal P((0, I}). Then the separatively ordered sets s i,, b, are xe2 isomorphic.
no
2532. R e i m r k . We shall denote by b, 0b , the complete Boolean algebra with the base 8, c $,. We first prove a theorem on uniqueness. 2533. T H ~ O R E (TSS’, M El). If I is a cut in s and if for any x E s both and d, are bases for a Boolean algebra b, then the complete Boolean algebras c and d which have bases n ’ c , and n ’ d y respectively*) are XES XES isomorphic. c,
*) We say that an ordered set (a, ’) is a base for a complete Boolean algebra 6 if a is a base for b and 5- is the canonical ordering of b restricted to u.
136
CHAP. 11 SEC. 51
2535
SEPARATIVELY ORDERED SETS
Proof. We may suppose w. I. 0.g. that d, = b, for any s. By Theorem 2438 it suffices to prove that nzc, is a base for the algebra d. If u is an XES
n'(6,)such
element of d then there exists an element f of the ordered set
XES
that f 5 u (in the canonical ordering of d). We let g be a function with the same domain as f such that g'x E cx and g'x S x f ' x for all x E D ( 9 ) ; there is such a g, since, for every x, c, is a base for b,. This g belongs to nl c, and we have g 5 f 5 u, so that c, is a base for d. XES
n'bx
2534. TIIEKOEM (TSS'). If and if t 5 s then
n"'b,
is a base for an algebra 8
XES
is a base for some subalgebra
6,
of
=
(6,cT,f)
b.
XES
Proof. We denote n'b, by ( a , 5 ) and n l " b , by ( a / t , S / t ) . Let 6, be the set of all joins Vz such that z E a/f. We shall prove that (6,, C 6, f P (6,)) is a subalgebra. The set 6, is clearly closed under joins, so it is sufficient to show that 6, is closed under complementation. For any f~ n we denote by f / t the unique element g of u / t which assumes in s the same values as f . We have f 5 f / t a n d f , 5 fi -,fl/t 5 f 2 / t . On the othcr hand, for any f E a and g E a / t we have f 5 g -+ f / t 5 g. It follows that
r
/"
f
A
g
=
06
-+
(f/t) A g =
for f
Ob
E u
and g E a/t ,
and
f
A
(Vq) = 0, -+ ( f / t )
A
(Vq) = Or
for f E a
and
q E a/t .
If u E 6, and u = Vz for some z c a / t we let Z be the set of all g E a/f such that g A u = OP We prove - u = VZ. We have Vz v VZ = 1,. Indeed, in the contrary case there exists f~ a such that f A (Vz v VZ) = $, so that ( f / t ) A (Vz v VZ) = h: hence ( f / t ) A Vz = OK and so f / t E 2, a contradiction. Further we have Vz A VZ = 06; for, otherwise there exist g E z and g E Z such that g A S OK, which contradicts the definition of 2. Thus 6, determines a subalgebra which evidently has ( a / t , as a base.
+
2535. THEOREM (TSS', El). Let
s/t)
IT' 6, be a base for a complete Boolean
XES
,
algebra 5 = (6, C, f); suppose that Norm (1) 5 K,+ and that 7 p( b,, K,+ If u E 6 is of power K, then there exists a set t of power
n'
XES
K, such that the algebra determined by the base
a subset.
nr''
b, contains u as
137
2536
SEPARATIVELY ORDERED SETS
n’
[CHAP. I1 SEC. 5
s).
be an exclusive Proof. We denote 6, by ( a , For any f E u let set of elements of a such that Var = f and for each z E a/. let fix ( 2 ) be the set of all fixing points of z. If we set a, = U ar and t = U fi ( z ) then f
r-
z=u
has power 5 Xz, since every set fix ( z ) has power at most Naand a, is the
union of at most Xu sets of power at most Nw It follows that u E where 6, is the algebra whose base is 6,.
n”’
6,
XES
2536. Example. Let b, = at (2) for x E o and let I = P,, (a).Denote by Cant the algebra with the base b = n ’ b , . Then Cant has no atoms. XEO)
Remark. This construction can be generalized; generalized Cantor algebras will be studied in Chapt. 6 Sect. 1.
138
CHAP. I11 SEC. I]
3102
REGULARITY A N D CHOICE
CHAPTER UI SECTION 1
The second and third axioms of regularity. The second axiom of choice. Axioms concerning urelements
We shall now present some axioms stronger than D1. The discussion takes place in TSS. Among other things we shall establish the consistency of Dl with TSS. 3101
(D2) ( 3 0 ) (Real (D) & Reg ( D ) & D ( D ) = On & W (D)
(E2) (3F) (Real (F) & Un (F) & D ( F )
= On& W
(F)
=
=
V) ,
V) .
Both axioms are similar; they postulate the existence of a certain real relation whose domain is On and whose range is the universe; axiom D2 requires this relation to be regular while E2 requires it to be a function. 3102. LEMMA (TSS). (a) Let D be a real regular relation; define F = = ( ( , y , x); y = Ext,(x)}. Then F is a real mapping, D(D) = D ( F ) and and F’x = Ext, (x) for every x E D ( D ) .
(b) Let F be a real mapping, define D = {(y, x); y E F ’ x } . Then D is a real regular relation, D ( F ) = D ( D ) and ExtD(x) = F’x for every x E D(F).
Proof. The only non-trivial part is the statement that the F defined in (a) and the D defined in (b) are real classes. (a) Let a be a set; we prove that a n F is a set. D ( u ) is a set and so D D ( u ) is a set (it is a real semiset). Put d = D D ( a ) and define f = ( ( y , x); y = E x t d ( x ) } (by Metatheoreml438fis a set; for u E f i f f (Iy, x ) (u = ( y , x ) & y = Extd(x)) and it is easy to see that the last formula is equivalent to a PUP-formula). Now u n F = a nf , so u n F is a set.
r
r
139
3103
REGULARITY A N D CHOICE
[CHAP. 111 SEC. 1
r
(b) Similarly we prove that u n D is a set for any set a. F D ( u ) is a set, f say, and d = x); y ~ f ’ x is ) a set; then u n D = u n d , so a n D ( ( ~ 7 ,
IS a
set.
3103. THEOKI~M (TSS). (a) Axiom (E2) implies (D2) and (El). (b) Axiom (D2) is equivalent to the statement
( 3 F ) ( Real ( F ) & Un ( F ) & D ( F ) = On & U(W ( F ) )
=
V)
P r o o f . (a) is obvious; (b) follows immediately by the preceding Lemma. 3104. THEOREM (TSS). Axiom (D2) implies (DI). Prooj. Let X be an arbitrary relation whose domain is a semiset and let D ( X ) = 4 and let D be a relation satisfying (D2). We prove that there is a semiset relation o such that o E X and D (X) = D(o). Define Q = [ ( r , x); Y E Q & Ext, (x) n D”r = O}, Q is regular; for if y is an arbitrary element of Ext, (x) then there i s a y such that y E D”y, hence Extv (x) 5 7. Further, D ( Q ) E Q so that D ( Q ) is a semiset. By the axiom (C2) it follows that W (Q) is a semiset, hence there is an a ~ O such n that W (Q) c x . Let n = D”(z + 1) ( u i s a set by the regularity and realness of D).Put o = X n (0 x Q ) . Obviously o is a semiset and o c X . We claim that D ( o ) = Q ( = D (Xj). Clearly D (0) c Q . If x E Q then Ext, (x) E r ; hence there is a y such that y E D”(a + 1) and ( y , x) E X . Consequently, (y. I-) E o and .Y E D (0).
3105. Remark. The main signiiicance of this Theorem lies in the fact that it allows us t o use Metatheorem 1453 as a test for realness of classes in (TSS, D2). I n particular the model %caI can be studied as a model in (TSS, D2). The latter theory will be denoted by TSS”. 3106. DEFINITION (TSS”). Let D be a relation satisfying the conditions (112) (or of (E2)): we define the rank of a set as follows:
Under the assumption of (D2), the rank of a set x is uniquely determined; the collection of sets which have the same rank as x is a set (by regularity and realness of D). Moreover, under the assumption of (E2), different sets have different ranks. In the following paragraphs we shall work in the theory T S S + (D2) with a constant D satisfying Axiom (D2). 140
CHAP. 111 SEC. I 1
3113
REGULARITY AND CHOICE
3107. DEFINITION (TSS”). For each real class X , set
(the set of elements of X of least rank). 3108. Remark. A(X) has the set images property in T S S TSS (D2) one can prove
+
Real(X)&X =!= 0 . + A ( X )
+ (D2);
in
+ 0.
3109. THEOREM (TSS”). If X is a real equivalence then there exists a real P E C ( X ) such that for any x E C ( X ) the class (y; (y, x) E X & y E P> is
a nonempty set. (We say that P is a regulator of x.) Proof. We define P as follows: yEP
E
(3x E C ( X ) ) ( y E A (Ext, (x)) .
P has the required properties; it selects the set of elements of least rank from each equivalence class. 3110. Remark. We have given instructions for constructing P from X ;
i.e. we have described an operation which assigns to each cquivalence X a class P satisfying the conditions of the theorem. 31 11. THEORliM (TSS”). For any real relation
relation Y c X such that D(Y)
=
D(X).
x there exists a real regular
Proof. We define (y, x)
E
Y
= ( y , x)
E X & y E A (Ext, (x)) .
Y has the required properties; each x E D ( X ) is related only to elements of least rank. 3112. Remark. We can again define an operation which assigns to each relation X a regular subrelation Y having the same domain. Since we shall need this operation i n the sequel, we define it explicitly: 31 13. DEFINITION.
141
31 74
REGULARlTY A N D CHOICE
[CHAP. 111 SEC. 1
3114. Remark. Notice that in T S S Theorem 31 11 follows from Theorem 3109 without using (D2). (Exercise.) 3115. METALEMMA. The predicate WOrdg is normal in TSS”.
Dernonstration. We prove in TSS” that the predicate WOrdg ( X ) is equivalent to the formula
(*)
ReaI(X)& LOrdg(X)&(Vu -+ ( 3 z E u ) ( v u E
If)
c C(X))(u 4
((u, z )
+ x)).
04
The statement (*) obviously follows from WOrdg ( X ) . Conversely, suppose that X satisfies (*). Let U be a nonempty real subclass of C ( X ) which has no least element: we construct a nonempty set without least element. Let xo be some element of U ; if xo, . .., x, are elemenls of U such that ( x i + xi) E E X (i < n ) , then we let u , + ~= A({x E U ; (x, x,,) EX}. Since x, is not least in U we have u,, I =/= 0. By (*) u,+, has a least element; we let x,+ be the least element of u,, The set {x,; n E a} has no least element and hence (*) implies WOrdg ( X ) . Consider now the theory TSS + (E2). Since we can prove D2 in this theory we can also prove all the theorems mentioned above. These theorems can be strengthened in the sense that one can prove in T S S + (E2) that the operation A(X) assigns to each nonempty real class a set having just one element. 31 16. DEFINITION (TSS {y,
+ E2).
X) E
AC E x
$:
O & { y > = A (x) .
Ac IS obviously a real function and for any x Ac the universal selector.
+ 0 we have Ac’x E x. We call
3117. TIITOKLM (TSS + E2). For any real equivalence X there exists P E C ( X ) such that every equivalence class has exactly one element in common with P. 3118. TIIFOREM (TSS + E2). For a n y real relation X there exists a real function Y c X with the same domain as X. Remark. The proofs of these theorems are similar to the proofs of Theorems 3109, 311 I : we can define corresponding operations in a similar fashion. 142
CHAP. 111 SEC. 11
REGULARITY AND CHOICE
3122
3119. THEOREM (TSS). Axiom (E2) implies (El). Proof. Let x be a nonempty set; we shall construct a well-ordering of x using the real mapping F of On onto V given by E2. We let H be thc real function which assigns to each y E x the least a E On such that ( y , a ) E F ; H is a one-one mapping of x into On. For y , , y, E x we let ( y , y , ) E r = 3 ( H ’ y , < H’y,). The relation r is a well-ordering of x. Another aim of the present Section is to demonstrate so-called Reflection Principles; this will be done both in TSS + D2 and (in a slightly dincrent form) in TSS E2. To do this we show first that the relation D given by Axiom (D2) can be assumed to have certain additional properties.
+
3120. DEFINITION (TSS). A reflecting system Q is a real function on On with thc following properties: (a) (b) (c) (d)
U(W(Q))
=V
,
Q’a is complete for any a,
P(Q’a) c Q’(a + 1) for any a, Q’I. = Q’a for any limit A.
u
U < l
3121. LEMMA(TSS). If Q is a reflecting system then for any limit I we have (1) Q’A is V-like, (2) if z E A and U(z) = A then Q’A = U Q’a. UEZ
Prooj. The set Q’A is complete and closed by (b), (c) and (d); hence it is V-like. If a < B then obviously Q’a E Q’B; this together with (d) implies (2).
3122. THEOREM (TSS + D2). Reflecting systems exist. Proof. Let D be the relation given by Axiom (D2). We define Q by recursion: Q’O
=
Q’(a
Q’A
(0) ,
+ 1) = Unv (P (Q’a) u D ” u ) , =
u Q’a
for limit 1.
UCA
+
We have D”{a} c Q’(u t) and hence u ( W ( Q ) ) = V. The remaining conditions are also satisfied. 143
3123
[CHAP. 111 SEC. I
REGULARITY A N D CHOICE
31 23. METADEFINITIoN. I . For any godelian term G (X,, 0 ) containing no constants we define in TSS:
Refl,(Z, X I , 0 )
= . Vlk(Z)& G (XI, 0 ) n Z
=
G ( X , n Z,
0 )
(There is one definition for each term; if Refl, (Z, X I , 0 ) we say that 2 is a G-rejecting class w. r. t. X , , 0 . ) 11. If G o ,. .., GLare all the subterms*) of G we define in TSS:
HRefl, ( Z , X I , 0 )
(z is a hereditarily
3
. Refl," (Z, X I , 0 ) & . .. & ReflGL( Z , X , ,
G-reflecting class w. r. t. X , ,
0)
.
0.)
3224. Remarks. 1) If X is a variable then TSS
(Obvious.)
+ HRefl, (Z, X) = Vlk (z).
2) If G does not contain the operation D thcn TSS
HRefl, (Z, X,
0)
= Vlk ( Z ) .
(Cf. 1304, 13 19.) To illustratc the meaning of the notion of reflecting classes we demonstrate the following 3 2 2 5 . M~IATIIEOKEM. Let cp(x-, X , 0 ) be a normal formula not containing any constants. By Metatheorem 1122 there is a godelian term H(X, 0 , Y ) without constants such that H(X, 0 , V) represents cp in TC w.r.t. x, i.e. TC I- x E H ( X , 0 , V) = cp(x, X , 0 ) . The following is provable in TSS, Vlk (M), JTC/c'Gt (M):
(X* =XnM&o)-,[Refl,(M,X,o,V)r
= I"*; (p*(x*, x*,0 ) } *
= (x; q ( x ,
X,
0))
n M]
*) It is clear what if meant by "subterms of a term"; the precise recursive definition reads as follows: (a) If G is an atomic term then G is its only subterm. (b) If P is an operation and t,, ..., t, are terms then the subterms of F(tl, ..., t,) are all subterms of all terms t , , . .., t, together with the term F(t,, . . ., t,) itself.
CHAP. 111 SEC. I ]
3128
REGULARITY AND CHOICE
Demonstration. We proceed in the theory indicated, assuming X* = In fact, {x; p(x, X , 0 ) ) n M equals H ( X , 0 V) n M and {x*; p*(x*, X*, 0 ) } * equals H* ( X * , 0 , M) (since M = V”). Refl, (X, 0 , V) is equivalent to H ( X , 0, V) n M = H (X n M, 0 , M) = H (X*, 0, V*). But, by 1322 (b), H* ( X * , 0, V*) = H ( X * , 0, V*). This completes the proof. = X n M, 0.
3126. The preceding metatheorem has some consequences for restricted formulas. Suppose that q ( x , X , 0 ) is a restricted TC-formula without constants and that H(X, 0 , Y ) is a godelian term without constants such that H(X, 0, V) represents p(x, X, 0 ) w.r.t. x in TC. 1) Refl, (M, X , 0 , V) is provable in TSS, Vlk (M) & Comp (M), dTC/dGt (M). This follows by the preceding Metatheorem and Metatheorem 1327. 2) Since the theory considered sub 1) is an conservative extension of TSS we have the following result:
In particular we have the following 3127. METACOROLLARY. Suppose that p(x, X , 0 ) is a restricted TC-formula and that H(X, 0 , Y ) is a godelian term without constants such that H (X, 0, V) represents p(x, X , 0 ) w.r.t. x in TC. Then
TSS
Vlk(Z)&Comp(Z).&(X c Z & o ) .
+
Z n H ( X , 0, V) = H(X, 0, 2 ) . Let us now consider arbitrary godelian terms. Our aim is to demonstrate metatheorems of the following form: If G(X, 0) is a godelian term without constants then it is provable that for arbitary X,0 there is a “nice” Z such that Vlk (Z) and Refl, (Z, X , 0 ) . By “nice” we mean that Z is big enough, i.e. contains an arbitrary given infinite set x and either is complete (this will be proved in TSS”) or is not too big i.e. has the same cardinality as x (this will be proved in TSS, E2).
3128. METATHEOREM (Reflection Principle). Let G(X,, 0) be a godelian term without constants. Then we can prove the following in TSS + D2: Let Q be a reflecting system and let XI, 0 be real classes. 1) For any x, there is a limit number A such that X E Q’A and HRefl,(Q’A,X,, 0 ) . 145
3128
REGULARITY AND CHOICE
[CHAP. 111 SEC. I
2) If {A,},
is an increasing sequence of limit numbers and if 0 ) then HRefl, (Q’A, X,, 0 ) where 1 = U1,. Dernonstrarion. If G is an atomic term thcn both statements are obvious, since in T S S + D2 we can prove that Refl, (Q’A,X,, 0 ) for any limit number A. If GI and G , are godelian terms then in TSS + D2 we prove: if A E Zi= HRefl,, (Q’A. X,, 0 ) then
(Va E 7) HRe& (Q’A,, X,,
d E Z 1 n Z2
HRefl,,-,,(Q’IL,X1,
0)
HRcfl,, pGz(Q’l,X,, 0 ) .
Only the implications from left to right are non-trivial; we use here the cqualitics from 1304, 1319 and thc fact that Vlk ( Z ) implies D ( Z ) = Z. If 1, is an ordinal number we let be the least number such that R2,,+l > A,, and HRefl,, (Q7?.2n+l,X,r 0 ) and we let L,,,, be the least number such that I,2n+2 > A,,,+, and HRefl,, (Q’&n+Z, X , , 0 ) ; we then set I = A,,. We have HRefl,, (Q’I-,X,, 0 ) and HRefl,, (Q’1, X,, 0 ) .
u n
Hence Z , n Z, is a proper class. Further we have
We again use the equalities from 1304, 1319; in the case of Cnv ( X ) we have Z n Cnv (X) = Cnv (X n 2’) where Z’ = Cnv (2) u { y E Z; y is not an ordercd pair). If Z is V-like then Z’ = Z. The case of Cnv, is treated similarly. Suppose now that 1 E Z
E
HRefl,,,,,
(Q’I., X I , 0 ) . Obviously:
(1) Let S E G , ( X , , 0 , V) be a real regular relation with the same domain as G,. If I,, is the least I. such that x E Q’A then we let A2,,+, be the least 1 greater than ,I2,, and belonging to Z, and we let be the least A greater than A,,,+, such that S Q’?L2,,+lc Q’AZnf2; we then set A = US. If n G , (Xl, 0 ) is denoted by R we have
r
D(R n Q’A) = D(R) n Q’A . The inclusion E is obvious; we prove the converse. If u E D (R) n Q’1 then u E D (S) n Q’A, i.c. u E D (S) n Q’122n+l for some n ; hence there is a v such that ( v , u ) E S n Q’A2n+2and so ( v , u) E R n Q’A and u E D ( R n 146
CHAP. 111 SEC. 11
3129
REGULARITY A N D CHOICE
n Q’l‘). Thus we have
and hence HRefl,,(,,, (Q’A, X , , 0 ) and A E 2. (2) Suppose that A, and A satisfy the conditions of the assertion; denote Q’Au and Q’A by xu and x respectively. We want to prove x n D ( G , (XI, 0)) = D (G, (x n X,,
0))
The right-hand side is equal to D (x n G , ( X , , 0 ) ) and is therefore contained in the left-hand side. If u E x n D(G1(X,, 0 ) )then there is a n a such that u E X , n D(Gl(Xl, 0 ) ) = D(x, n G ( X , , 0 ) ) C D(x n G,(X1, 0)). Let us consider Axiom (E2) again. In addition to Metatheorem 3128 which of course also holds for TSS + (E2), we shall prove another slightly different Reflection Principle. 3129. METATHEOREM. The following may be proved in TSS + E2 for any godelian term G ( X , , 0 ) without constants: Let XI , 0 be real classes. (I) For any x, there exists y 2 x such that =
y = max (2, KO) and
HRefl, (y, X , ,
0)
.
(2) If {xu}El is a sequence of sets such that x, E xg whenever a < /Iand such that HRefl, (x,, X,, 0) for any a < , I ,then HRefl, (uxu,X I , 0 ) . a
Demonstration. In the case of atomic terms we prove the following in TSS + E2: (Vx) (3y) (Vlk ( y ) & x E y & = Max KO)). Let A be a universal selector. We set u g = x, U n + 1 = u, LJ{ z ; ( 3 p , 4 E u,) ( z = { p , q } v z = p - q v z = A’p} and y = uu,; obviously y is V-like and = Max (2, KO).The induction
5
n
(z,
r
step is treated in the same way as in the preceding Metatheorem. In the case of G1 - G , and G, G, we proceed as follows. Suppose that 2 = K,. Set
147
3130
If y
REGULARITY AND CHOICE
= uyn n
and if Zi
= {z;
HRefl,,(z,
[CHAP. 111 SEC. I
X,o)>then we have
y
E 2,
n Z2.
In the case of D ( C , )we proceed as follows. Suppose that x’ = h’, and that S 5 G , (x,, 0 ) is a real function with the same domain as G1(XI, 0 ) . We let
Reinark. We may impose additional conditions on the reflecting set. As an example which will be useful in Section 5 we demonstrate here the following modification of the Reflection Principle: 3130. METATHEOREM. The following can be proved in TSS + E2 for any godelian term C ( X , , 0 ) : Let X I , 0 be real classes. If h’, is a regular cardinal then, for any x such that 2 < X u there exists y 2 x such that < K,, Refl, ( y , X , , 0 ) and y n o, is a complete set (a fortiori, an ordinal number).
>
Deimizstration. In TSS
+ E2,
we let yo
= x,
y z n + l = A ( { z 2 yzn; n + n l oa)and~ =UY~;
~ ~ e f l G ( z , x I , ~= )Y & 2 n )t ) , ~ Z n + 2 = ~ z n + l u U ( ~ 2 we have < K,, HRefl, ( y , XI, 0 ) and Comp ( y n 0,). We have seen that Axiom D2 is equivalent to the existence of reflecting systems. We shall now present the third axiom of regularity, D3, which is stronger than D2 and is equivalent to the assumption that a certain welldefined class is a reflecting system. 3132. DI~PINITION (TSS).
8’0
=
9‘(x
0,
+ I)
=
P(9’a) ,
8’1= (J P’p if 1 is a limit number, tJ
Ker
=
u
P’cc
=
U(W (9)).
aPOn
(The class Ker is called the regular kernel.) Reinark. This definition is correA by Metatheorem 2130. We take G;x = = P (x), G;x = u(x). (Notice that G , , G2 are definable by PUP-formulas.) By the Metatheorem P is a real class. I48
REGULARITY A N D CHOICE
CHAP. 111 SEC. 11
3/35
3132.
Ker = V .
P3)
3133. THEOREM (TSS). Axiom (D3) implies (D2). Proof. By Theorem 3103 (the second formulation of (D2)). Thus (D3) also implies (D1). More generally we define: 3134. DEFINITION (TSS). S i O = x, P i ( a for A limit; Ker(x) = U S i a .
+ 1) = P(P:r),
= U9 a
;~
R€On
Thus, Ker = Ker (0). We shall write pz instead of S i a and pa instead of pp. In order to prove the consistency of (D3) with TSS we shall investigate the sets p:. 3135. LEMMA (TSS). Let x be a complete set.
(a) p: is complete for every a, and the class Ker (x) is complete. (b) a 6 p implies p z E p i for all a, B. (c) Ker (x) is an almost universal class; Ker (x) is not a semiset; Ker (x) is a real class. (d) For every u E Ker (x) there is a minimal a such that u E p z ; we denote this a by T(U) and call the rank of u. (Note that no rank is a limit number.) ) 0 and u E u implies T(u) < ~ ( u ) . (e) For any u, u E Ker (x), ~ ( u > (f) For every non-void z E Ker (x) there is a u E z with the minimal rank. (g) For any u, u E Ker (x) iff u E Ker (x).
Proof. (a) Let a be an arbitrary ordinal number. Suppose that p z is not complete. Let A = { p a; p i is not complete}. Note that Pxis a real mapping and so Pxr ( a 1) is a set; hence A is a set. Let y be the minimum of A; then y > 0. Either y is limit and p ; = p: or y = 6 , + 1 and p; = P(P:~);
+
u
&
in either case we obtain a contradiction. (b) Let a < pandsupposepz $ p i . Let A = ( 6 5 p; (37 < 6) ( p , $ pa)}. A is a set; let 6 be minimal in A and let y < 6 be such that p: $ p:. If 6 = 6 , + 1 then p: E p:o and P (p;) E P (pt0) = p:; but p; E P (p;) by the completeness of p;, hence p; G pf. If 6 i s a limit then p: = U p:, ll. R is regular and D (R) is a semiset, hence W (R) is a semiset of ordinal numbers. Let y be such
3136
[CHAP. 111 SEC. 1
REGULARITY A N D CHOICE
that W(R) C_ y ; then CJ c p: and p : ~ p : +c~ Ker(x). Hence Ker(x) is an almost universal class. If Ker(x) were a semiset then there would be a y such that Ker (x) c pis, hence Ker (x) = p:. But = P (p:) > p; and c Ker (x), a contradiction. Real (Ker (x)) follows by 1440. (d) Let u E Ker (x), u E p:. Put A = { f i =( a ; zc E p z } . Obviously fi E A iff (3q E YPx (Y 1)) (30) ( q = ( u , f i ) & u E P ) and 9, (a + 1) is a set; hence A is a set (by Metatheorem 1431). Let a. be minimal in A ; then z(x) = = zo has the required property. (e) If Z(O) > 0 and u E u then z(u) = y 1 for some y, hence u E P (p:), 11 c p: and u E p:. Consequently, z(u) 5 y < z(u). ( f ) Let z E Ker (x). Then z s p: for some y. Put A = {a;(3q E 9, (7 1)) ( 3 u E z ) (3u) ( q = ( u , a)& u E u ) } . A is a set and has a minimal element a ; there is a u E z such that u E p:; then z(u) = a and no element of z has a smaller rank. (g) The implication -+ is the completeness, see (a). If u c Ker (x) then I I E p: for some y (by the proof of (c)), hence u E P : + ~ , c Ker (x).
1.'
r
+
+
r
r
+
3136. THEOREM (TSS). Axiom (D3) is equivalent to either of the following statementh:
(a) For every non-void x, there is a y E s such that y n x = 0. (b) For every non-void real class X , there is a y E X such that y n X
=
0.
Proof. Suppoce (D3). Let y be an element of x with the minimal rank. ~ ( y )cannot be 0 because there are no elements, of the rank 0. Hence, for any z E y, z(:> < z(y) and therefore z 4 x. We show that (a) implies (b). Suppose that X is a real class such that
n X $. 0 for every u E X . Let u be an arbitrary element of X and set x = = Unv ( u ) n X . Then for any y E x we have y n x $. 0, which contradicts
II
(4.
Suppose (b). If V - Ker += 0 let y E ( V - Ker) be such that y n n ( V - Ker) = 0; we have y E Ker and hence y E Ker, a contradiction. 3137. THEOREM (TSS). If x is complete then Ker (x) is a model class.
Proof. Ker(x) is a complete and almost universal class. Further more Ker (x) is a closed class. Indeed, suppose z , y E Ker (x); then there is a y such that z , y E p;; moreover we may suppose y to be a limit number; hence E (z), z - J,, D ( z ) , z f y , Cnv (z), Cnv, ( z ) E p;. 3138. METATHEOREM. The specification of Gt(M) by M mines an essentially faithful model of T S S + (D3) in TSS. 150
=
Ker deter-
CHAP. I11 SEC. I ]
3140
REGULARITY AND CHOICE
Demonstration. Consider M = lDir (aTSS/aGt (M)) in TSS, M = Ker, aTSS/aGt (M). By Theorem 3137 and Metatheorem 1445 SJZ is a model of TSS-, in TSS. By 1327 every R F is absolute. We prove (Cl)*. Let a be a set satisfying (Cl) and let u' = u n Ker (a' is a set because Ker is real). We have a' G Ker, hence a' E Ker, 0 E a' and if x E a' then x E pr for some y ; hence (x) E a, (x} E pu+ (x) E a'. Denote the theory TSS, M = Ker, aTSS/aGt (M) by TSS,. We now show that
,,
TSS, I- (Vx*) (x* 9 O*
-+
(3y* E* x*) (y* n* x* = 0.)).
The last formula is equivalent to (Vx E Ker) (x $. 0 -+ (3y E x) ( y n x = 0) in TSS,, and this formula can be proved in TSS, in the same way as the implication (D3) --+ (a) in Theorem 3L36. Consequently (D3) holds in the model and fm is a model of TSS + (D3) in TSS,. Finally,'consider fm as a direct model in TSS, D3, M = Ker, dTSS/aG;t(M) and call it fm'. This model is evidently equivalent to the identical model and is also essentially faithful. is therefore essentially faithful; consequently 3139. COROLLARY. (D3) is consistent with TSS. (A fortiori, (D2) and (Dl) are consistent with TSS.) We shall sometimes make use of the so-called transfinite non-empty powers defined as follows:
3140. (1)
DEFINITION
p;
= x,
(TSS). =P(
jj:+,
u pi) - ( 0 )
86a
3
jj; =
u F:
a
if ;1 is a limit;
Ker' (x) = U j z aEOn
(2) LEMMA (TSS). (a) Ker' (0) (b) 0 E x
-+
Ker' (x)
(c) 0 $ x
-+
0 $ Ker' (x).
=
=
0.
Ker (x).
Proof. (a) Every p,"is empty. (b) For every a, p z element of any p:.
=
U pi. (c) 0 is not an
854
We shall now present a generalization of Axiom D3. We first give the following 151
3141
REGULARITY AND CHOICE
[CHAP. I11 SEC. I
3141. DEFINITION (TSS). A set x is an urelement (Urel (x)) if x We denote by Ur the class of all urelements.
= {x}.
EX)(^
= x)&
3142. Remark. Ur is a real class; for x E U iff ~ (b'y & ( 3 y E x) (y = x) and the last formula is restricted. 3143. Axioms concerning urelements. (UI) V = U Ker ( x ) , xCUr
(U2) V = U Ker (x) & (D2), xsur
(U3) M (Ur)& V
=
Ker (Ur) .
3144. Remark. U1states that for every set y there is a set x of urelernents such that y E Ker (x). Note that the following can be proved in TSS: (U2)
= .(UL)&(3R)(Real(R)&Reg(R)&D(R) c O n & W ( R ) =Ur).
3145. LEMMA(TSS). (a) (U3) (b) 033)
+
+ (U2) -+
(Ul)
(u3).
Obvious. 3246.
METATHEOREM.
The formula Ord (x) is a RSF in TSS
Demonstration. We can prove in TSS
+ (Ul) that
+ (Ul).
Ord(x)=Comp(x)&(Vu,uczx)(uEu v u = u v u E u & x n U r = O ) . For, if we assume the right-hand side and if 0 =+= y c x and x E Ker (z) then there exists an element of y of least rank in Ker ( 2 ) which is disjoint from y. The formula on the right is restricted since x n Ur = 0
=
1 (3y E x) ( y = { y } ) .
3147. METATHEOREM. Axiom (Cl) holds in IDir (dTSS/dGt (M)) as a model in TSS, U1, Mcl (M), dTSS/dGt (M). Demonstration. By 1327 all RF's are absolute and therefore, by the preceding Metatheorem, Ord* (x) = Ord (x) whenever x E M. But if a E On then a E M; for c1$ M implies (Vg 2 a) (P I$ M) by the completeness of M. Hence On - M =k 0 implies M(On*). But this implies M*(On*) by the almost-universality of M, which is a contradiction since 1 M ( 0 n ) is 152
CHAP. 111 SEC. I ]
REGULARITY AND CHOICE
3147
provable in TSS without using ((21). Hence EM, Ord*(o), o ~ O n : , (obvious). We have (Onll 0)* which implies ((21). by 2135. In Section 4 we shall strengthen the axiom (E2) to (E3) in such a way as we strengthened (D2) to (D3) in the present Section. We complete this Section with the following Figure depictingthe interdependence of the axioms of groups D, E and U:
+
1' 7 9 U2-
02-
U3-
D2--E3
t
f
E2
t
153
3201
[CHAP. In SEC. 2
MODEL-CLASSES
SECTION 2
Model-classes and set-universal relations. Independence of Axiom (D3)
The remaining Sections of Chapter 111 will be devoted to model-classes and some relations in TSS’ and certain stronger theories. (In the rest of this book we shall seldom work with theories weaker than TSS‘.) In the present Section, in addition to giving several general theorems on modelclasses, we shall show how to construct certain types of model-relations; in this way we shall prove the independence of (D3) and the consistency of various strengthenings of (Ul) concerning the size of the class Ur. First we generalize the operation Ker (x); we define Ker ( X ) also for real classes. 3201. LEMMA (TSS’). x
c y + pz s d.
Proof. Suppose not; let a E A = p: $ p i . A is a real class so let a be the smallest element of A. Clearly a is not 0. If a = p + 1 then p ; c p i and p,“ = P(p;) G P(&) = p,’. If cc is limit then p,” = u p ; c u p ; = p:. In a a either case we obtain a contradiction. 3202. COROLLARY (TSS’). (I) x
E
y
-+ Ker (x)
c Ker (y),
(2) Ker (x) = U Ker ( y ) . ycx
3203. DEFINITION (TSS’). For any real class X , Ker ( X )
= (J Ker (y). YEX
Remark (TSS’). It follows by the preceding corollary that in case X is a set this definition coincides with Definition 3134. 3204. THEOREM (TSS’). Let X be a real class. Then Ker ( X ) is the smallest real class Z such that X c Z and P ( Z ) c Z . Proof. Suppose X c Z & P ( Z ) E Z & Real (2).Then by induction we have x c X -+ p: E Z so Ker ( X ) E Z . On the other hand X c Ker ( X ) and we 154
CHAP. 111 SEC. 21
3208
MODEL-CLASSES
show that P(Ker(X)) e Ker(X). Indeed, let z
c Ker(X) and set
Q = {(u,u);u c X & u ~ z & u ~ K e r ( u ) ) By (Dl.), there is a semiset relation Q , such that Q, c Q and D(Qo) = = D ( Q ) ;then W (Q,) is a subsemiset of X and there is a U c X such that U W (Q,) c ii. It follows that z E Ker (U), hence z E Ker (U) E Ker (X). Thus P (Ker (X)) E Ker (X).
u
3205. COROLLARY (TSS'). (I.) For any real X,
(2) Real (X) -+ Ker (X) is almost universal. Proof. (1) is obvious. (2) If 0 c Ker (X) and 0 E a, then 0 E Ker (X) n n a and Ker ( X ) n a is a set, b say. (Ker (X) is real.) Hence it suffices to show that every subset of Ker (X) is (a subset of) an element of Ker (X). But this was shown in the preceding theorem. 3206. THEOREM (TSS'). For every real complete class X, Ker(X) is a model class. The proof is by the preceding corollary and by 3137. Thus, the fixing of constants X, M by Real (X) & Comp (X)& M = = Ker (X) is a specification for Gt (M) in TSS'. 3207. METATHEOREM. Consider the model IDir (i?TSS/dGt (M) as a model in TSS', (Real (X)& Comp (X) & M = Ker (X)), i?TSS/aGt (M).
is a PUP-formula (cf. 1437) then q is absolute in this model. (2) Moreover, the formulas M(X) and Sm (X) are absolute. Demonstration. (1) It suffices to show (in the theory indicated) that P* (x*) = P(x*), U*x* = Ux* and (x", y*)* = (x*, y*) for all x*, y * . The last two equalities are evident (cf. 1327); the first follows from the fact that z c Ker (X) -+ z E Ker (X) is provable in TSS'. (2) With regard to Sm (X) see 1445; M* (X") = M (X") follows from the provability of z G Ker(X) = z €Ker(X). (1) If
3208. COROLLARY. Gt (M) with Real (X)& Comp (X) & M = Ker (X) determines a model of (TS, D1) in (TS, Dl). Demonstrution. The formula Sm (X) -+ M (X) is provable in (TS, Dl); hence by the preceding Metalemma, Sm* ( X " ) -+ M* (X*) is also provable. 1.55
3209
[CHAP. 111 SEC. 2
MODEL-CLASSES
3209. Remarks. (1) The same discussion of Ker ( X ) is possible in TSS (without Dl); but proofs are longer. As we have already mentioned, we shall in future always assume D1; nevertheless the reader may prove all preceding theorems in TSS and demonstrate all the preceding metatheorems for TSS. (2) The reader may give a definition for Ker(X) (in TSS’) as a class iteration of the operation P ( X ) using Metatheorem 2145. 3210. METATHEOREM on €-induction. The following can be proved in TSS’ for any normal formula q ( x , X , , e):
Demonstration. Use Theorem 3204. 3211. THEOREM (TSS’)(Construction by erecursion). If F is a real function on a complete real class X and if K is a real function then there exists a unique real function G such that D (G) = Ker ( X ) , G X = F and G’x = = K’(G”x) for any x E Ker ( X ) - X . Proof. We fix F , K and X . We let ”’(0) = F and
r
H”(a)
=
((z, x); x c D (H”a)& x 4 D ( F ) & z = K’[(H’’a)” x])
It may be shown that the operation T (Y - X )
= ((z, x); x G
D ( Y - X ) & z = K’ ( Y ” x ) }
is iterable, Y being e.g. a variable for a regular real relation. (Exercise.) Further we let G = W ( H ) , so that Un (G, Ker (X), V), G r X = F and G’x = K’(G”x)) for every x E Ker ( X ) - X . 321.2. DEFINITION (TSS’). A real relation R is well-founded (WF( R ) ) if for any nonempty x E: C ( R )there is some y E x such that EXtR ( y ) n x = 0.
3213. Remark (TSS’). The relation E n Ker (0) is well-founded.
3214. LEMMA (TSS’). If R is a real regular well-founded relation then for any real nonempty X c C ( R ) there is a y E X such that EAR( y ) n X = 0. Proof. Suppose there i s a real nonempty X c C ( R ) such that (Vy E X ) (Ext, ( y ) n X ==! 0). If J’ E X then, by the regularity and realness of R , 156
3217
MODEL-CLASSES
CHAP. 111 SEC. 21
+
Unv, (y) is a set. If we let Unv, ( y ) n X = x then x 0 and for any u E x there is a u E E d R ( u ) n X . Hence u E X and so u E Ext, (u) n x. The existence of such an x contradicts well-foundedness. 3215. THEOREM (TSS’). If R is a real regular, strongly extensional and well-founded relation, then there is a unique complete real class X E Ker (0) and a real isomorphism G between X and C ( R ) w.r.t. E and R . Proof. There.exists a unique u E C ( R ) such that Ext,(u) = 0; we let u = 0, and define
H”{o) H”{a)
=
{(O,
OR>>
9
= {(X, y ) ; J’ E
E D (H”a) & x
c ( R ) & EXt, =
(y) C (H”N)” Ext, (y))
for a > 0 and
G
=
H” O n .
G is a morphism w.r.t. R and E; G is a function and is one-to-one by strong extensionality. We have D ( G ) = C ( R ) ; for if u E C ( R ) - D ( G ) is such that ExtR(u) n ( C ( R ) - D(G)) = 0 then ExtR(u) E D(G) and, since Ext,(u)isaset,thereisanasuchthatExt,(u) c D(H”a)andu E D ( H ” ( a } ) , a contradiction. Hence G is a n isomorphism between W(G) and C ( R ) w.r.t. R and E. If G G is another real isomorphism then there is a least a such that G D (H”{a)) G D (H”(cc)) and there is a n x E D (H”(a)) such that G’x ==! G’x. Since Ext, (x) E H”a, we have G’x = G” (ExtR(x)) = = G”(Ext, (x)) = e’x, a contradiction. Hence C is unique.
r
+
+ ‘/
3216. COROLLARY (TSS‘). If R is a real well-founded E-like relation then there is a unique complete class A E Ker (0) such that E n A and R are isom orphic. We shall now consider the independence of Axiom D3. We shall do this in a rather general way; to speak imprecisely, we show that if the existence of some regular and strongly extensional relation is consistent then it is consistent to assume that this relation describes the membership relation between certain sets in an exhaustive way. The following consideratioins take place in TSS’(or in an arbitrary extension of TSS’). 321 7 . DEFINITION (TSS’). Tf R and S are relations we say that R is a complete subrelation of S ( R E c S ) if R c S and (x, y ) E S implies (x, y ) E E R for any x E C (S) and y E C ( R ) ; hence E x t R ( y ) = Ext, ( y ) for any Y E c (R). 157
3218
[CHAP. 111 SEC. 2
MODEL-CLASSES
3218. Example. If P is complete then (E n P’) E
E
E.
3229. LEMMA (TSS’). (a) If R is a real regular relation and if x E C ( R ) then there exists r E c R such that x E C ( r ) . (b) I f R E c S and S c E T t h e R c 5 T . (c) If r E c R for all r E 2 then E cR.
u(2)
Proof. (a) Let r
=
R f Unv, ( x ) ;
(b) and (c): Exercise. 3220. DEFINITION (TSS‘). A relation R is said to be set-universal (denotation: SetUnvr (R)) if for each x E C (R) there is a u E‘C (R) such that x = = Ext, ( u ) (every subset of C (R) has an R-code). Note that (I) E is a real set-universal relation and (2) if R is a real setuniversal relation then R is almost universal (cf. 141.5). (Both remarks are provable in TSS’.) 3221. METATHEOREM. ( 1 ) The F-definition %(R) with the specification
Real (R) & sExtl (R) & Reg (R)& SetUnvr (R) determines a model of TSS’ in TSS‘. Moreover the following is provable in TSS’, (Real (R)& sExtl (R) & Reg (R) & Set Unvr (R)), dTSS/d% (R): (2) There is a real isomorphism Is, of On, On* w.r.t. E,R. (3) There is a real isomorphism Is, of Ker, Ker* w.r.t. E, R.
(4) M ( X * ) G M* ( X * ) , Real ( X * ) E Real* (X*). (5) If X * , Y* are real subclasses of C(R) then the following are equivalent : (i)
( I F ) (Real ( F ) & Un, ( F , X * , Y*)) ,
(ii)
(W*) (Real* (F*) & Unf (F*, X * , Y*)) .
(6) Consequently if X * is a real subclass of C (R), A is a real ordinal (i.e. either an ordinal number or On) and A* = ZsyA, then the following are
equivalent: (i) (ii)
( I F ) (Real ( F ) & Un, ( F , X * , A ) ) ,
(W*) (Real* (F*)& Un: (F*, X * , A*)) .
Demonstration. (1) By I416 it suffices to prove Axioms (Cl) and (A2) to (A7) in the sense of Dir(dTSS/a%(R)). We proceed in TSS’, (Real(R)& 158
CHAP. I11 SEC. 21
MODEL-CLASSES
3221
& sExtl (R)& Reg (R) & SetUnvr (R)), dTSS/d% (R). Denote C (R) by P. There is a unique x, E P such that Exta (x,) = 0. For any n E o there is a unique x . + ~such that ExtR (x,+ 1) = {x,). Hence there is a unique x E P such that Ext, (x) = {xn; n E a}.It follows that (C1)* holds. We now prove (A2)*-(A7)*. We let o i ( u , xl, 0 ) be a SF equivalent to u E Fi (xl,0 ) ( i = 1, .. ., 7); for xl, 0 E P we let Z = { u E P ; o;(EdR(u), EXtR (xl), 0 ) ) . Z is included in P and is a set; for, if i = 1 then 2 has at most two elements, if i = 2 , 3 , 5 then Z c EXt, (xl) and if i = 4,6,7 then we can define a set-mapping of a subset of Ext,(x,) onto Z . By set-
universality there is a w E R such that Z = ExtR (w)and hence we have M* (2). We have x* E* Z = o:(x*, ExtR (xl), Ext, (x2)) which completes the proof of (1).
(2) The class On* is ordered by R; by regularity and realness of R each segment of On* is a set. Let x be a nonempty subset of On*; by universality x = EdR( z ) for some z. Using properties of ordinals we have (3y E x) (Vz E x) ((2, y> $ R), so that On* is well-ordered by R; On* is obviously a proper class. Hence On* is isomorphic to On. ( 3 ) Let F = { (u,, 0 ) } where u, E C (R) is such that ExtR (u,) = O* = 0, and let y = K’x = x = ExtR(y) for and x C C(R). By 3211, there is a real function G on Ker such that G (0) = F and G’x = K’(G”x) for any x E Ker - (0). The function G is obviously an isomorphism w.r.t. E and R; we prove that W (G) = Ker*. If x E Ker is an element of least rank such that G‘x $Ker* then G”x _c Ker* and M* (G”x) and we have G”x = = Ext, (Gx) E* Ker*; hence G’x E Ker*, a contradiction. Jf x is in Ker* and if x* = Ext, (x) is an element of least *-rank such that x $ W (G) then x* E W (G) and x* = G”y foi some y; hence x = G’y, a contradiction.
r
(4) M* ( X * ) + M ( X * ) follows by the regularity of R, the converse implication foIIows by the set-universality of R. Suppose Real* ( X * ) and consider x n X*. This semiset equals (x n C (R)) n X * but C (R) is a real class, hence x n C(R) is a subset of C(R); by set-universality, there is a y* such that x n C (R) = y*. It follows that M* (y* n X * ) and hence M (x n X*). Thus we have Real (X*). Conversely, if X* is a real class then, for any y*, y* is a set, X * n y* is a set and X * n y* E C (R); by set-univenality, there is an x* such that y* n X * = x* and so we have Real* (X*). ( 5 ) If F is a real one-one mapping of X * onto Y* then we let H be such that F = Dee, ( H ) (cf. 1308); H is a one-one real mapping of X* onto Y* in the sense of !J (R).lConversely, if Real* (F*) & Unr (F*, X * , Y*) then Real (DeCR (F*)) & Un2 (Dee, (F*), x*,Y*). 159
3222
[CHAP. I l l SEC. 2
MODEL-CLASSES
(6) follows from ( 5 ) because ( 3 F ) (Real ( F ) & Un, (F, X*, A)) is equivalent to ( I F ) (Real (F)& Un, ( F , X * , A*)). 3222. COROLLARY. (TSS‘, (Real (R) & sExtl (R) & Reg (R) & SetUnvr (R)),8TSS/d% (R)).
(a) (E2) implies (E2)* ; (b) (D2) implies (D2)* . This follows easily from (6). The following Metatheorem says that we can define various real, regular, strongly extensional set-universal relations in TSS’; in fact, we show how to construct such a relation starting from an arbitrary real regular strongly extensional relation and extending it “upwards”. 3223. METATHEOREM. There are godelian terms RUpw (Q), Imb (Q) such that (1) the following is provable in TSS‘: If Q is a real regular strongly extensional relation then RUpw (Q) is a real regular strongly extensional set-universal relation; (2) the following is provable in the theory TSS’, Real Reg (Q) & sExtl (Q)& R = RUpw (Q), 8TSS/d% (R) : One can define a real complete subrelation Q of R such that Imb(Q) is an isomorphism of Q, Q, and such that V* = Ker* (C (Q)). RUpw(Q) is called the real upward extension of Q . (Cf. Chapter V Section 1.) Demonstration. In TSS’ we define Imb (Q)’ x = (0, x) for x E C (Q). Set Q = (((0, x), (0,y ) ) ; (x, y ) E Q } . Evidently, Imb (Q) is a n isomorphism of Q, Q. Further, define a real relation RQ c [(O, 1 ) x VJ2 as follows:
(a)&
We shall find a real class P, G C (R) such that RUpw (Q) a relation with the required properties.
For x
c (0, 1)
(a) COdp ( x ) (b) CodQ(x)
= =
=
RQ n P i is
x V we define CodQ( x ) (the code of x) as follows:
y , if
X = Exty ( y ) and ( 1 , x) otherwise.
J’ E
c (Q),
CHAP. 111 SEC. 21
3223
MODEL-CLASSES
The term T ( X ) = { CodQ(x); x a relation H as follows: H”(O>
=
c (Q) ,
H”{Ct)
=
(Coda (X);
PQ = W ( H ) ,
c X } is iterable; hence we can define
X
c H”a) for
RUpw(Q)
=
CL
>0,
RQ n P : .
Claim 1 . Q is a complete subrelation of RUpw ( Q ) . Indeed, @ is a subrelation of RUpw(Q) and if x E C ( Q ) then E d n ( . ) = Ext,, ( x ) = c E RUpw ( Q ) . = Ext,”,,(Q, (x); hence Claim 2. The relation RUpw (Q) is real and regular. Indeed, RUpw (Q) is real by 1454. We show that RQ is regular. If y = ( 0 , x) then ExtRQ( y ) = = Extg ( y ) ; since Q is regular, @ is regular and so ExtRp( y ) is a semiset. If y = (1, X ) then EXtR, ( y ) = X. Claim 3. The relation RUpw (Q) is strongly extensional. Indeed, denote RUpw ( Q ) by S and suppose that (i, x) and ( j , y ) have the same extension; we prove (i, x) = ( j , y ) . If i = j = 0 then Ext,((O,x)) = Extn((O, x)) and E d , ((0, y ) ) = Exto ((0, y ) ) , hence (0, x) = (0, y ) . It follows by construction of P , that i f j is impossible. Hence let i = j = 1 . Then Ext, ((1, x)) = x and Ext, ((1, y ) ) = y which implies x = y and (1, x) = = ( 1 , Y>. Claim 4 . The relation RUpw ( Q ) is set-universal. Indeed, using properties of iterable terms we get x s P, + CodQ(x) E P,. If x E P, then x =
(XI).
(CodQ Thus (1) is demonstrated. Now let us proceed in the theory described in (2). We already know that Imb (Q) is an isomorphism of Q, Q and that Q is a complete subrelation of RUpw (Q) = R. C (Q)is trivially a class in the sense of % (R). It remains to prove V* = Ker* (C (Q)). For this purpose suppose C (Q) E X * , Real* ( X * ) and (Vx*) (x* G X * -+ x* E* X * ) ; we shall prove X * = P,. If cc is the least ordinal number such that H”(cc} 4 X * then we let x E H”(cc} - X * . We have u > 0 and Ext, ( x ) c X * . Hence Ext, (x) E* X * and CodQ(Ext, (x)) = x EX*, a contradiction. It follows that X * = V*. This proof completes the demonstration of the whole Metatheorem. The preceding Metatheorem enables us to demonstrate the consistency of several statements contradicting (D3). As an important example we demonstrate the following =
WtUpw(Q)
161
3224
[CHAP. III SEC. 2
MODEL-CLASSES
3224. METATHEOREM. Let T ( X ) be a godelian term with one class variable and without constants such that (1) TSS’ (2) TSS’
T (V) is an ordinal ,
+ T (V)
=
T (Ker) .
Then the following axiom is consistent with TSS‘: (3F) (Real ( F ) & Un, ( F , Ur, T (V)) & V = Ker (Ur) .
Demonstration. We proceed in the theory TSS’. In this theory we define Q = { ( a , a); a E T (V)] and consider Bir (dTSS/d% (RUpw (Q)) as a model in TSS’, aTS.§/d% (RUpw (Q)). We proceed in the latter theory. The relation Q is isomorphic to Q and we know that V* = Ker* ( C ( Q ) ) holds. We prove Ur* = C Evidently, x E C -+ x E Ur* holds because ExtR,,,(Q,(x) = Exto(x) = {x}. Hence C ( 6 ) E Ur*. On the other hand,
(a).
(0)
TSS’ t- V = Ker ( X ) & Real ( X ) & Comp ( X ) . + Ur
cX
and hence the following is provable in the theory in question:
V*
=
Ker* ( X * ) & Real* ( X * )& Comp* ( X * ) . -+ Ur*
E. X *
.
Thus we have Ur* = C(@. Furthermore, Ker and Ker* are isomorphic w.r.t. E, RUpw (Q). Denote by A* the image of T (Ker) in the isomorphism of On, On* (A* = Is,”T (Ker), cf. 3221 (2)) . Thus we have (3F*) (Real* ( F * ) & Unf (F*, Ur*, A*) by ( 3 F ) (Real ( F ) & Un, ( F , C T (V)) and by 3221 (6). By (2) we obtain A* = T* (Ker*) = T* (V”) and the Metatheorem is demonstrated.
(a),
Model-classes in TSS‘, U1. We shall consider the model a i r (aTSS/aGt (M)) in the theory TSS’, U1, Mcl (M), dTSS/dGt (M).
3225. THEOREM (TSS’, U1, Mcl (M), aTSS/dGt (M)). (1) O n c M & O n * = O n ; (2) V* = Ker* (Ur*) ;
(3) P,* = P 162
a n
M.
CHAP. 111 SEC. 21
3227
MODEL-CLASSES
Proof. (1) First note that Ord(x) is a formula restricted in TSS‘, U1, and hence Ord* (x*) = Ord (x*). Since Sm* ( X * ) = Sm ( X * ) , On* is not a semiset. If x* E On* then x* E On by absoluteness (cf. Section 1);hence On* E On. If a E On then there is a b > a such that b E On*; by completeness we have a E M and hence On E On*. (2) Since the formula Urel (x) is restricted in TSS’ we have Urel* (x*) = = Urel (x*) and hence Ur* = Ur n M. Suppose now that Real* (X*), that X * 2 Ur* and that x* E X * -P x* EX*; we prove X * = V*; if x* is an element such that x* # X * then x* $ X * ; x* E Ker (u) for some u and we may suppose that for each y* E Ker (u) of a smaller rank (w.r.t. Ker (u)) we have y* E X * . But then x* s X*, a contradiction. Hence X * = M, i.e.
x* = v*.
(3) We have p o
=
0
=
O* = p x . If
is the least number such that p x =!=
+ p . n M then a is not a limit since u is absolute; hence c1 = p + 1 and CI
we have p; = ps n M and p,* = P* ( p z ) = P (p;) n M = P(p, n M ) n n M = P (ps) n M = p . n M, a contradiction. 3226. DEFINITION (TSS‘, Ul).
Mcl, ( X ) = . Mcl ( X ) & (Va,
a) (a= D
-+ (3f
EX
) (Un,
(f,a, P)) .
( X is a model-class with absolute cardinals.) Note that TSS’, U1 k Mcl,, (V). We defined the predicate Mcl,, in TSS’, U1 because in this theory we know that On is a subclass of every model-class. (This cannot be proved in TSS”.) The name “model-class with absolute cardinals” is justified by the following 3227. LEMMA (TSS’, U1, Mcl (M), dTSS/aGt (M)).
Mcl,(M)
= Cn
=
CnGt‘M’
Proof. The right-hand side is equivalent to (Va) (Card (a) = Card* (a)). The lemma follows by absoluteness of Un2 ( f , x, y ) . We shall consider model-classes with absolute cardinals on various places in this book.
Model-classes in TSS, D3. Finally we consider air (aTSS/aG;t (M)) in the theory TSS, D3, Mcl (M), 163
3228
[CHAP. 111 SEC. 2
MODEL-CLASSES
dTSS/dGt (M). Since (D3) is stronger than (U1 & Dl), Theorem 3225 is provable in this theory. Moreover, we have the following
3228. THEOREM (TSS, D3, Mcl (M), dTSS/dGt (M)). (1) (D3)*.
(2) For any x, x is an element of M iff there is an f and a relation r E M such that f is an isomorphism of r and E Unv ((x)). (3) Mcl*(X*) = Mcl(X*).
r
Proof. ( I ) is obvious.
r
r
(2) Suppose first that x E M; if we let r = E Unv ((x)) and f = I Unv ((XI) then the condition obviously holds. Suppose conversely that the condition holds. The relation r is regular, strongly extensional and wellfounded; hence I' is regular etc. in G t (M). By 3215, there is a complete real X * E Ker* and a real isomorphism G* of E r X * and r . It follows that the composition of G* and f is an isomorphism of X * and Unv ((x]) w.r.t. E, E. By completeness, X* = Unv ({x}) and x E M.
(3) Suppose first that Mcl* ( X * ) ; then Comp* ( X * ) implies Comp ( X * ) and Clos* (X*) implies Clos ( X * ) . It remains to prove AUncl (X*). Suppose c E X * . Then Sm* (0) and, by AUncl* (X*), there is an x * E X * such that CT _c s*. Thus we have proved Mcl (X*). Suppose conversely that X * is a model-class. Then we have Comp* ( X * ) and Clos* (X*) since Comp ( X * ) and Clos(X*). AUncl(X) is evidently absolute from above and hence AUncl* ( X " ) follows from AUncl ( X * ) . 3229. COROLLARY (TSS, D3). If X is a model-class then x E X iff there is an f and a relation r E X such that f is an isomosphism of X Unv (x) and I'. Indeed, this is provable in TSS, D3 iff the following is provable in TSS, D3, Mcl (M) (M being a constant): .y
E
M
= (3f) (3r E M)
(f is an isomorphism of E
'/
Unv (x) and r) .
But this is equivalent to the provability of (2) in Theorem 3228 (in the theory indicated i n this theorem). 3230. DEFINITION. X is a model-class with E l (denotation: Mcl,, ( X ) )
M c l ( X ) & ( t " x ~ X ) ( x+ O + S e l ( x ) n X + O ) . 164
CHAP. I11 SEC. 21
MODEL-CLASSES
3232
3231. LEMMATSS, D3, Mcl (M), aTSS/aGt (M).
(El)*
E
Mcl,, (M) .
Obvious by absoluteness. We conclude this section with a theorem showing that different modelclasses with E l have different sets of ordinals. This theorem is interesting but will not be used anywhere in the book. 3232. THEOREM (TSS, D3). If Mcl,, (X) and Mcl (Y) and P (On) n X = = P ( O n ) n Y t h e n X =Y. Proof. Let us proceed in TSS, D3, Mcl(A), aTSS/aGt(A), Mcl(B), aTSS/dG,t (B) . This is a conservative extension of TSS, (D3.) Denote the notions defined if by aTSS/aGt (A) by superscript A and similarly for B. Suppose (El)”. Notice first that A, B are real, OnA = On = OnB, On x ” On = On x On = On x BOn, PiU(On) = P,,(On) = P& (On) and SdA = Sd = SdB (cf. 2226) We define a well-ordering R of On2 as follows:
we have R* = R = RB and every segment is a set (the same holds also in the sense of both A and B). Hence there is a unique real isomorphism F between On2 and On w.r.t. R and E; we have FA = F = FB.It follows that PA(On’) = PB(On’); for, if x E PA(On’) then F”x c On, F”x E A n B, (Cnv (F))”(F”x) E B and hence x E PB(On2) (the converse is treated similarly). We shall now prove A c B. If x E A then by (El)” there exist a E On and aone-one mapping f E A of a onto Unv(x). We define r c a2suchthat (y, 6) E E r = f ’ y ~ f ’ 6 f; is a n isomorphism between a and Unv (x) w.r.t. r and E. Since r E P (On2) n A we have r E B and hence, by 3229, x E B, which completes the proof that A E B. Moreover, we have proved that f E B; for f is the unique isomorphism between E Unv ((x}) and r, and since both sets are in B, f is also in B. We shall now prove that B E A. Suppose that this is not the case and let a be the least number such that p t c p f . It is obvious that a > 0 and that a is not a limit; hence a = B + 1 for some B and p t = p i . There exists 5 E On and a one-one mapping f E A of p i onto by the first part of the proof we havefEB.IfxEBandx E pithenwelety =f”x;wehaveyEP(On)n B and hence y E A; thus x = (Cnv (f))” y E A and p t = p:, a contradiction.
r
r;
165
3301
SYMMETRIC SETS
[CHAP. 111 SEC. 3
SECTION 3
Symmetric sets and the independence of Axiom (El) in TSS’
We shall now present a classical method for constructing model-classes in the theory TSS, U3; this method enables us to construct various models in which Axiom (El) is false. The method does not enable us to demonstrate the consistency of 1(El) with T S S + (D3); however, it will appear later, when we have a method for proving the consistency of (El), that we can automatically obtain various consistency results for TSS + (D3) using only the consistency results for T S S + (U3) (proved in the present Section). The basic idea is that a set which cannot be well-ordered is in a sense symmetric. The notion of symmetry will be precisely defined. Throughout this Section we shall work in T S S + (U3). 3301. DEFINITION(TSS + U3). A one-to-one mapping of a set a onto itself is called a pernzutation of a . The set of all permutations of a is denoted by g ( a ) . The identical permutation e, is the permutation I a.
+
r
3302. DEFINITION (TSS U3). If p and q are permutations of a then p . q (the composition of p and q ) is the permutation defined by y = = ( p . q)’ x = (3z) ( y = p’z & z = q’x) for x, y E a ; in other words, ( p . q)’ x = p’(q’x). The converse of p is denoted by p - l . 3303. LEMMA (TSS + U3). The following identities hold for any permutations p , q and r of a :
(a) ( P . 4 ) . I’
=
P .(4 . r)
9
(b) p . e a = e a . p = P , (c) p . p - l = p
-1
(d) ( p . q ) - l
4-l. p-l.
166
=
. p =en,
3307
SYMMETRIC SETS
CHAP. 111 SEC. 31
3304. DEFINITION (TSS
+ U3).
(a) A set h c g (a) is called a group (of permutations of a) if e,, if p . q E h and p - l E h whenever p , q E h. (b) A group h, hz G h i .
c g(a) is called a subgroup of a group h,
E
E
h and
g(a) if
(c) If p is a permutation and h is a group then we write
p . h . p-’
=
{q E g(a); (3r E h) (q = p . r . p - ’ ) )
(d) The commutator of a permutation p is the set
the commutator of a group h is the set
3305. LEMMA (TSS
+ U3).
(a) The set g (a) is a group. (b) If h E g(a) is a group and p is a permutation then p . h . p - l is a group. The group p . h . p-’ is said to be conjugate to h. (c) The commutator of a permutation or group is a group; every group is a subgroup of its commutator. (d) The intersection of a non-empty set of groups is a group. Proof. Obvious.
+
3306. DEFINITION (TSS U3). A non-empty set z of groups of permutations of a is called a group-jilter (or a g-jilter) on a if it satisfies the following conditions: (i) h,, h, (ii) h ,
E
E
z -+ h, n h,
E
z,
z & h , E h 2 . -+ h ,
Ez
for any groups h, and h, of permutations of a. (“g-filter” means “g-filter on Ur”.)
+
3307. DEFINITION (TSS U3). A function F is called an automorphism of a real class X (Aut (F, X)) if F is a real one-to-one mapping of X onto X and if x E y = F’x E F‘y for all x, y E X . 167
3308
[CHAP. 111 SEC. 3
SYMMETRIC SETS
3308. LEMMA (TSS + U3). A real function F is an automorphism of the universal class if and only if F is a one-to-one mapping of V onto V and F’x = F”x for all x.
+ U3). If F is a n automorphism of V then
3309. LEMMA (TSS
a) z ( F ’ x ) = z (x) for every x , b) F f U r E g ( U r ) . 33fO. METATHEOREM. For any TC-formula qo(x,, 0 , X , , constants the following formula is provable in TSS + U3:
0)
without
Demonstration by induction. 3322. COROLLARY. For any Codelian term F (XI, 0 ) without constants the following formula is provable in TSS + U3: Aut ( F , V)
-+
(VXl,
0 ) (F
(F”X,,
0)
= F” F (XI,
0))
We shall now be interested mainly in permutations of Ur which we shall refer to simply as permutations. We write g instead of g (Ur).
r
3322. THEOREM ( T S S + U3). For every permutation p there exists a unique automosphism G of V such that G Ur = p . Proof. This follows by Theorem 3211 on setting F = p and K = I. 3313. DEFINITION (TSS + U3). For any p E g, we denote by Aut ( p ) , or simply by p, the unique automorphism of V which extends p .
+
3314. LEMMA (TSS U3). For any x we have ( ( p . 4)”)’ x = ;’($x) ( ( p - ’ ) ” ) ’ x = (Cnv (j))’x.
and
Proof. By inducrion.
3325. DEFINITION (TSS + U3). Inv (x) = { p E g; $’x = x> (the invariant of x), PInv (x)
=
n Inv ( y ) . YEX
3326. LEMMA(TSS groups;
+ U3).
a) For every x, Inv(x) and PInv(x) are
(b) For any group h we have h 168
G
Inv (x)
= (Vp E h) (p”x
G x).
CHAP. 111 SEC. 31
SYMMETRIC SETS
3317. LEMMA (TSS
3320
+ U3).
(a) If x E Ker then Inv (x)
=
g,
(b) Inv (E (x)), Inv ( D (x)), Inv (Cnv (x)), Inv (Cnv, (x)) 2 Inv (x), Inv (x - y ) , Inv ( x y ) , Inv ({x, y } ) 3 Inv (x) n Inv ( y ) .
r
(c) Inv (j’x)
=
p . Inv (x) . p - l , for every p E g.
(d) Inv(p) = [ p ] , Inv (h) = [ h ] for every permutation p and every group h. (e) For any x and each p E g we have Inv (p‘x) = p‘ Inv ( x ) . Proof. (a) Since $0 = 0 for every p E g, we obtain by transfinite induction $x = x for every p E g and x E Ker. (b) By corollary 3311. (c) We shall prove i ’ p = q . p . q-’ for any p , q E g. We have ( u , v) E p E (q’u, q ’ v ) E i’p. Similarly we have (u, u ) E p
=
= (v, q ’ u ) E p - 1 . q - I = . q-1 = (q’u, 4’0) E q p q - 1 .
(q’u, u ) E q . p
= (q’u, q ’ u ) E q . p This gives both c) and d).
3318. LEMMA (TSS + U3). If Ur x a for some a E Ker then for every group h there exists x such that h = Inv (x). Proof. Let f be a one-to-one mapping of a onto Ur and for any p E g let p be the composition of p and f;we have p’u = p ’ ( f u ) for every u E a . If we define h = { p ; p E h } then since 4.p = i’p we have q E Inv (h) whenever q E h; for, if q E h and p E h then i ’ p = 4.p E h. Conversely, if i”t5 = h then G’G E h; i.e. q’e = 4.e = ij and hence q E h. Thus we have Inv (I;) = h. 3319. DEFINITION (TSS
+ U3). For
any g-filter z we define the class
S ( z ) of all symmetric sets and the class HS ( z ) of all hereditarily symmetric sets as follows:
S ( z ) = {x; Inv (x)
Ez}
,
HS ( z ) = {x; x E S ( z ) & Unv (x) c S ( z ) } . 3320. THEOREM (TSS + U3). For any g-filter z the class HS(z) is complete, closed and real. 169
3321
SYMMETRIC SETS
[CHAP. 111 SEC. 3
Proof. Completeness follows immediately by definition; the fact that HS (2) is closed follows by Lemma 3317, (b). HS (z) is obviously real. 3321. LEMMA(TSS + U3). If z1 and z2 are g-filters such that zl c z 2 then HS(z,) c HS(z2). 3322. DEFINITION (TSS (Sym (z)) if Inv (z) E z.
+ U3).
A g-filter z is said to be symmetric
3323. THEOREM (TSS + U3). If a g-filter z is symmetric then HS(z) is a model-class. Proof. By 3320 it suffices to prove that HS(z) is almost universal. If x E HS (z) then we let y = { 2 u ; u E x & p E Inv (2)). We have y c HS (z): for, if p E Inv (z) then i ” S (z) = S (z) and ;”HS (z) = HS (z). Moreover, we have Inv ( y ) 2 Inv (z) and hence y E S (z); thus y E HS (z) and y 2 x. 3324. THEOREM (TSS a g-filter z such that
+ U3).
If P is a model-class then there exists
(i) z is symmetric,
(ii) P
C_
HS (z),
(iii) if z1 is a g-filter such that P E HS (z,) then z G zl. Proof. Let zo be the set ( h ; (3x E P ) (h = Inv (x))} and let z be the set { A ; (3h, E z,) ( k , c k)}. The set z contains the intersection of any two groups in z; for, if x, y E P thenInv (x) n Inv (y) = Inv (<x, y)) and <x, y) f E P. Hence z is a g-filter. For every g-filter zIwe have P c HS ( z l ) E zo c c z , and hence P s HS ( z l ) E z c z,; in particular, P c Hs (z). This proves (ii) and (iii). It remains to prove that z is symmetric. Since P is almost universal there is a set x E P such that (Vh~z,) ( h e x ) (k = In.(.)). We have Inv (x) E z,; we show that Inv (x) G Inv ( 2 ) . Suppose that p E E Inv (x) and h E z , we shall prove F’h E z. There is a n h, E zo such that h, c k ; we choose u E x such that Inv (u) = h,. We have i ’ u E X and Inv ( i ’ u ) E z,; since Inv ( i ’ u ) = p . ho . p - ‘ = i’k,,we have j ’ k 2 j ’ h , and i’/z E z. This completes the proof. 3325. COROLLARY ( T S S + U3). If HS (2) is a model-class then there exists a symmetric g-filter z 1 c z such that HS (zl) = HS ( z ) ; if HS ( z 2 )= = HS (z) for some g-filter z 2 then zI c z2.
+
3326. DEFINITION (TSS U3). A symmetric g-filter z is said to be well-symmetric (WSym (2)) iff Inv (x) E z for each x E Ur. 170
CHAP. III SEC. 31
3331
SYMMETRIC SETS
3327. LEMMA (TSS + U3). A symmetric g-filter z is well-symmetric iff for each e E Ptin(Ur) we have PInv (e) E z . (Obvious.) 3328. LEMMA (TSS, U3, Sym (z), aTSS/dGt (HS(z))). Ur* = Ur iff z is well-symmetric. Proof. If Ur* = Ur then, for each x E Ur, x E HS (z), hence Inv (x) E z. On the other hand, if Inv(x)Ez for each X E U then ~ M*(x) for each x E Ur; hence (Vx) (Urel* (x) = Urel (x)) (since Urel (x) is RF).
We now state several theorems which concern the possibility of wellordering elements of the model-class HS (z) in the senseof HS (z). Consider a fixed well-symmetric g-filter z and denote HS (z) by P. 3329. THEOREM (TSS, U3, Sym (z), aTSS/dGt -(HS (z))). If an element x of P is equivalent to an element y of Ker then
x z * y rPInv(x)Ez. Proof. Since the concept of a mapping is absolute the condition x x * y is equivalent to ( 3 f P ~ ) (Un, ( f , x, y)). Suppose first that x M * y and let f~ P be a one-to-one mapping of x onto y. We have Inv ( f ) E z; we shall prove Inv ( f ) E PInv (x). If p E Inv ( f ) and u E x we show that i ’ u = u. There is a v E y such that ( v , u) E f . Since u E Ker we have i ’ v = v and $((u, u)) = (p’o, i ’ u ) = (u, i ’ u ) . Since p E Inv ( f ) we have E;’f = f and (v,i;’u) E f . The function f is one-to-one so that E;’u = u. Suppose conversely that PInv (x) E z and let f be a one-to-one mapping of x onto y. We shall prove Inv(f) I> PInv (x). If p E PInv (x) and (0, u) E f then $ ’ ( u , u ) = ( 0 , u ) ; hence i y = f . This completes the proof.
3330. THEOREM (TSS, U3, Sym (z), dTSS/dGt (HS (z))).
(Vx*) [(3r*) (WOrdg* ( r * ) & C* ( r * ) = x*)
=
= (3r) (WOrdg (r) & C ( r ) = x* & PInv (x*) E z)] . Proof. A set can be well-ordered iff it is equivalent to some ordinal. Since the concepts of ordinal and one-to-one mapping are absolute, the result follows from the preceding Theorem. 3331. THEOREM (TSS, U3, WSym (z), dTSS/dGt (HS (z))). (El)*
= (El)& (el E z . 171
3332
[CHAP. III SEC. 3
SYMMETRIC SETS
Proof. We again denote HS (z) by P . If Axiom (El) holds and if {el E Z then the g-filter z contains all groups; hence P = V and (El)* holds. Conversely, if (El)* holds then there is an CI such that Ur x * CI. By 3329, (e) = PInv (Ur) E z which gives z = g and hence V = P ; ( E l ) follows by (El)*. 3332. DEFINITION (TSS, U3). If I is an ideal on the set Ur then we call the set of all groups h such that h 2 PInv ( u ) for some u E 1 the g-filter determined by 1. 3333. LEMMA (TSS, U3). If z is the g-filter determined by an ideal 1 on the set Ur then (i) z is a g-filter; (ii) (e) E z iff Ur E I v (3x
E
(iii) z is symmetric iff Inv (I)
Ur) (Ur - {x} E
E
I) ;
z.
Proof. (i) and (ii) are easy. (iii) If ( e } E z then the equivalence trivially holds. Jf {el 4 z we show that Inv ( I ) = Inv(z). For u E Ur let h, = = PInv (u). We have i’h,, = 17;” for any permutation p . Let p E Inv (1) and h E z so that h 2 h, for some u E 1. We have p’h 3 ;’h,, = h2, and p u E 1. Hence p h E z. Conversely, let p E Inv ( z ) and u E 1, so that = ph, E z . There is a v E 1 such that 112~ 3 h,. We have p u G v ; for if p’u - u =k 0 then there are x E ;’u - u and y x such that y 4 v. Consequently, if p is the transposition of x and y then p E h, - h;,, a contradiction. Hence
+
p’u E
I.
3334. METATHEOREM. Axiom ( E l ) is not provable from the axioms of U3). Dentonstration. By Section 2, the axiom Ur = coo is consistent with TSS, (U3). In the theory TSS, (U3), (Ur = coo) we can prove that the g-filter z (P,, (Ur)) determined by the ideal P,, (Ur) (which is obviously well-symmetric) does not contain the group { e } . (Obvious.) Hence the F-definition Gt (M) with the specification M = HS (z (P,, (Ur)) determines a model of TSS, (U3), 1 (El) in TSS, (U3), (Ur M coo) (more generally in TSS, (U3), 1 Fin(Ur)). We have the following more precise result: (TSS
+-
K,, if I 3335. THEOREM (TSS, U3, Sym (z), JTSS/dGt (HS (2))). If Ur is the ideal of all sets of urelements of power less than K p and if z is determined by I then the following holds in the sense of Gt (HS (2)): 172
CHAP. 111 SEC. 31
3335
SYMMETRIC SETS
(1) Ur cannot be linearly ordered by any set relation. (2) If x E Ur and 1 (x linearly ordered. (3) K, < K,
< K,)
then (Ur - x)
< K,
and x cannot be
-,K, < Ur;
(4) neither K,
< Ur nor Ur < K,.
Proof. (1) follows from (2) since Ur < K, cannot hold in the model. (2) We first prove that for each x c Ur, x is in the model iff x E 1 or (Ur - x) E 1. If x c Ur and x E HS ( 2 ) then Inv (x) z PInv ( u ) for some u E 1 and hence either x c u or Ur - x c u . (Otherwise let u E x - u and w E Ur - (x u u); the transposition of u and w is in PInv ( u ) but not in Inv (x), a contradiction.) The converse is obvious. Hence x < K, or Ur - x < K, whenever x E Ur and x E HS (2). By 3329, x < K, v Ur - x < K, holds in the sense of Gt (HS (z)). To prove (2) it remains to show that if (Ur - x) < K, then no linear ordering r of x is symmetric. Suppose, on the contrary, that there is some e E 1 such that PInv (e) E Inv (r). Let u , u be two different elements of x - e ; we may suppose that ( u , u ) E r. i f p i s the transposition of u and u then p E Inv(r) and p’
3336
SYMMETRIC SETS
[CHAP. 111 SEC. 3
3336. METADEFINITION. Let T ( X ) be a godelian term without constants and with exactly one variable X . The term T (V) is a good deJinition of an ordinal number in a theory T(stronger than TSS, U3) if (1) T C- T (V) is an ordinal number,
(2) T t- (VX) (Mcl,, (X)
+
T(V)
=
T(X)).
T(V) is a good definition of a cardinal number if instead of (1) we have (1') T t- T (V) is a cardinal number. Remark. (2) is equivalent to the following:
T, Mcl,, (M), dTSS/dGt (M) I- T (V)
=
TGtCM) (V'*(M)) .
3337. METADEFINITION. (Relativizing quantifiers.) Let q ( x , 0 ) be a SF and let T be a term such that no variable occurring in T is bound in q(x, 0 ) . We define q T ( x , 0 ) (cp with the quantifiers relatiuized to T) by the following induction :
(a) (x
E
y)'
is (x
E
y ) , (x
=
yT) is x
(b) ( v & I I / ) ~ ~c psT & I I / T , ( l c p ) T
=
y ,
is 1 ( q T ) ,
(c) ((3x) q)Tis (3x E T) qT . Note that if T is a constant, M say, then one can prove the following for any closed SF cp: TSS', Mcl (M), dTSS/dGt (M) t (x* = x &
0)
4
( cpM(x,0 )
E
q*(x*,
0))
.
But we shall not make use of this fact. We need qTfor the following 3338. METADEFINITION. Let cp(x) be a SF with one free variable x; suppose that a i s a constant which is defined in a theory T by a good definition of an ordinal number. We say that q ( x ) is a-boundable if Tt ~ ( x = ) E qpux(x).(For the definition of p: see 3140.) Until now we have mainly been concerned with permutations of urelements; this can however be generalized. 3339. DEFINITION (TSS + U3). A set x is called a set of indioiduals (Ind (x)) if all elements of x have the same rank. 3340. LEMMA (TSS + U3). If x is a set of individuals and if p is a permutation of x then there exists a unique automorphism F of Ker'(x) such that F r x = p . 174
CHAP. I11 SEC. 31
SYMMETRIC SETS
3344
The proof is left t o the reader. If, for any y E Ker’(x), we denote by Inv, ( y ) the set of all permutations of x such that the corresponding automorphism keeps y fixed (i.e. F’y = y ) , then we can prove in a similar way that Inv, ( y ) is a group of permutations of x. If x is a set of individuals and if z is a g-filter on x then we define:
3341. Remark. Ur is a set of individuals; the set of all infinite subsets of o is a set of individuals etc. Notice that if x and y are sets of individuals, if x w y and if neither x nor y contains urelements, then for each CL the sets p: and pi are isomorphic w.r.t. E and E. Urelements must be excluded because the singleton of an urelement is this urelement itself. In what follows we specify a certain rather general kind of description for g-filters. 3342. METADEFINITION. Let o(x, z ) be a SF with two free variables and let T (V) be a godelian term which represents o (w.r.t. x, z in TC). We say that o(x, z ) well describes a symmetric g-jilter (or that T (V) well defines a class of symmetric g-Jilters) in a theory T stronger than TSS, U3 if the following is provable in T:
(1) T (V)
E ((x,
(2) T ( V )
+ 0,
(3)
z ) ; z is a symmetric g-filter on x & (Vy E x) (Inv, ( y ) E z)},
Z~&XEW(T(V)).+~EW(T(V)),
(4) T (V) n Ker
=
T (Ker).
3343. Remark. (1) is equivalent to w(x, z ) z is a well- symmetric g-filter on x; (2) is equivalent to (jx, z ) o(x, z ) ; (3) is equivalent to x x y & & (3.) o(x, z ) . (3.) o ( y , z). -+
-+
3344. METADEFINITION. We say that an a-boundable formula q ( x ) has a permutation model in a theory T stronger than TSS, U3 if there exists a formula o(x, z) which well describes a symmetric g-filter and such that T k Ind (x)& w(x, z ) . + qF5x(”’(x). (We also say that o determines a permutation model of q.) 175
3345
SYMMETRIC SETS
[CHAP. 111 SEC. 3
3345. In the following two examples we shall suppose that a formula
q ( x ) is a-boundable (for some a defined by a good definition) and q ( x ) has a permutation model determined by w(x, z ) (everything w.r.t. T S S U3).
+
We shall show that then the formula ( l x ) q ( x ) is consistent with T S S + + U3. (In Chapter VI we shall show that it is consistent even with T S S + D3.) 3346. Example 1 . The axiom (32) w(Ur, z ) is consistent with TSS
+ U3 + E2.
’
+
+
We proceed in the theory T S S D3 + E2. The formula (3x, z ) o(x, z ) is provable; choose a fixed x, z such that o(x, z) and define the relation r = { ( t i , u>; u E x}. Consider the relation R = RUpw (r) and add aTSS / / 3% (R). By Theorem 3221 there exists a unique real isomorphism F of Ker and Ker* w.r.t. E, R; therefore we have F”(x, z) E* T* (Ker*), hence, by (4) i n 3342, F”(x, z) E* T* (V*) and therefore w*(F”x, F”z). Since x = Ur* and x z F”x, we have x NN* F”x by Theorem 3221 and, by (3) in 3342, there exists some z* which is a set in the sense of %(R) and which satisfies w*(Ur*, z*). This completes the proof and hence the specification of ‘3(R) by w(x, z) & R = RUpw (I x) determines a model of the theory T S S + U3 + E2 + (3z) w(Ur, z ) in T S S + D3 + E2.
r
+
3347. COROLLARY. The axiom q(Ur) is consistent with T S S U3 + E2. This follows from the fact that q(Ur) holds in Gt (HS(z)) as a model in TSS U3 E2 o(Ur, z); qFQwr(Ur)holds in the sense of HS (z) since ,ij:’(z) is rjp in the sense of HS(z).
+
+
+
3348. Example 2 . We say that a is a set of a-collections of urelements (col ( a , a)) if a is a disjointed family of sets of urelements, u = Ur and every element of a is of power X,, where k’, is the least regular cardinal greater than the cardinality of pz. The axiom (3z,a)(col(a, a) & o ( a , z ) ) is consistent with T S S + + U3 E2. Proceeding in the theory T S S + D3 + E2 and fixing w(a, z) & Ind (a) we define
u
+
w, = Min {ay;coy regular &
{<<x,u>, (x, a>>; x R = RUpw (r) .
r
=
rjt < coy) ,
E a & u E cop},
Further we add the definitions dTSS/d%(R). We have Ur* = a x cog and (V = Ker (Ur))*. If F is a real isomorphic embedding of Ker into C (R) 176
CHAP. 111 SEC. 31
SYMMETRIC SETS
3349
then we have F”(a, z) E* T* (Ker*) E T* (V*) and hence w*(F”a, F”z). If x E: a let h’x be the unique element u E C (R)such that EdR(u) = {<x, p); p c as>and let a* = h”a. Then a* x a so that a* x*F”a and therefore there exists z* such that w*(a*, z*). Furthermore, if a * is defined by the same definition as a but in the sense of % (R)then we have a* = F”a and F”p: is equivalent to p$ in the sense of ‘ill (R);hence F”afl, is, in the sense of ‘92 (R), the least regular cardinal >jj$ and in addition &’”afl M * o* for u* E* a*. Thus we have co1* (a*, a*). In the following we continue to assume that a has a good definition, that w well describes a symmetric g-filter and that cp is a-boundable and has a permutation model determined by m. 3349, (1) DEFINITION (TSS, U3, E2, col (a, a)). If 4 denotes the extension of q E g (Ur) to a (i.e., 4 = ij /‘a) then we define hl = { q E g (Ur); 4 E h] for each group h of permutations of a. If w(a, z ) let z-1 = { k ; k is a group of permutations of Ur& (3h E z) (3e E Pk (Ur)) ( k 3 hl n PInv, (e))).
(2) THEOREM (TSS, U3, E2, col (a, a)). If w(a, z) then ZJ is a well-symmetric g-filter on Ur such that a E HS ( z l ) and jjt n HS ( z l ) = pt(z).
(3) COROLLARY (TSS, U3, E2, col (a, a) & w(a, z), dTSS/aGt (HS (zi))). Put a* = a; then we have cp*(a*). Proof of ( 2 ) . hl is the set of all permutations of Ur whose extensions to a belong to h. Clearly, z l is the least g-filter containing all the groups h i where h E z and in addition all the groups PInv,, (e) where e E Psn(Ur). It is not difficult to show that p . ( h l ) . p - l = (fi . h . @-I) -1 for each p E E Inv”, (a) and that Inv, ( z ) 1 c Inv”, ( z l ) ; hence from Inv, ( z ) E z we obtain Inv, ( z l ) E zJ so that 2-1 is a well-symmetric g-filter. Further we have h l E k l = h E k ; in fact, the implication t is trivial and conversely, if q E h - k then any permutation of Ur which extends to q is in h i but not in k l . It follows that
(*)
hEz
3
hJ
E z 1 for
any group h of permutations of a.
Indeed, if 12 E z then h l E zJ; on the other hand, if hJ E 21 then for some k E z and e E P,,,(Ur) we have h l 3 k l n PInvu, ( e ) .Put 6 = ( u E a; u n e =/= =I0). = Then 6 is a finite subset of a, whence PInv, (t?) E z ( z being well-symmetric). Consequently, we may suppose without loss of generality that k is included in PInv, (&)(if not, replace k by k n PInv, (6)). We prove h i 2 3 k l . If p E k l then fi E k and @ is identical on &. We can change p to p1 177
3350
[CHAP. In SEC. 3
SYMMETRIC SETS
and p1 is identical on e. Then pI E kl n PInv,(e), hence such that 8 = p1 E h i and fil E h. We have hl I> k l and so h 2 k. This proves h E z. Now observe that the following holds true:
6.)
x E Ker’ (a)
+
[Inv, (x)J
=
Invu, (x) n Inv,, (a)] .
Indeed, if p E Inv, (a) then p extends to Ker’(a) in the same manner as fi does. Consequently, we have the following: ( w * ) x E Ker’ (a)
+. Inv, (x) E z = Inv,,
(x) E
zl .
In fact, if Inv, (x) E z then Inv, (x)l E z.1 but, by (**), Inv, (x)J c Invur (x) and so Inv,, (x) E z l . On the other hand, suppose Inv,, (x) E z l . Obviously, Inv,, (a) = g (a)J E zl; hence Inv, (x)l = Inv, (x) n Inv,, (a) E z.1 and, by (*), Inv, (x) E z . For every x E Ur we have Invu, (x) E z l . The same holds for every u E a (since z is well symmetric) and Inv,, (a) E z l so that a E HS (z.1). Finally, we obtain from (***) by an easy induction that, for every p E On, pi n HS(zl) = = &z). This completes the proof. The proof of (3) is analogous to the proof of 3347. 3350. LEMMA.Under the assumptions of the preceding Theorem let x be a set of urelements such that x < K,, and let k = h 1n PInv (e) for some h E z and e E Pfi,,(Ur).If p E k then there exists q E k such that fl = 4 and for every i E x, q’i = L v 9’1 4 x. Proof. Let u E a and let fi’u = o. Consider (x n u) - e and let y E 0 be a set of the same cardinality as (x n u) - e and disjoint from x u e u p”((x n u) - e). Finally, let f be a one-to-one mapping of p”((x n u) - e) onto y. We define q’i
-e, = f - ’ ( p ’ i ) for i E p - ” ’ ( y ) , = p’i otherwise in u .
=f’(p’i)
for
i
E (x n u )
The mapping q has the required properties. We shall now investigate certain relationships between complete Boolean algebras and complete Boolean algebras in the sense of a permutation model-class HS (z) where z is a symmetric g-filter on Ur; similarly for 178
CHAP. 111 SEC. 31
3352
SYMMETRIC SETS
r
separative orderings. We first observe that if p is a permutation of Ur and b is a complete Boolean algebra then p E Inv,, (b) iff b is an automorphism of b as a complete Boolean algebra; a similar remark holds for separatively ordered sets. For a given symmetric g-filter z let us denote by P the modelclass HS (z) of all sets hereditarily symmetric w.r.t. z. Let a be an ordered set such that Inv (a) E z (i.e. a E S (z)). If we set a1 = ( a n P, I n P ) then Inv(a,) 2 Inv (a) n Inv(z); hence a, E S(z) and, since a , c P, we have a, E P . Clearly a, is an ordered set (in the sense of Gt(HS (z))). Suppose a is separatively ordered. The ordered set a, need not be separatively ordered (in the sense of Gt(HS (z))), but, if it is, it determines a complete Boolean algebra in the sense of Gt(HS(z)) and it is important to determine the relationship between this algebra and the algebra determined by a. We therefore make the following definition:
3351. DEFINITION (TSS + U3). Let a be a separatively ordered set and let z be a symmetric g-filter on Ur. As in the preceding paragraph we let a, = ( a n P, n P). We say that a is fruitful w.r.t. z if it satisfies the following condition: (1) Inv (a) E z ;
(2) a I is separatively ordered; (3) For every u E a and every h E z such that h -C Inv (u ) there exists V E a , such that v 5 V $u, the join being taken in an arbitrary complete Boolean K h
algebra with a as base. 3352. THEOREM (TSS, U3, WSym [z), aTSS/d(HS(z))). Let a be a separatively ordered set which is fruitful w.r.t. z. If b is a complete Boolean algebra with the base a and if 6 , is a complete Boolean algebra in the sense of Gt(HS(z)) with the base a, then there exists a one-to-one operation-preserving mapping of b , into b, i.e. a one-to-one mapping g such that g’( - u) = = -g’u whenever u E b , and g’(A,q) = Ag”q whenever q E P* ( b , ) (where and A, are the operations in b, and -, A are the operations in b). Proof. Without loss of generality we may assume that the algebras b and b, are constructed in the same way as in the proof of Theorem 2442 ; thus, in particular, the elements of b and b , are saturated subsets of the corresponding bases and the meet coincides with intersection. Every permutation in Inv (a) induces an automorphism of b as a complete Boolean algebra; a similar remark holds for b,. We recall that u E b is said to be
,
-,
179
3353
[CHAP. I11 SEC. 3
SYMMETRIC SETS
symmetric if Inv ( u ) E z. If u is a symmetric element of b we set f’u = = 11 n P. We shall show that g = Cnv ( f ) is the required mapping. 1) We prove that W ( f ) c b,, i.e. u n P E b , whenever u is symmetric. Since Inv (u n P) 2 Inv (u) n Inv (z), we have (u n P) E P . It is clear that if y* 5 x* E u n P then y* E u n P . Suppose that (Vy* 5 x*) (3t* 5 y*) (t* E u n P); we prove that x* E u n P . If we suppose the contrary then x* r# u and there exists y x* such that the segment determined by y is disjoint from u . If we set h = Inv (x*) n Inv ( u ) n Inv (a) then for each p ~ h the segment determined by i ’ y is disjoint from u and i ’ y x*. Since a is fruitful there exists y* 5 V i ’ y and, since the segment determined by y* pEh
is disjoint from u , we arrive at a contradiction. 2 ) We now prove that b , c W ( f ) . If u E b, then we define XELlZE
.xEa&(tJy
5
x)(3
5
y)(3w
>= t ) ( W E U ) .
Clearly u is a saturated subset of a and hence u E b. We shall prove that u n P = u. Let x* E u and let y* be an arbitrary element such that y* 5 x*. There exist t 5 y* and w 2 t such that w E u. If we let h = Inv (y*) n n Inv ( w ) n Xnv ( a ) and t* 5 V i;’t then t* 5 y and t* 5 w so that t* E u . peh
This proves that z4 n P E u ; the converse inclusion is trivial. Evidently, u is symmetric since Inv ( u ) 2 Inv (a) n Inv (zI). Hence u E b and u = f ’ u . 3) We now prove that f is one-to-one. Suppose that u , u E D ( f ) and that u 1’. Then there exists x such that x E u - u, say. If we let h = Inv ( u ) n n Inv ( u ) n Inv (u) and x* 5 V i ’ x then x* E u and x * 4 D. (Exercise.)
+
pSh
4) Finally we prove that f preserves the operations. If k E P* (b,) and k’ = Cnv ( f ) ” k then A , k = n u = 0 ( u n P ) = (nu ) n P = Ak‘ n P = uek
uok‘
uEk*
= f’(Ak’), since Inv ( n k ’ ) 2 Inv (k‘) 2 Inv ( k ) n Inv ( a ) n Inv (z). larly, if u is symmetric then - u is also symmetric; we show that
Simi-
u)} n = {x* E a , ; 7 (3t* 5 x*) (t* E u n P ) } . The inclusion c is obvious. Let x * be an element of a , such that 7 ( 3 t * 5 - x”) (t* E u). Suppose there exists an element t of u such that t 5 x*. 5 There exists t* 5 V i ’ f , where h = Inv (u) n Inv (x*) n Inv (a); hence { x E a ; 1( 3 t
5 x* and
5
x) ( t
poh
E
,
which is a contradiction. Hence - (u n P) = ( - u ) n P and the theorem is proved. t*
t* E u
3353. Examples. Let 6, ~ K e rbe a complete Boolean algebra having more than two elements and let 1 be an ideal on Ur. We shall investigate 180
CHAP. 111 SEC. 31
the product
Ur
3356
SYMMETRIC SETS
b,
of
b, over 1 (as usual b,
is the set of all elements of 6 ,
except for 4, and lb0, with the canonical ordering). Let z be a well-symmetric g-filter on Ur; we denote b, by a = ( a , 5 ) and HS ( z ) by P.
n’ Ur
=
3354. LEMMA(TSS + U3). Inv(1) c Inv(a); if Inv(1)Ez then a , ( a n P, 5 n P) is a symmetric separatively ordered set.
=
Proof. (We proceed in TSS, U3, WSym (z), aTSS/aGt (HS (z)).) Put = HS (z). If p E Inv ( 1 ) and f~ a then D (i’f) E 1 and ( u , i ) ~f = = ( u , p’i) E i’f;hence Inv (1) E Inv (a). If Inv (I) E z then a is symmetric and a , is also symmetric. It remains to prove that a, is a separatively ordered set. Let f *, g*, h* and k* denote elements of a,. Suppose that (Vg* 5 f*) (3h* 5 g*) (h* 5 k*); we prove that f* 5 k*. To do this we first prove that D (k*) _C D ( f * ) . Suppose on the contrary that there exists i E D (12”)- D(f*),andlet-,k*’r = x ~ b , . I f w e l e t g * = f *u { ( x , i ) } t h e n g * ~ P , since Inv ( 1 ) E z. The inequality h* 5 k* does not hold for any h* 5 g* and so we have a contradiction. We shall nGw prove that f *’i k*’i for any L E D (k*), where 4 , is the canonical ordering of b,. If we let x <,f*’i and g* = ( f * - ( ( f * ’ i , 1 ) ) ) u {(x, 1 ) ) then g* E P and there exists h* 5 g* such that IT* <= k*; hence h*’i 5 x and h*’i 5 k*’i. Thus we have (Vx < , f * ’ i ) ( 3 y x) ( y k*’i) and, since 4, is separative, f * ’ c k*’i.
P
<,
<,
<,
<,
<,
3355. THEOREM (TSS, U3). Let 1 = I,, be the ideal of all subsets of Ur of power less than K,, where K, is the power of Ur and let z be the g-filter determined by this ideal. If b, E Ker is a complete Boolean algebra and if a = H‘h, then a is fruitful w.r.t. z. Ur
Proof. By Lemma 3354 a is symmetric and a1 is symmetric and separative. Since every element of a is hereditary symmetric we have a , = a and the last condition for fruitfulness is satisfied vacuously.
3356. METATHEOREM. Let a be a well defined constant for an ordinal number, let o(x, z ) well describe a symmetric g-filter and suppose that q ( x ) is a-boundable and has a permutation model determined by w . Then the following assertion is provable in TSS,U3, E2, col (q, a)& o(q, z): Let b, E Ker be a complete Boolean algebra, and let zl be the filter on Ur defined in 3349. If 1 is the ideal of all subsets of Ur of cardinality less than K,, where Ur is of power K,, then a = ( a , 5 ) = b, is fruitful w.r.t. zl.
n’ Ur
181
335 7
[CHAP. I11 SEC. 3
SYMMETRIC SETS
Demonstration. We proceed in the above-mentioned theory. Let b be an algebra with base a. Let k = hJ n PInv (e) where h E z and e E Pfin(Ur). Let u E a and put v = u e ; we prove v E a , and u 5 V $u. Evidently, v
r
PEh
is symmetric. If w E a and w 5 v then we shall find a p E k such that i ’ u A A w =I 0,; = this will prove v 5 5’u. Let p be a permutation such that (1)
v
psk
is identical on q, (2) p E PInv (e) and (3) D(w) n p”(D(u) - e) = 0. (On each x E q, p maps D(u) - e onto a subset of x disjoint from D(w) u e; recall x % &!) Then p E k and w A i ’ u =/= 0,. Finally we prove a lemma which will be useful later on:
3357. LEMMA (TSS, U3). Let 1 = I,, be the ideal of all subsets of Ur of power less than N,, where N, is the power of Ur. Let b, E Ker be a complete Boolean algebra and let b .be an algebra with base a = For e E I
n’bo. Ur
let b/e be the subalgebra of b with base ale = n f t e b o (We . suppose that the Ur
algebra b is in “canonical form”, i.e. its elements are the saturated subsets of its base.) ( 1 ) If u E b and Inv (u) 2 PInv (e) then u E b/e. ( 2 ) If q 5 b is a disjointed system such that Inv(q) 2 PInv(e), then q E b/e. Proof. (1) We prove that u e u whenever v E a , u 5 u, u E b . Given v E a such that u 5 u, let w be an arbitrary element of a such that w 5 v e. Set e , = D ( u ) and e2 = D (w);let p be a permutation identical on e and such that the image of (el u e 2 ) - e by p is disjoint from (el u e2) - e. Then p E PInv (e) so that $’u 5 u, and w A g ’o =!= 0,. Hence setting J = = w A 5’v we have: for each w g u e, there is a J w such that W u. If follows that u e S u and hence u is the sum of all v e such that u 5 u. Consequently u E b/e.
r
s
r
/”
r
r
r
( 2 ) Again it suffices to prove u e iu for u E q, u 4 u, v E a. We set e , = D (u); let p be a permutation which is identical on e and maps e , - e onto a set disjoint from el - e. We have A v =!= 0, and $u E q . Since $’u A u =# 0, and q i s disjointed, we have $u = u and i ’ u u. We prove that u e 5 U where U is the join of all $u for which p has the property just described. Let w be a n element of a such that w 5 D r e ; we shall find some p having the property and such that w A i;‘v 0,. To do this it suffices to take a permutation which is identical on e and which maps the set (el u D (w)) - e onto a disjoint set.
r
i’u
3v
+
182
s
CHAP. 111 SEC. 41
3401
DEFINABLE SETS
SECTION 4
Definable sets and the consistency of the axiom of choice the third axiom of choice
m);
In the present Section we shall prove the consistency of the axiom of choice. Most of our work will be done in the theory TSS,(D3); we shall define a certain model-class HDf such that the axiom of choice holds in the seDse of Gt (HDf), i.e. there is a relation R such that Realmf ( R ) and R is a well-ordering of HDf. We start with the class So,of all sets pn such that I is a limit; all of these sets are V-like. The closure of the class So,under the godelian operations will be denoted by Df; the elements of Df will be called definable sets*). can be well-ordered and so the same is true of Df. In fact The class Son we can define a well-ordering of Df. To speak imprecisely, Df is the maximal definable class with a definable well-ordering; a precise version of this statement is given below. The class Df itself need not be a model-class; however, the class HDf of all hereditarily definable sets (i.e. definable sets of definable sets of definable sets ...) is a model-class. The construction of this class can be generalized in a natural way and we can define the class of all (hereditarily) definable sets relative to a given real class X . We now carry out in detail the program sketched above. 3401. DEFINITION (TSS’).
*) Although the term “definable” is not unnatural in this context, the notion Df is not intended as an explication of the intuitive notion of definability.
183
3402
DEFINABLE SETS
Cgn+l (x)
=
Cg (x)
=
Cgn (x) u (Pi(u, 0); U , V
E
[CHAP. III SEC. 4
Cgn (x), i
=
1, ...) 7)
9
U Cgn (x)
nEm
(closure under the godelian operations). 3402. LEMMA(TSS’). If Proof. By induction.
X E
3403. DEFINITION (TSS’). Df
sets).
C g ( y ) and y =
E
C g ( z ) then x~Cg(z).
(J Cg ( { p n } ) (the class of all dejinable
nsoolr
3404. LEMMA(TSS’). If X is real and closed and if Po,, c X then Df c X . Proof. Obvious. 3405. LEMMA (TSS‘). The class Df is closed.
Proof. Lct x and y be in Df; suppose that x E Cg ( { p a } )and y E Cg ( ( p s } ) ; (throughout the present Section, a, p, y ..., will denote limit ordinals). By Lemma 3402 it suffices to prove that pa and p s are both in C g ( { p , } ) for some y. We prove this by the Reflection Principle. Suppose that there exist pG and p s for which there is no y such that both p a and ps are in Cg ((p,,}). There is a godelian term T (V) such that T (V) = {<pa,p s ) } where a < p and (a, S } is the Sd-least pair of ordinals with the property 1 ( 3 y ) ( p a , p s E E Cg ( ( p , , } ) ) . By the Reflection Principle there exists 6 > CI, p such that T (p,) = T (V) n pa = ( ( p a , p s ) } ; this is a contradiction, since then pa, ps E E
cg ( { P b H 9.
3406. METATHEOREM. There is a godelian term T ( X ) without constants such that the following is provable in TSS’ :T (V) is a one-to-one mapping of Df onto On. Demonstration. I t suffices to define in TSS’ a 1-1 mapping of C g ( { p , } ) into o. It is then easy to define a mapping S = S(V) which enumerates all elements of Df by means of pairs (a, n } as follows: S’x = = (a, n} = x is the n-th element in C g ( { p , } ) C g ( { p , } ) . Using the
u
y
ordering Sd we can define a 1-1 mapping T(V) of Df onto On. We shall now define in TSS’ a 1-1 mapping of C g ( ( p , } ) into o.First we define a set t = t(V)together with a well-ordering o f t . Let to = {0},t n + l = ( ( i , u , u ) ; *) Cf. Metatheorern 3128.
184
CHAP. 111 SEC. 41
3408
DE.FINABLE SETS
i = 1, ..., 7, u, u E
u ti},
isn
f, = t,, - (J ti and t i
=
u ti. The
icm
sets f, are
mutually disjoint. The set to is well-ordered by r , = 0. Suppose ti as that t i is well ordered by r,; we define a well-ordering r , + l of
u
u
ijn
ijnfl
follows: (u,u)Er,+l
= (u,u>Er, v
( u ~ U t ~ & u ~ t , , v, )
v (u,
(3j, k, 211,
E f,+
1 %6
isn
( u = ( j , u,, u 2 ) & u = ( k , u l , u z ) & [ j
u2, u 1 , u 2 )
< k v (‘j= k &
( u , , u,> E r,) v
v ( j = k & u1 = u1 & (u,, u 2 ) E r,)])) .
Thus the elements of t are well ordered first according to the indices of the sets t, to which they belong; if two elements are in the same set t, thenwe order them according to their first members (number 1, .. ., 7); if the first members are the same then we order them according to their second members which are already ordered by hypothesis; finally, if the second members are the same then we order them according to their third members. The relation r = r, well orders t and since each t, is finite, r is iso-
r
u
,El0
morphic to E w,; hence there is a godelian term i (V) which defines a 1-1 mapping i of wo onto t. For a limit number a we define a mapping f of t onto Cg ( ( p u } ) as follows: f ’ 0 = pu and f ’ ( i , u , u ) = Fi( f ’ u , f ’ u ) . Finally, there is a godelian term R (V, a) which defines a one-to-one mapping R of Cg ( ( p a } ) into wo as follows: R’x = n E n is the least number such that f ’i’n = x. (The reader should verify in detail that R is a godelian term and that we can prove in TSS’ that R (V, a) is a one-to-one mapping of Cg ( ( p l l } ) into no.)This completes the demonstration.
3407. DEFINITION (TSS’).A set x is in HDf if x E Df and Unv (x)
c Df.
3408. THEOREM (TSS,D3, aTSS/dGt (HDf)). HDf is a model-class and Axiom (E2) holds in G,t(HDf). Proof. 1) HDf is closed, since Df is closed.
2) HDf is complete by definition. 3) HDf is almost universal. If x c HDf then x E HDf n pu for some a. HDf is defined by a normal definition; i.e. HDf = T(V) where T(X) is a certain godelian term. By the Reflection Principle there exists 1. > a 185
3409
DEFINABLE SETS
[CHAP. 111 SEC. 4
such that z = HDf n p A = T (pa). Hence z E Cg ( { p A } ) and z c HDf; hence x E z and z E HDf. 4) If T(V) is the one-to-one mapping of Df onto On constructed by Metatheorem 3406, then T(V) HDf is a one-to-one mapping of HDf into On; hence there is a godelian term ? (V) which defines a one-to-one mapping of HDf onto On since HDf is a proper real class. By the following Metatheorem we get RealGt(mf)(? (V)):
r
3409. METATHEOREM. The following is provable in TSS godelian term T (X) without constants:
+ D3 for any
(Vx E HDf) (T (V) n x E HDf) .
Demonstration. Let T (X) be such a godelian term; we prove the above formula in TSS + D3. If x E HDf then T (V) n x E HDf; hence it suffices to prove T (V) n x E Df. By the Reflection Principle there exists pu such that x c paand T(V) n pa = T(pa); hence T(V) n x = T(V) n pa n x = = T (pa) n x E Df. 3410. COROLLARY. Axiom (E2) is consistent with TSS, D3; the model IDir (dTSS/dGt(HDf)) is a model of TSS, D3, E2 in TSS, D3, dTSS/aGt (HDf).
The above construction may be generalized. We start now with elements of some fixed real class X (in addition to the pa). 341 1 . DEFINITION (TSS').
3412. Remark. (TSS'). Df = Df (0) = Df (Po,,). 3413. LEMMA(TSS'). If Df(X) E Df(Y) then HDf(X) E HDf(Y).
Proof. Obvious. 3414. THEOREM (TSS, D3). HDf (X) is a model-class.
Proof. 1) Df (X) is closed and hence HDf (X) is closed. If
186
CHAP. 111 SEC. 41
DEFINABLE SETS
3418
then pA, and pL2 are both in some Cg ( { p l ) ) and Fi (x, y ) E Cg (el u e2 u u ( P A J n PA}) since x n PA, = (x n Pn) n P A , . 2) HDf (X) is complete by definition. 3) HDf (X) is almost universal by the Reflection Principle. Indeed, if x E c HDf (X) n pQ and HDf (X) = T (V, X) then there exists pn 3 p a such that p A n T (V, X) = T ( p l , X n p r ) and pn n T (V, X) 2 x; hence p A n n T ( V , X ) E Df(X). Since p A n T(V, X) c HDf(X) we have p l n n T (V, X) E HDf (X). 3415. THEOREM (TSS'). Let X, Y be reaI classes and suppose Y c X. Then Df (Y) E Df (X) iff p a n Y E Df (X) for every limit number a. Pruof. If Df (Y) G Df (X) then the condition obviously holds. If the condition holds then for any e E Pfin(Y) and any c1 E On,, we have e u u { p a ,Y n pa} c Df (X) and hence Cg (e u { p a , Y n pa>) -C Df (X); thus Df(Y) E Df(X). 3416. COROLLARY (TSS'). 1) Df(0) E Df(X). 2) If z E Df (X) then Df (X n z ) E Df (X). 3417. METATHEOREM. Let 0 (Y, X , Z , , 0 ) be a godelian term (containing no constants). The following is provable in TSS, D3, Real (X), dTSS/dGt (HDf (X)):
Let Z,, 0 c HDf (X) and 0 (V, X, Z , , 0) E HDf (X). If the classes Z , , 0 are real in the sense of Gt (HDf (X)) then 0 (V, X, Z , , 0) is also real in the (X1sense of Gt (HDf (X)). In particular, if X c HDf (X) then RealG*(MH(X)) Denionstration. We proceed in TSS, D3, Real (X), dTSS/dGt (HDf (X)). If x € H D f ( X ) then we prove x n O ( V , X , Z , , O)EHDI(X). By the Reflection Principle there exists p A such that x n 0 (V, X, Z,, 0 ) c pi. and 0 (V, X, Z , , 0 ) n pn = 0 (PA, X n pi, Z , n p i , 0 ) . Since x, p l , X n pn, Z, n p i , 0 E Df (X) we have x n 0 (V, X, Z , , 0) = x n 0 (pn, X n p i , 2, n p A , 0 ) E Df (X); moreover, x n 0 (V, X, Z , , 0 ) E HDf (X) since x c c HDf (X) and therefore x n 0 (V, X, Z , , 0) E HDf (X). 3418. THEOREM(TSS, D3, Real (X)& X E On, dTSS/dGt (HDf (X))). Axiom (E2) holds in the sense of Gt (HDf (X)). Proof. We construct a real mapping of HDf (X) into On. We have a regular real well-ordering Sd of Prin(On) and hence a real regular wellordering of Pfin(X). For every e E Pfin(X) and every I E On,, we can define a one-one set-mapping R (e, A) of Cg (e u {pn. X n p i ) ) onto w. The theorem follows in the same way as Theorem 3408. 187
341Y
DEFINABLE SETS
[CHAP. 111 SEC. 4
We shall now return to the class HDf = HDf (0). We know that HDf is defined by a godelian term (in TSS + D3) and we have defined a constant S such that TSS D3 t (S is a 1-1 mapping of Df onto On). We shall show that the class Df is, in a sense, maximal among classes having this prdperty.
+
3419. METATHEOREM. If T(X) and S ( X ) are godelian terms without constants then the following can be proved in T S S + D3: If S (V) is a well-ordering of T (V) then T (V) E Df. Demonstration. We proceed in T S S + D3. Suppose that T (V) - Df =# 0 and let A (T (V) - Df) be the set of elements of least rank in T (V) Df. 0) From S (V) we may define a selector, i.e. a mapping (V) such that (Vx (x E T (V) -+ 9 (V)’ x E x) & Un ( S (V)). Set z = S (A (T (V) - Df)). Clearly z = o (V) where o is some godelian term; by the Reflection Pririciple there exists 2 E On,, such that z E pn and pi n o (V) = o ( p i ) = z . Hence 2 E Df, a contradiction.
+
3420. AXIOM OF DEFINABILITY (E3). V 3421. COROLLARY (TSS (b) (E3) implies (E2).
=
Df.
+ D3). (a) (E3) implies V = HDf
3422. Remark. The consistency of Axiom (E3) will be established in the next Section. The above Metatheorem shows that Axiom (E3) is the weakest of the so-called strong axioms of choice. By a strong axiom of choice we mean a n axiom of the form “T(V) is a one-to-one mapping of V onto On” where T ( X ) is a godelian term without constants. Every strong axiom of choice implies V c Df and hence Axiom (E3). The authors feel that the notion of a strong axiom of choice corresponds exactly to the original intuitive conception of the axiom of choice: a rule can be presented which assigns to each nonempty set one of its elements.
To conclude this Section we show that there is a godelian term T(X) without constants and such that the following is provable in T S S + D3:
(Vx E Df) (3a E On,,) (x
=
T (p,)) .
We may call such a term universal f o r dejinable sets. (This result will hot be used anywhere in the sequel.) 3423. METALEMMA. There is a godelian term S ( X , Y ) such that TSS 188
+ D3 k (Vx
E
Df) (3x, fi E On,,) ( c i >
p & x = S (pm,ps)) .
CHAP. 111 SEC. 41
3424
DEFINABLE SETS
Demonstration. Clearly, we can define a godelian term G ( X ) such that TSS
Hence TSS
+ D3 t Un, (G (V), Po,, Df) .
+ D3 k (Vx E Df) (IF E On,,) (x = G (V)’ p s ) .
By the Reflection Principle we have in TSS + D3: There exists a > such that x E p a & G (V) n p . = G (p,) and hence
/3
3424. METALEMMA. There are godelian terms R ( X ) and Q ( X ) such that
Sketch of the demonstration. (1) We proceed in T S S = ( ( a , p); a,P E On& a > B},
+ D3. Let Pa =
SdP = (<<.I, PI>, (a22 Pz>>; ( a , , B,>, (a29 P z > E Pa& @ I 2 BI} S d {EZ, B z ) } (see 2226 for the definition of Sd). Sdp is a real regular well-ordering of Pa; hence we can define a unique isomorphism F between Pa and On w.r.t. Sdp and E. The isomorphism F has the following properties:
Further, let 9 be as in 3131 i.e., @’a = pa. (2) There are godelian terms Ti ( X ) (i = I , . .., 4) such that the following is provable in TSS D3:
+
(c)
TI (V)
(d)
for every A E On,,, Ti ( p n ) = p A A T i(V) .
=
Pa, T, (V)
=
Sdp, T3 (V)
=
F, T4 (V)
=
9,
(Hint: for i = 1 , 2 there are RF’s pi(x) such that TSS
+ D3 t x E Ti (V) = q i ( x ) ; 189
3425
DEFINABLE SETS
[CHAP. I11 SEC. 4
for i = 3 , 4 there are RF’s qi(x,f, z ) such that TSS
+ D 3 t - A ~ O n , , & x ~ p , .. x e T i ( V ) = ( 3 f ~ p ~ ) q ( x , f , p * ) . -+
Hence the result follows by 3126. (3) Let To ( p A ) = D (T, (p,)). Then the following is provable in TSS + D3: Let ( a , f?)E Pa, a, fi E On,, and let F’ (a, f?)= I . Then I E On,,, a = U(W (To (pA))) and f? = U[((a> x A) n D(To(pA))]. Further, if x E _C p i then u x = W (E (p,) f x). Hence we can find godelian terms Ro (X), So ( X ) such that
+
TSS + D3 t- (a, f?)€ P a & a, PEOn,, . -+ E On,,) (a =
Ro (PA)8L P
=
so (PA)) -
(4) The following is provable in TSS + D3: Let and let I = F (a, p); then 9 n p , = T4 (p,), {pa}
= (p*a} =
(PSI = IY’P) Pa =
=
(a, f?)E Pa, a,
(9n PA) n ( P A x (Ro (PA)))
7
(9n PA) n (PA x (So (PA)))
7
U(PaI3
PS =
fi E OnIl
U{P~}.
Hence we can find godelian terms R ( X ) , S ( X ) such that TSS
+ D3 I-
(a,
fi> E Pa
+
(31 E On,,) ( p a = R ( p i ) 8L p s
=
S (P,)) .
This completes the demonstration.
If S, R and Q are as in the two preceding Metalemmas and if we let T ( X ) = S (R( X ) , Q ( X ) ) then we have the following 3425. METATHEOREM. There is a godelian term T ( X ) such that TSS
+ D3 I- (Vx E Df) (3a E On,,) (x
=
T (pa)) .
3426. COROLLARY. There is a natural number p such that the following can be proved in T S S + D3: ( { pEa U })). ( V X E D ~ ) ( ~ ~ E O ~ I ,C) g(iX i5P
The number p is determined by the complexity of the term T(X) and can actually be calculated. 190
3502
CONSTRUCTIBLE SETS
CHAP. 111 SEC. 51
SECTION 5
Constructible sets and the axiom of constructibility. The consistency of Axioms (33) and (Cont)
In the present Section we shall demonstrate the consistency of the axiom of Continuum and of the axiom of Definability. For this purpose (and others) we shall prove in TSS, D3 that for any set there is a least modelclass containing this set; in particular there exists a model-class which is included in all other model-classes. We shall show that a certain axiom of constructibility (Constr) holds in the sense of this least model-class; from this axiom we shall derive the axiom of definability and also the axiom of continuum. This is the principal aim of our work in the theory TSS, D3. We shall conclude this Section with a more general construction; we shall define for any real class X a model class which in a certain sense is least w.r.t. X . 3501. DEFINITION (TSS’). In accordance with Section 4 we define
Cg, (x) = x u {z; ( 3 ~ u, E X) ( Z =
u-u v z
=
D(U) v z
=
u u‘/
=
{u,U] v z = E ( u ) v
=
v z = Cnv(u) v z = Cnv,(u))).
3502. METALEMMA. The formula y = Cg, (x) is restricted in TSS‘. Demonstration. The formula y E Cg, (x) is clearly restricted; hence the formula y c Cg, (x) is restricted. To show that Cg, (x) E y is restricted notice that this formula is equivalent to the formula x c y & (Vu, D E x) ( g z ~ y ) ( z= { u , u } v ... v z = Cnv,(u)). In order to find the least model-class containing a set x we ask: given a model-class M such that x is an element of M , what else must belong to M? The notion of absoluteness helps us to show that some sets must necessarily belong to M . First note that since we are interested in TSS, D3, we may use Theorem 3225 which tells us that all ordinal numbers are in M . Furthermore we have the following 191
3503
[CHAP. 111 SEC. 5
CONSTRUCTIBLE SETS
3503. LEMMA (TSS, D3, Mcl(M), aTSS/JGt(M)). If x* is an element of M then (a)
U*(x*) = u(x*)
(b) (c) (d)
Unv* (x*)
cg: (x*) Cg* (x*)
=
= =
9
Unv (x*) ,
c g , (x*) > Cg (x*) .
Proof. (a) The fxmula y = u(x) is restricted in TSS.
+
(b) Unv (x*) = UW ( H ) where H’O = x*, H’(n 1) D(H) = o.By induction we get H*’n = H’n for n E o.
(c) y
=
=
U ( H ’ n ) and
Cg, (x) is restricted in TSS’ by 3502.
(d) Is proved analogously to (b). This lemma leads US to the following 3504. DEF~NIT~ON (TSS’). The consfructing f u n c t i o n YX is defined inductively as follows: S:O = Unv({x>), 9qCI
9;J.
+ I)
=
c g , (9iCI)u {%;a> ;
U Y ~ Cfor I A limit;
=
a
Cstr (x)
=
U~LcI.
aeon
Cstr (x) is the class of all sets constructible f r o m x. 3505. LEMMA (TSS, D3, Mcl(M), JTSS/dG.t(M)). If x* is a n arbitrary element of M, then Cstr* (x*) = Cstr (x*) .
Proof. It follows by induction (using Lemma 3503) that for every = hence
CI,
9:;u
Cstr* (x*)
=
u* 9:;. On
=
u %i*x
=
Cstr (x*) .
On
3506. THEOREM. (1) (TSS’). For any x, Cstr(x) is a model-class containing x. (2) (TSS, D3). If X is an arbitrary model-class such that x E X then Cstr (x) c X . 192
CHAP. I11 SEC. 51
3511
CONSTRUCTIBLE SETS
Proof. (1) Since x E Unv ( { x ) ) , we have x E Cstr (x). The class Cstr ( x ) is complete, since, by induction, %:A is complete for every limit A. Also Cstr ( x ) is closed, since C g , (%:a) c %:(a + 1). Finally, Cstr ( x ) is almost universal, since if z s Cstr (x) then for some a we have z E $:a E Cstr ( x ) . (2) By Lemma 3505 x E M + Cstr ( x ) E M is provable in TSS, D3, Mcl (M), aTSS/dGt (M). By 1257 the last formula is provable in TSS, D3, Mcl (M) and therefore, by 1245, (b) is provable in TSS, D3.
3507. COROLLARY (TSS, D3). For every modelclass X , Cstr (0)
cX.
3508. Remark. Cstr ( x ) is usually denoted by L, and Cstr (0) by L. 3509. THEOREM (TSS, D3, Mcl (M), dTSS/JGt (M)). If M = Cstr (x*) then V* = Cstr* (x*); in particular if M = Cstr (0) then V* = Cstr* (O*). Proof. V* is M, hence V* = Cstr(x*); by Lemma 3505 Cstr(x*) = = cstr*(x*). We are led to the following axioms: 3510. (Constr) V = Cstr(O), (Sconstr) (3x c On)(V = Cstr ( x ) ) .
(Constr) is called the axiom of constructibility and (SConstr) is called the axiom of constructibility relative to a set of ordinals. 3511. METATHEOREM. (Constr) is consistent with TSS, D3; in fact, a i r (aTSS/dGt (Cstr (0))) is a faithful model of TSS, D3, (Constr) in TSS, D3, aTSS/dGt (Cstr (0))Demonstration. By 1445, 3147, 3228 and 3509 air (dTSS/dG;t (Cstr (0))) is a model of TSS, D3, (Constr); to prove the faithfulness consider the following diagram:
By Theorem 3509 we obtain V* = V in the third theory; i.e. the composition 1 * 2 is equivalent to the identity model of the first theory in the third one 193
3512
CONSTRUCTIBLE SETS
[CHAP. 111 SEC. 5
and obviously the third theory is a conservative extension of the lirst. Hence 1 * 2 is faithful and consequently 1 is also faithful.
3512. Remark. It is not difficult to show that TSS’ t (SConstr) + (D3) and that air faTSS/dGt (Cstr (0))) is a (faithful) model of TSS’, (Constr) in TSS‘, aTSS/aGt(Cstr(O)). Moreover, one can prove in TSS’ that Cstr (0) is a subclass of every model-class containing all ordinal numbers. But we shall not need these facts; their easy proofs are left to the reader. Let us consider the axiom (SConstr). It is weaker than (Constr); instead of saying that the universe equals Cstr (0) it only says that V equals Cstr ( y ) for some set y of ordinal numbers. The question why y is assumed to be a set of ordinal numbers is answered by the following 3513. THEOREM (TSS, D3). (a) If V HDf ( y ) and hence (E2) holds.
=
Cstr(y) and y E On then V
=
=
(b) If V = Cstr(x) and (El) holds then there is an y c On such that V = Cstr ( y ) . Proof. (a) If y c On then HDf ( y ) is a model-class containing y (evidently, y E Df ( y ) ; if y c On then y c HDf ( y ) and hence y E HDf (y)). Consequently, Cstr ( y ) 5 HDf ( y ) and so V = HDf ( y ) . (E2) follows by 3418. (b) Suppose V = Cstr (x). By (El), Unv ({x}) can be mapped one-one onto an ordinal number (by a fhction which is a set). Therefore there is a relation r E On2 isomorphic to E Unv ({x}) (there is a set-isomorphism). Let r be a fixed relation with these properties and let Is be the isomorphism of On2, On w.r.t. Maxlex and E (see 2241; note that Is can be defined by a formula restricted in TSS‘). Finally denote 1s”r by y; we prove that Cstr (y) = V. It suffices to prove x E Cstr (y). By absoluteness, Is is a class of 3)ir (aTSS/aGt (Cstr (y))) and Is = Is*. (* denotes the notions of the direct model in question.) Moreover, Real* (Is”) holds since Is is defined by a RSF and consequently, by a NSF. Hence r = Cnv (Is)” y is a set in the sense of Gt (Cstr (y)). By 3229, x E Cstr (y).
r
3514. Remark. Let q ( x , r) be a formula saying “ r c On2 and r is isomorphic to E Unv We proceeded in TSS, D3, El, V = Cstr (x), q ( x , r), y = IS”;, aTSS/aGt (Cstr (y)) and proved V = Cstr (y). Hence by 1245, 1257 the following is provable in TSS, D3, El (x, r, y now being variables): For any x, r, y , if V = Cstr (x), q(x, r ) and y = Is” r then V = Cstr ( y ) holds. Consequently (b) is provable in TSS, D3. 194
({XI)”.
3518
CONSTRUCTIBLE SETS
CHAP. 111 SEC. 51
3515. COROLLARY (TSS, D3). (1) (SConstr) implies (E2); (Constr) implies (E3). (2) ).3( (V = Cstr (x)) implies (El) = (E2).
3516. METACOROLLARY. (E3) is consistent with TSS, D3. (By Metatheorem 3511 and Corollary 3515.) We now study powers of cardinals in TSS, D3, SConstr.
3517. LEMMA (TSS, D3, El). If Unv (x) is subvalent to Nuthen (a) for any
/3 < mu, 3:/3 is subvalent to K,; (b) for any p and y, if p 2 K, and B w K, then 9ifl w K,. 3518. THEOREM (TSS, D3). If V = Cstr(x) and x G o, then 2’@ = =
NB+lfor any p 2
CI.
Proof. To prove the theorem it suffices to show that P(9$u) NN ma+,. We prove this by finding for each y c %;ma some y < such that y E 3iy. First some metamathematical considerations. Consider the formula x c On& u = gxf (t 1) (with free variables u , x, t). We claim that it is restricted in TSS, D3. Indeed, it is equivalent to
+
x E On& Un(u)& u’0 = {x] =
u x u u x & D (u) =
c + 1& (Vp E D (u))[(p + 1) E D (u) U’ (p + 1) = = Cg, (u’p) u (u’p) .& . p E On,, -+ u’p = U U ” ~;] -+
cf. the Metalemmas at the end of Chapt. I Sect 3 and Metalemma 3502. Hence let cp(u, x, y) be a restricted TSS-formula such that TSS, D3 t. t q ( u , x , y ) 3 x c O n & y E O n & u = 3, r ( y 1). Further, let H(X) be a godelian term without constants such that H(V) represents q ( u , x, y ) in TC w.r.t. u, x, y . Hence TSS, D3 k ( u , x, t) E H(V) E u = gX (t l ) & x E On. We return to our proof. Let’y c 9:o, and let t be such that y E KJ. Set co = 3iouu { y , gX (t l), t}. Hence co contains all elements of 9io, and, in addition, the three elements y, 9, l), 5. We have co z K,. By the Reflection Principle there is a set c of power Na such that co E c and HRefIH(c, V) (in particular, Vlk (c) and H (V) n c = H (c)). By Theorem 3215 there exists an isomorphism f w.r.t. E, E which maps c onto a complete set d, again of power K,. By Lemma 1321 we have f”(H(c)) = H(d) and by 3126 we have H(d) = H(V) n d. Since Unv (x) E c and x E c we havef’x = x; similarly we havef’y = y. Further-
+
+
r
+
r
r(t +
195
3519
[CHAP. 111 SEC. 5
CONSTRUCTIBLE SETS
/‘(c
r)
r + r (f’z
more,f’(gx + I), X, = (f’(s,(t + I), x , f ’ o E H ( d ) = H (V)n n d , hence f’(9, (t + 1)) = YX 1) and since (y, 0E 93, f (5 1) we have ( y f ’ { ) E 9, (f’5 1). Since d % K, and gxrf’t + 1 E d we have f ’5 < a,+ q.e.d.
r
+
r
r
+
+
3519. Remark. 1) The same assertion can be proved under the assumption that V = Cstr (x) where x is a subset of some ordinal ( of power K,.
2) The preceding theorem describes properties of the operation 1(2, K,) for fi 2 a. Properties of this operation for /3 < a can be derived from the following 3520. THEOREM (TSS + D3). Let x G 0, where K, is regular and let V = Cstr (x). If y E cog and /3 < a then there exists 6 such that a, j /3 < a, and j*E Cstr (x n 6). Proof. Set co = cog u {v, 9.y ( 5 + l), 5, x] where y E Sit. Since co x = S, < Ng there exists a V-like set c 2 co such that c < a,, c n w, is complete and c is H-reflecting (H having the same meaning as in the proof of 3518; cf. Metatheorem 3130). We let f be an isomorphism of c onto a complete set d (w.r.t. E, E). Since c n w, is complete, we have f ’ x = = x n 6 where 6 = c n a,. We also have (f’(gX (( + l)), f ’ x , f ’ ( ) E E H ( d ) = H (V) n d. It follows that f’(gX ( 5 1)) = ?Ix f (f’t n, 1). Furthermore y E d , f ’ y = y and f ’ 5 E d so that y E S,,g(f’t).
r
r
+
r
+
3521. THEOREM (TSS + D3). If x E co, where k’, is regular and V = Cstr (x) then for every p < a,2’p N,. Proof. For any 6 < co, we let k, = {y E as;y E Cstr (x n a)}; we have P (cob) = k,. For any 6 let Nt mean the cardinality of 6 in the sense of
<
=
u
<
mp56
Cstr (x n 6). Since N: < K,, we have N:+i N, and by the remark 3519 2’”* = Nz+ holds in the sense of Cstr (x n 6). Hence k, K6,, K, and so P(cop) X,, q.e.d.
<
<
We shall now present a more general notion of constructibility relative t o a class. 3522. DEFINITION (TSS
+ D3). For any real X ,
Cstr, ( X ) =
u Cstr ( X n pa) .
asOn
3523. THEOREM (TSS, D3). For each real class X , Cstr, ( X ) is the least model-class Y such that (Vy E Y ) ( X n y E Y ) and X s Y. 196
CHAP. III SEC. 51
CONSTRUCTIBLE SETS
3529
Proof. (a) We prove that Cstr, ( X ) is a model-class. If CI < j? then X n c X n ps E Cstr (X n ps) and p , n Cstr (X n pa) E Cstr ( X n pa) (cf. 3225 (3)) so that X n pa = p , n Cstr ( X n pa) n X n pa E Cstr ( X n ps). Hence Cstr (X n p,) E Cstr (X n ps). This implies that Cstr, ( X ) is closed. To prove that Cstr, (X) is an almost universal class we suppose x E c Cstr, (X). Then we have x c Cstr(X n pa) for some CI and by what we proved there exists y such that x c y E Cstr (X n p,). Thus Cstr, ( X ) is a model-class. Further we have X c Cstr, ( X ) and for all a, X n pa E E Cstr, ( X ) ; hence, for any y E Cstr, ( X ) , X n y = X n pa n y for some a and so X n y E Cstr, (X). (b) If Y is a model-class and X c Y & (Vy E Y ) ( X n y E Y ) then, for all CI, X n pa = X n (Y n pa) E Y (since Y n pa E Y). Hence Cstr (X n pa) c Y and so Cstr, ( X ) c Y. n p,
3524. LEMMA (TSS, D3). If x is a set then Cstr, (x) = Cstr(x). Proof. There is an a such that x = x n p,; hence x E Cstr, (x) and so Cstr (x) E Cstr, (x). Conversely, since x E Cstr (x), we have (VJJ E Cstr (x)) (x n y E Cstr (x)) and so Cstr, (x) c Cstr (x).
3525. COROLLARY (TSS, D3, Mcl (M), aTSS/dGt (M)). For any X* such that Real* (X*) we have Cstr: ( X * ) = Cstr, (X*). Proof. By 3225 (3) and 3505, Cstr (X* n pa) = Cstr* ( X * n p,*) for all a. 3526. COROLLARY (TSS, D3, Real (X), aTSS/aGt (Cstr, (X)). V* = CstrT (X)
.
3527. THEOREM (TSS, D3). (E2) implies that there exists a real class Y = Cstr, (Y). Proof is analogous to the proof of (b) in Theorem 3513 (note that V = Cstr, (v)).
G
c On such that V
=
3528. The axiom of constructibility reIative to a class:
(Cconstr)
(3Y c On) (Real ( Y ) & V
=
Cstr, (Y)) .
3529. THEOREM (TSS + D3). Let X be a real class. If V = Cstr, ( X ) and X c On then V = Df(X). Proof. HDf ( X ) is a model-class and if X G On then by Theorem 3417, X is a real class in the sense of Gt (HDf (X)); hence Cstr, (X) c HDf (X) and SO Df(X) = V. 197
3530
[CHAP. I11 SEC. 5
CONSTRUCTIBLE SETS
+
3530. COROLLARY (TSS D3). Axiom (CConstr) is equivalent to Axiom (E2). Proof. (E2) --+ (CConstr) by Theorem 3527; (CConstr) + (E2) by the preceding theorem and Theorem 3418. 3531. Remark. air (dTSS/dGt (Cstr, (X))) is a faithful model of TSS, D3, (CConstr) in TSS, D3, Real (X) & X c On, dTSS/aGt (Cstr, (X)).
Finally we give some results which will be used in demonstrating the independence of the axiom of constructibility from the axiom of definability (E3), i.e. the consistency of V = HDf =!= L. We show that V = HDf $: L is a consequence of certain statements whose consistency will be established in Chapter VI. 3532. METADEFINITION. Let T ( X ) be a godelian term without constants. We make the following definition in TSS, D3: A relation r is called T-determining (Det,(r)) if
Mcl ( X ) & Mcl ( Y ) & X E Y . + T ( X ) c T ( Y ) for each X , Y.
(3)
3533. DEFINITION. A relation F is called an estimate of a relation r (Est (r, )F if
L) D(r)
=
D(F) ,
2) r c F , 3) (va E D ( r ) ) (33E D (r)) (vE 4)
cstr (0))(un, (L(i:
r
a), 8))
F E cstr (0) .
3534. METATHEOREM. For every godelian term T(X) without constants there is a godelian term It (r, F) with set images property such that (E3) & & 1 (Constr) holds in air (dTSS/dGt(Cstr (r It (r, i)))) as a model of (TSS, D3) in the following theory:
r
TSS, D3, Det, (r), Est (r, i), r dTSS/aGt(Cstr 198
(C
r
r ( 0 ) Icstr (0),
It(r, i))).
CHAP. 111 SEC. 51
3534
CONSTRUCTIBLE SETS
<
Demonstration. We proceed in the latter theory. Let = D(r) = D(F), = 1, xo = (O}. We define constructible sequences (p,; n E o} and (g,; n E o} by the following conditions: 3/, < /3,+1 for each n; each g, is the first constructible one-one mapping of F 8, onto p,+ - 8,. We have /3, < t for each n. We let x , + ~= g y ( r f x,) and x, = U x,. It is obvious
r
nE(Q
that x, is defined from i: and r by a godelian term It(r,F). We let M = = cstr (It (r, f)). We first prove x, = T(M). If a E X, then r ( E } E M, Cstr (r (a}) c M and hence a E T(M) follows from E E T(Cstr (r (a})). On the other hand, suppose a $ x, and let M,,, = Cstr (r (D(r) - (a})). Then r x, s M(,). Indeed, E = 0 is impossible and we have r xo EM,,,. Moreover, if r x, E M(,, then X, + = gy(r x,) E M,,, since r f X, = (r (D(r) - (a})) x,. Evidently, the whole sequence (x,; n E o) is an element of M(,) and hence x, E M(,,. This implies Cstr (r x,) E Cstr (r (D(r) - (a})) and tc Q T(M). We have proved x, = T(M) and consequently x, E DfGt(M'. But x , c On so that we have x, E HDfGt'M'. It follows that the sequence {x,; n E o} is an element of HDf'*(M). Since r x, = Cnv (gn+ x,+ we have (r x,; n E o] E HDfG*(M).Consequently, r x, E HDfGtCM) and hence HDfGt(M) 2 Cstr (r x,) = VG*(M). We have proved (E3)Gt(M).Evidently, r (0) E M. This implies M =t= Cstr (0) or, equivalently, 1 (Constr)e*(M'.This completes the proof. To establish the consistency of V = HM L it remains to show that for a certain T it is consistent to assume the existence of an estimated T-determining relation. This will be done in Chapter VI.
r
F
r
r
r
r
r
r
r
r
r
r r
r
r
F
r
+
199
4101
[CHAP. IV SEC. I
AXIOMS OF SUPPORT
CHAPTER IV SECTION 1
Axioms of support
We consider the theory TSS’. The basic concept throughout the chapter is the dependence of one class on another, defined in Chapter I, Section 4. We recall that X is said to be dependent on 2 (Dep ( X , 2)) if X = r”2 for some r ; if X is dependent on Z then X is a semiset. We shall consider another more special concept of dependence; we first define 4101. DEFINITION (TSS’).
Dr ( R ) E Re1 ( R )& (VX, y
E
D ( R ) ) (X ==! y
+
E X ~ R (x)
n E ~ ( YR) = 0)
( R is a disjointed relation). 4102. LEMMA (TSS’). R is a disjointed relation iff Cnv ( R ) js a function.
4103. DEFINITION (TSS’).
Dep, ( X , Z )
E
(3) (Dr ( r )& X
=
r”Z)
( X is disjointedly dependent on 2). 4104. LEMMA (TSS’). Dep (a, 2 ) = (31) Dep (0,Z n u). Proof. If u = r”Z then a = r ” ( 2 n u ) where a = D ( r ) . Conversely, Dep (2 n a, Z ) because 2 n a = (I a)” 2, hence Dep (a,2 n u ) implies Dep (0,Z). 4105. LEMMA (TSS’). For every a, 2, Dep (P (u )
- P ( u - Z), 2 n u).
Proof. We let ( u , x) E r = x E u & u c a. We have r”(2 n u ) = P(u) - P ( u - Z ) since P(a) - P(a - 2) is the class of all subsets of Q which are not disjoint from 2. =
200
CHAP. IV SEC. I ]
AXIOMS OF SUPPORT
4108
4106. THEOREM (TSS’). If A is a real class then Dep (a,2 n A) Depd (0, P (A) - P ( A - 2)).
=
Proof. (1) Suppose Depd (a,P (A) - P (A - Z)). Then there is a disjointed relation r such that a = r”(P (A) - P (A - 2)) = r”(x n (P ( A ) - P ( A - 2))) for every x 2 D ( r ) . We may suppose D ( r ) c P(A). Let u = u ( D (r)); then P ( u ) 2 D ( I ) and P (u) n (P (A) - P (A - 2)) = = P ( u ) - P (a - (2 n u)). Hence Dep (a,P (u) - P (u - (2 n a))), furthermore Dep (P (u) P (u - (2 n a)), 2 n u ) and Dep (2 n a, 2 n A), and so we obtain Dep (a, 2 n A).
-
(2) Conversely, if a = r”(Z n A ) then we let ( x , u> E S = x E W ( r ) & & u = { y E D ( r ) n A ; x E Ext, (y)}. Since the converse of S is a function, S is a disjointed relation; since S E W ( r ) x P(D(r))and Sisreal, S i s a set. We shall prove that a = S ” W where W = P (A) - P (A - 2 ) is the class of all subsets of A which are not disjoint from 2. If x E a then there exists u such that ( x , u ) E S and u n Z 0; clearly u E Wand so x E S”W. Conversely, if u E W and (x, u) E S then there exists y E Z n u such that (x, y) E r ; hence x E r”Z n A = a.
+
4107. COROLLARY and DEFINITION (TSS‘).
We recall that in Chapter I, Section 4, we defined a support to be a class 2 such that Dep (X, 2) and Dep (Y, 2) implies Dep (X - Y, 2) for all X and Y. Let us further define: suppd (2) 5 Supp (2)& (Va)(Dep (a,Z) + Depd (a,2))We also defined the model G11pp(Z) determined by the F-definition Gapp (2) with the specification Supp (Z) (see 1463). We shall sometimes write Supp, (2) instead of Supp (2). We have shown in TSS‘, Supp, (Z), aTSS/aGupp(Z) that every x is a set of the model; hence every normal formula is absolute. In particular, the formula Dep (X, Z) is absolute and the formula Supp (2) is absolute from above. Clearly (Vx) Dep (x n Z, Z); hencc 2 is a class of the model and so it is a support in the sense of the model. Since (Va)Dep (a,Z) holds in the model, we say that Z is a total support in the sense of the model. This leads us to the following definition: 4108. DEFINITION (TSS’). Z is called a total support (TSupp(2)) if (yo)Dep (a,Z). 201
4109
AXIOGS OF SUPPORT
[CHAP. I V SEC. 1
4109. The first axiom of support (Sl)
Note that every total support is a support. Axiom (Sl) states that a total support exists. 4110. METATHEOREM. The model air (aTSS/dGupp (Z)) is a faithful model of TSS‘, SL in TSS’, Supp, (Z), dTSS/dGupp (Z). Demonstration. In the following diagram the composition 1 * 2 is equivalent to the identity model of the first theory in the third and the third theory is a conservative extension of the first.
Remark. By Theorem 4106 Axiom (Sl) is equivalent (in TSS‘) to the following formula: (32)(Vo)Dep, (o,Z). At the end of the present Section and in the next Section we shall study consequences of the axiom of support and of some stronger forms of this axiom. Before doing this, we shall devote our attention to supports in TSS‘. Supports which are semisets are of special importance. We shall find simple necessary and sufficient conditions for a semiset to be a support. 4111. LEMMA(TSS’). If a is dependent on 2 n a then, for each c such that o E c, c - o is disjointedly dependent on P (a - Z ) . Proof. By 4107 there exists a disjointed relation r such that o = r”e where Q = P(a) - P(a - Z). If we let b = r” P(a) then a c b, and, since r is disjointed, b - a = r” P ( a - Z). Put s = ((c - b ) x ( 0 ) ) u u ( r n ( ( b n c) x V)); s is disjointed and c - cr = s” P ( a - Z ) . 41 12. THEOREM (TSS‘).
(a)
tb) 202
= (va) Dep suppd (z) = (va)&Pd
SUPP (2)
(p ( a
- Z), z)
(p (. - z), z,
9
CHAP. IV SEC. I1
4116
AXIOMS O F SUPPORT
Proof. (a) Let Z be a support. By 4105 and 4104 we have Dep (P ( a ) P ( u - Z), Z) and since P (a) is a set we have also Dep (P (a), Z). Hence Dep (P(a - Z), Z). Conversely, suppose (Va) Dep (P(a - Z), Z). It suffices to prove
If Dep(a, 2) then by 4104 and 41Ll there are sets b r> a and a such that Dep,(b - a, P ( a - 2)). Using Dep(P(a - Z ) , Z) we obtain Dep (b - a, Z). Hence Z is a support by 1466. (b) follows from (a) by 41 11. By definition, Z is a support if every pair of semjsets satisfies a certain condition, while by the preceding Theorem it is sufficient to consider a particular semiset. Indeed, the theorem implies the following 4113. COROLLARY (TSS’). If a E a then
(a) (b)
= Dep SUPP, b) = k P ,
SUPP
(0)
(P ( a - a), 0) , (p( a - a), a) .
Proof. The implication -+ follows immediately by the preceding theorem. And it is easy to modify the above proof in order to prove the converse implications of the present corollary. 4114. DEFINITION (TSS’).
Sim ( Z , , Z,)
E
. Dep ( Z , , Z , ) &
Dep (ZZ,Zl)
( Z , , Z, are similar if they depend on each other). 4115. LEMMA (TSS’). If Z,. Z , are similar then they are semisets. 4116. LEMMA (TSS’).
(a)
Sim ( o , ~, ) Sim (a,Q) -+ Sim (Q, a), Sim (a1,a,)& Sim (a2,a3).-+Sim (aL,C T ~ ).
(b)
If al, a2 are similar then Dep (e, al)zz Dep (Q, a2) for every
(c)
If al,a, are similar and al is a total support then a, is a total support.
Q.
203
4117
AXIOMS OF S U P P O R ~
[CHAP. IV SEC. 1
4117. THEOREM (TSS’). If Z is a disjointed support and o is dependent on Z then there is a set b such that o and 2 n b are similar. Prooj. Let r be a disjointed relation such that D = r”Z and set b = = D ( r ) . Then o = r”(2 n b) and 2 n b = (Cnv (r))” O.
4128. DEFINIIION (TSS’). A relation R is totally unioersal if each semiset is the extension of some element. Totunvr ( R )
= (YO) ( 3 ~ (0 ) = ExtR (x)) .
4119. TIIEOREM (TSS’). Axiom (Sl) is equivalent to the existence of a totally universal relation. Plooj’. Suppose Axiom (St) holds and let Z be a total support. The relation defined by (x, r ) E R = x E r”Z is totally universal, since each semiset X is an extension of some r . Conversely, if R is a totally universal relation then we show that R is a total support. If X = ExtR( u ) and X 5 a we let
Clearly r”R
=
ExtR(u)= X .
We shall now establish an extremely important Metatheorem concerning the theory TSS’, S 1. 41 20. METADEFINITIOK. (a) The formulas x
E
y, x E o and x E X are srmi-
iiornial (SNF).
(b) If cp and $ are S N F then cp& J/ and l c p are SNF.
( c ) If cp is a SNF then (3.w) cp and (30) cp arc SNF.
(d) Every SNF is obtained by a finite number of applications of the rules (b) and (c) to SNF sub (a). 4121. METATHEOREM. Every S N F is normal in TSS’, S1.
Dentonstration. Every atomic S N F is a NF. Let q and $ be SNF and let R be a variable for regular totally universal relations; if $(R) and $(R) are normal formulas equivalent to cp and JI in TSS’, St then (p & I$ and l$ are normal formulas equivalent to cp & J/ and l q in TSS’, S1 while (3x) $ is a 204
CHAP. I V SBC. I 1
4127
AXIOMS OF SUPPORT
normal formula equivalent to (3x) cp in TSS’, SL. If z is a set-variable which does not occur in cp then the formula ( 3 z ) (cp(R”(z})) is equivalent to (3a)q (a) in TSS’, S1. 4122. COROLLARY. Every SNF is representable by a Godelian term in TSS‘. S t
We have already mentioned that those supports which are semisets are of special importance. Before considering such supports more closely, we shall study so-called locally semiset supports. 4123. DEFINITION (TSS‘). Z is a localljv semiret support (Supp,(Z)) if
Supp, (Z)& (Va) (36) (Va E a) (Dep (a,Z ) + Dep (n, Z n b)) ; Z is a locally semiset total suppport if (Va) (3b) (Va E a ) (Dep (a,Z n b))
4124. The second axiom of support (S2)
(32)(Va) ( 3 6 ) (Va c Q) (Dcp (a,Z n b)) 4125. LEMMA (TSS’). If Z is a locally scmiset total support then 1) Z is a locally semiset support,
2 ) Z is a total support. Proof. Obvious.
4126. Remark. A support 2 is “locally semiset” iff for every set a there exists a semiset a (of the form b n Z ) included in 2 such that all semisets dependent on Z and included in a are dependent on a. 4127. THEOREM (TSS’, S2). (VU)
(30) (Ve E Q)
(e + 0
-+
(3x
E
D (a)) (e
=
Exto (x))) .
This formula will be denoted by (Pot). Proof. It suffices to put (x, r> E a
= r c (a
x b ) & x E r”Z 205
4128
AXIOMS OF SUPPORT
[CHAP. IV SEC. I
where b has the property given in Axiom (S2); a is a semiset, since it is included in a x (P(a x b)).
4128. THEOREM (TSS’).(S2) = . (S1)& (Pot). Proof. It suffices to prove the implication from right to left. Let Z be a total support; we prove that 2 is locally semiset. Let a be a set and let0 be a semiset relation which codes all subclasses of a, i.e. (Ve E a ) ( 3 x ) ( e = o”{x)). Suppose that a = r”Z and b = D ( r ) . We have r~ = r”(2 n b ) ; we show that (VQ c 0 ) (Dep (e, Z n b)). There exists xo such that e = Ext,(x,); setting e = S”a where ( y , ( y , x,)) E S = y E a, we have Dep (g, c). This completes the proof. 4129. METATHEOREM. air (ZTSS/dGupp (Z)) is a faithful model of TSS’, S2 in TSS’, Supp, (Z), dTSS/dGiipp (Z). Demonstration. (1) We prove that (S2) holds in the model. We have Sm* ( X ” ) E Dep ( X * , Z); since the predicate Dep is absolute it follows by the definition of a locally semiset support that the formula (Vu) (3) (Vo* E u ) Dep (a*,Z n b)
is provable in TSS’, Supp, (Z), dTSS/dGupp (Z). (2) Consider the following diagram (where TSupp, (Z) means TSupp(2) 8L SUPP, (z)):
r
.~
TSS,k2, T Sup2CZ), d TSSI a GUPP(Z)
The demonstration is analogous to the demonstration of 4110. 4230. DEFINITION. (TSS‘) Z is a semiset support (Supp, ( Z ) ) if Supp, ( Z ) & & Sm (z). 4132. The third axiom of support (S3)
206
CHAP. IV SEC. 11
AXIOMS OF SUPPORT
4136
4132. LEMMA (TSS’). If (Ve) (Dep (e, a)) then
1) a is a semiset support, 2) a is a total support.
Proof. Obvious. 4233. DEFINITION (TSS’). A set a is called bounding (Bd (a)) if for every semiset a there exist a set b -< a and a semiset e c b such that a = U(g); in other words, CJ is the union of a collection of sets which is subvalent to a.
Axiom of boundedness (Bd)
4134. THEOREM (TSS’).(S3) = (Pot)& (Bd). Proof. Since a semiset total support is locally semiset we have (S3)
--f
+ (Pot).
We prove that (S3) + (Bd). Let 2 c a be a total support, suppose 2 to be a disjointed total support. We shall show that a is bounding. Let a = r”2; we set u E b = (3x E a) (u = Ext, (x)). Clearly b a; if we set u E e E (3” E D ( r ) n Z ) (u = = Ext,(x)) then g E b and a = U(e). Conversely, suppose that (Pot) & (Bd) holds; we shall prove (S3). Let a be bounding and let a code all subclasses of a, i.e. (‘Ye E a)(3x) (g = Ext,(x)). We claim that a is a total support. Let e be a semiset and suppose e = U(Qo) where Qo c b and b =$ a; let f be a 1-1 mapping of b into a and let el = f”eo = Ext, (x). It follows that Dep (el, a), Dep (go, g,) and Dep (e, eo);hence Dep (e, a), q.e.d.
<
4135. METATHEOREM. air (aTSS/aGupp (Z)) is a faithful model of TSS’, S3 in TSS’, Supp, (Z), aTSS/dGupp (Z). The demonstration is analogous to the demonstration of Metatheorem 4110 and may be left to the reader.
4136. THEOREM (TSS’, Supp, (Z,), aTSS/dGupp (Zl)). ( S 2 ) implies that Z, is a locally semiset total support in the sense of Gupp (Z,). (In other words, if we assume that there is a locally semiset total support and Z, is an arbitrary fixed support then it follows that Z, is a locally semiset total support in the sense of Gupp (Z,).) Proof. Let Z be a locally semiset total support and let Z, be an arbitrary support. To prove that Z, is a locally semiset total support in the sense of 207
4137
AXIOMS OF SUPPORT
[CHAP. I V SEC. 1
Gupp (Z,) it suffices to prove (Pot)6uPP(Z1).Let a be a set and let T = = { I ; Re1 ( r ) & D (r) c b} ( b as in 4123). If we let ( y , r) E R = r E T & & y E r ” Z , , then R is a class of the model. Let 0 code all subclasses of a, i.e. (Ve c a ) (3x) (e = Ext, (x)). We define (r,z)ES
3
z ~ D ( a ) & r ~ T & E x t , ( z=) ExtR(r).
D ( S ) is a semiset and hence by D1 there exists eo c S such that D (eo)= = D ( S ) . Thus W (Q,) is a semiset and W (eo) c T; hence W (e,) c q for some q . Let R , = R q ; R , is a semiset of the model. If r”Z, E a then r”Z1 = Ext, ( z ) ; hence there exists s E D (e,) such that Ext, (s) = r”Z1; if ( r , , s) E eo then Ext,, (rl) = r”Z1. Hence. in the sense of Gupp (Zl), R , codes the subclasses of a .
r
4137. THEOREM (TSS’, Supp, (Zl), dTSS/dGupp (Z,)).
(Bd) 4 (Bd)GUPp‘Z’’ .
Proof. Let a be a bounding set; we shall prove that a is bounding in the sense of Gupp (Zl). If a is a semiset in the sense of Gupp (Zl) then there exist b a and Q E b such that a = U(e). We let eo = {x E b; x c a}; we have Sm6upq(Z1)(Qo). Since a = U(eo)the set a is bounding in the sense of Gupp (2,).
<
4138. COROLLARY (TSS’, Supp (Zl), dTSS/dGupp(Zl)). (1j
(2)
(s2)-+ (
S ~ ) ~ ~ P Q; ( ~ ~ )
(s3) + ( S ” ~ ” P P ‘ Z ’ ’ .
We shall prove a stronger result:
4139. THEOREM (TSS’, Supp, (Zl), dTSS/dGupp(Zl)). If Z is an arbitrary disjointed semiset total support then there is a set b such that Z n b is a semiset total support in the sense of Gupp (Zl). Proof. We know that (Pot) and (Bd) hold in the sense of Gupp(Z,); hence there exists a such that a is a semiset total support in the sense of S u p p (Z,). By 4117 there is a set b such that a and Z n b are similar and therefore Z n b is a semiset total support in the sense of Gupp (Z,). 4140. METATHEOKEM. IDir (dTSS/dGupp (2)) is a faithful model of TSS’, = I, 2, 3).
(Si) in TSS’, (Si), Supp, (Z), dTSS/dGupp (Z) (i 208
Demonstration. Denote the theory TSS’, Supp, (Z), aTSS/aGupp (Z) by T S S & , and consider the following diagram:
t
1I
I
11”
Arrow 1. is lDir (aTSS/aGupp (2)).By 4109 and 1223 it is a model. Arrow2 is an identity; by 4138 and 1230 it is a model. The model we are interested in is the composition 1. * 2. Now, arrow 3 is also a model because evidently TSS’, TSupp, (Z) !- (Supp, (Z)& (Si)). By 4109, 4129 and 4135 1 * 2 * 3 is faithful; hence 1 * 2 is also faithful (cf. 1233).
209
4200
[CHAP. IV SEC. 2
BOOLEAN SUPPORTS
SECTION 2
Boolean supports
In the present section we shall consider a special type of supports, namely complete ultrafilters on generalixd complete Boolean algebras. Generalized complete Boolean algebras are like complete Boolean algebras except for the fact that the equality relation need not be the identity; such algebras will be called Boolean embryos. We shall find necessary and sufficient conditions for the existence of a total support which is a complete ultrafilter on a Boolean embryo. In particular, we shall give necessary and sufficient conditions for the existence of a total support which is a complete ultrafilter on a complete Boolean algebra b. Furthermore we shall show that in the theory T S S + D2 the existence of a total support which is a complete ultrafilter on a Boolean embryo is equivalent to the existence of a total support which is a complete ultrafilter on a complete Boolean algebra. We also prove various properties of Boolean embryos which are of importance for set theory; in particular, we shall prove in TSS D2 the existence of a universal complete Boolean algebra, i.e. complete Boolean algebra Bsuch that every complete Boolean algebra is a homomorphic image of B. We recall the definition of a complete Boolean algebra: Let B, C and F be nonempty real classes; suppose that C is a mapping of B into B and that F is a mapping of P ( B ) into B. Then B = [B,C , F ] is a complete Boolean algebra if thc following conditions are satisfed, where F’O = 1,. C’x = -s, - 1, = O,, F ‘ ( . Y , J.1= S A j‘:
+
(I) x
A
--Y
(2) x
A
- y = 0,
=
0, for all
=s
A
X E B ,
y = s for all x, y
(3) F’z = l;’(F’’4) for all z
c B and
We shall now give a more general
q
E
EB
,
P(B) such that z
= U(q).
CHAP. I V SEC. 21
4204
BOOLEAN SUPPORTS
4201. DEFINITION (TSS’). A real class B = [B, C , F ] is a Boolean embryo (Bemb ( B ) ) if B is nonempty, C is a 1-1 mapping of B onto B such that C’C’x = x for each x E B and if I; is a mapping of P ( B ) into B.
We again write C’x = -x, F’z = Az and F’(x, y} = x A y . Remark. If B is a complete Boolean algebra then B is a Boolean embryo. 4202. DEFINITION (TSS’). If B and B are Boolean embryos then H is callcd a homomorphism of B onto if H is a real mapping of B onto B such that H’C’x = ~ ’ H ’ xfor all x E B and H’(F’z) = F’(H”z) for all z c B. Remark. If B and % are complete Boolean algebras then the concept of homomorphism coincides with that defined previously. The same remark applies also to other concepts to be defined below.
s
-
--
4203. DEFINITION (TSS’). Let B be a Boolean embryo and let R be a real equivalence on B; we write x y if (x, y) E R and z if (VxEz)(3yEw)(x
-
y)&(VyE1v)(3xEz)(X
-
?’).
R is called a congruence if the followjng conditions are satisfied: (i) x (ii) z
-
y
N N
-+ C’x
-
w + F’z
C’y for any x, y E B , N
F’w for any z , w
G
B.
A congruence R on B is called canonical if it is the weakest congruence (the least according to the ordering by inclusion) having the following properties: (I) x
A -X
-
0, if X E B ,
( 2 ) x ~- y - O , ~ x ~ y - x i f x , y ~ B , (3) F’z
-
F’(F”q) if z
cB
and z
= U(q).
4204. THEOREM (TSS‘). The canonical congruence exists on every Boolean embryo B. Prooj. We use Metatheorem 2145 and define a certain real relation I1 by induction; in what follows we use the notation x y rather than (x, y) E E H ” ( a } : We let x w O y if one of the following conditions is satisfied:
-..
l)x=y,
2) x
=
0, and y
3) x
=
F’z and y = F’F”q for some 4 5 P ( B ) and z
= ti A - u
for some u E B, =
u(q). 21 1
420.5
[CHAP. 1V SEC. 2
BOOLEAN SbPPORTS
-
-
If I > 0 then we let x - a y if one of the following conditions is satisfied (where x < a J’ denotes (3p < a) (x - B J’) and I - < a w denotes (VS E z ) (3y E w ) ( 3 p < a ) (x -0 y ) & (vs E w ) ( 3 y E z ) (3p < a) (x - p y ) ) : 1) x -<,y
2) (3u) (x
-
--
v y - < = x v -x
85. u
3) ( 3 ~HI , c B)(2
-<= 9
x = F’z&
4) ( ~ u , u ) ( uA
-U
-
5) (gu, U ) ( U
u
<= u
A
-
We now define
--
-
x
- J’,
=
u
&x = u
A
- I’ =
= F’w), =
A U&J’
-u&
(3a)(s
y
u),
= On).
-.)I).
-- -
--
The relation is obviously an equivalence and we have x A - x 0, and s y -+ -s -)I. Suppose z w ; we shall prove (3a) ( z <= w). We define a real function G on ( ( 0 ) x I) u ((I) x w ) such that G’(0, x) is the least a such that ( 3 y E w) (x <, y ) and similarly for (1, y). The domain of the function G is a set and so the range of G is also a set; hence there i s some a which is greater than all elements of W ( G ) . Thus we have z <= w and F ’ i -a F’w, so that F’z F ’ w . If s A - y 0, then x A A -y 0, for some a; hence x A J’ - a x and x A y x; analogously, x A .I. x implies x A - y 0,. Finally, if z = U(q) then F’z -,, F’F’’q and hence F‘z F’F”q. By Metatheorem 2146, is the least congruence satisfying the above conditions. Remark. If B is a complete Boolean algebra then the canonical congruence coincides with the identity. (Obvious.)
-- -
- -
4205.
DrFlNrTroN
-
-
--
-
(TSS‘). If B is a Boolean embryo then we define
The relation 5 is called the canonical quasi-ordering of B.
4206. LEMMA (TSS‘). The quasi-ordering defined above has the following properties: x 5 x; if x 5 y and y 5 I then x 6 z . 4207. Remark. As in the preceding lemma, we may prove results corresponding to each of the theorems of Chapter 11, Section 4 (replacing = 212
CHAP. IV SEC. 21
4212
BOOLEAN SUPPORTS
-
and 5 by and 5 respectively); we shall leave this to the reader. The followjng is of special interest: if, for x E B and z E B we define inf (x, z)
=(
~ Eyz ) (x
5
y) & (VX) [ ( ~ yE z ) (F
5 y)
-+
x 5 x] ,
then we have inf ( A z , z). 4208. DEFINITION (TSS’). If Bemb ( B ) then for all z
In particular, x v y
=
-(-x
A
-y).
4209. LEMMA (TSS’). If z c B then
Az
-
E
B we define
- VC”z.
4210. Similarly if, for x E B and z E B, we define sup (x, z ) = (Vy E z ) (y 5 x ) & (VZ) [(Vy E z ) ( y 5 X) --f x 5 X] for all x E B and z c B, then we have sup ( V z , z).
-
421 1. THEOREM (TSS’). Let B and B , be embryos, and let and be the corresponding canonical congruences. If H is a homomorphism of B onto B , then x
-
y -+ H’x
-(,)
H’y for all x,
E
B
Proof. Let u be the least ordinal for which there exist x and y such that x - = y and l ( H ’ x -(,) H’y). We first prove that u > 0. If u = 0 then there are two possible cases: a) x = u A - u and y = OB = -F’O, so that H’x = H’u A(,) H’u and H’y = -(1) ACI)O = OBI, whence H’x -(,) H’y; b) x = A z and y = A (At) where z = U(q), so that H’ Az = fEq
A(,) H”z - ( I ) A(I) (U H”t) ~ ( 1 A) ( 1 ) (A(i+”t) - ( I ) A (1) (H’ At) fE9 f=l -(,) H’( A(At)).Thus we have x > 0, we shall investigate Cases 1 to =
fE9
fEq
5
--
from the proof of 4204. If one of the conditions 1 to 5 is satisfied then H’x -(,) H’y; we establish this for Case 3). If x = Az, y = A w and z H”w by induction hypothesis; hence <(I w , then we have H”z A ( l ) H ” z -(1) A ( l ) H ’ ’ w and H’x -(,) H’y. This completes the proof.
--
-
4212. METATHEOREM. There is a godelian term B(V) such that the following is provable in TSS’: 213
4212
[CHAP. IV SEC. 2
BOOLEAN SUPPORTS
B (V) is a triple of real classes; if we denote B (V) by B, (or [B,, C,, F,]) then (a) B, i b a Boolean embryo and (b) Every Boolean embryo is a homomorphic image of B,. (B, is called the f r e e Boolean embryo.) Demonstration. Define a real relation H by the following condition:
where s(z) = Min (a; 1 (@ 2 a) ( 3 4 , u) ({u, u, b) E z)). The definition is correct in TSS’ by Metatheorem 2145. Indeed, if X’ is a variable for real classes, it suffices to show that the term
T(X”) = {(x,
E,
s(x));
x E X”} is iterable
,
We further define
Bo
=
w(H)
>
Cb(=, 0, a ) = ( z , 1, a ) ,
C X z , 1. a )
=
( z , 0, a ) ,
Fb4 = (4,0, s (4))
.
B,, Co and F , are real classcs. Let B, = [Bo, C,, Fa]. We call Bo the free Boolean embryo. Evidently, B, is a Boolean embryo. We now prove that every Boolean embryo is a homomorphic image of B,. Thus, if B is a Boolean embryo then using Metatheorem 2145 we suppose 0 6B and let (x, y )
E
K”(0) v
(x
3
. (X E B & ( y = (x, 0 , O )
= O g & ( 3 z $ B ) ( z #=
o&y
v y = ( -x, 1,O))) =
v
( ~ ~ 0 ~ 0v ) ) )
v ( x = 1B&(3z~B)(z*O&y=(z,1,0))) a
> 0 -+
. (x, y ) E K ” ( x } =
(32 c D (K”a)) [(y = ( z , 0, s(z)) & x
v ( y = ( z , I, ~(2))& x
=
=
A(K”a)”
Z)
v
- A(K”~)”z)] .
We claim that H m = W ( K ) is a homomorphism of B, onto B. The class is real and is a function, since every K”a is a function; this can be proved by induction. The function Hni obviously preserves intersections and complements.
Hiii
214
CHAP. IV SEC. 21
421 7
BOOLEAN SUPPORTS
+
4213. In the theory TSS (D2) we shall prove the existence of a free complete Boolean algebra, i.e. a complete Boolean algebra 3 such that every complete Boolean algebra is a homomorphic image of B. Let w O be the canonical congruence on B, and let R = Rg (- o ) (cf. Definition 31 13). For any x E B,, we let E be the set { y ; (y, x> E R). We define y E B = = ( 3 x E B,) (j= X), C‘X = C T for x E Bo and F’q = Fb u(q)for q E B, B = [B,C,F ] . It follows from the definition of the canonical congruence that B is a complete Boolean algebra; we call it the free complete Boolean algebra. If B is a complete Boolean algebra then there is a homomorphism H of Bo onto B: we define W’Z = H’x for x E B,,.To prove that B is a homomorphism of B onto B, it suffices to show that ff is a mapping of B onto B, i.e. that X = j7 + H’x = H’y; this follows from Theorem 4211. Thus we have 4214. THEOREM (TSS, D2). Every complete Boolean algebra is a homomorphic image of B. 4215. D E F I X I T I ~ N(TSS’). If B is a Boolean embryo then we say that a class 2 E B is a complete ultrajlter of B if it satisfies the following conditions: X E Z - - x # Z forany ~ E B ,
a
c 2 + Aa E 2 for any set a c Z ,
x ~ Z & x S y - + y ~forany 2 s , y ~ B . 4216. LEMMA (TSS’). If B is a Boolean embryo and 2 is a complete ultrafilter on B, then
a) O B 4 z b)/\uEZ--+uEZforall u c B . 7
Proof. a) Since 0 E 2 we have 1, = A0 E 2 and 0, = - 1, 4 2. b) Suppose Au E 2 and let y E u; it suffices to prove Au 6 y . If we let s = Au then we have x A y = F’{F’u, y } F’{F’u, F ’ { y } } = F’F”(u, { y } ) = = F’(u u { y ) ) = F’u = x; this follows from the fact that y F’{y) = = A y, since y A - y 0,.
-
-
-
4-317. LEMMA (TSS‘). If B is a Boollean embryo and 2 is a complete ultrafilter on B then for all u c B we have V u E 2 iff u n Z 0. Proof.
+
V U$ Z
-AAC”U # 2
AC”U E Z
= C”U c Z = u n Z
’
=0. 21 5
4218
y
-
BOOLEAN SUPPORTS
[CHAP. I V SEC. 2
4218. LEMMA (TSS’). Let Z be a complete ultrafilter on B. If x E Z and x then y E Z . Proof. Sincex y, we have - x -)’and hencex A - j x A --x OB, SO that x 5 J’.
-
-
-
-
-
4219. THEOREM (TSS’). If Z is a complete ultrafilter on a Boolean embryo B then Z is a disjointed support. Proof. We shall prove Dep, ( P ( u - Z), Z ) for every a . Evidently, it suffices to suppose a c B. If (y, x) E r = ( y c a & ?I = -Vy), then P(a - Z ) = r”Z and r is a disjointed relation. Hence Z is a disjointed support by 4112. 4-320. DEFINITION (TSS’). (1) Z is a Boolean support on B iff B is a Boolean embryo and Z is a complete ultrafilter on B. (2) Supp, ( Z ) = (3B) ( Z is a Boolean support on B ) . ( 3 ) Z is a total Boolean support iff there is a B such that Z is a Boolean support on B and 2 is a total support. 4221. AXIOM (S4). (32) (Z is a total Boolean support.) 4222. METATHEOREM. Dir (aTSS/dGupp (Z)) is a faithful model of TSS’, S4 in TSS’, Supp, (Z), dTSS/dGupp (Z). Demonstration. The following is provable in TSS‘, Supp, (Z), dTSS / / dGupp (Z): Z is a class of the model and, in the sense of Supp (Z), Z is a complete ultrafilter on a Boolean embryo. Hence (S4) holds in the model. A standard argument shows that the model is faithful. 4223. THEOREM (TSS‘, S4). Every nonempty class of ordinals has a least element. Proof. It suffices to prove that every nonempty semiset of ordinals has a least element. Suppose that Z is a total support and that Z is a complete ultrafilter on some B. Let X be a nonempty subclass of some ordinal number 5. We have X = r”Z = r”(Z n a ) wherc a = D ( r ) ; we may suppose that W ( r ) E l. Obviously a n Z += 0 ; for any x E a we let f’x be the least element of the set Ext, (x). If a 5 t then we define u, = V{x;f’x = x ) - Vlu,: /? < Y]. If fi $. ct then u, A u, 0, and if f ’ x = ct then x 5 6 V{u,; /I 5 x } ; hence V a 5 V{u,, a 5 t}. Since V a E Z , we have V{u,; a 5 t}E Z and (3ct) ( u , E Z ) . This a is unique and there is some X E Z such that cr~Ext,(x). The ordinal ct is the least element of X ; for, if p < ct and P ~ E x t , ( y )then y 5 V{uy; y 5 p) I$ Z since u y $ 2 whenever ‘r’ < a.
-
216
CHAP. 1V SEC. 21
4227
BOOLEAN SUPPORTS
4224. The axiom of standardness (St)
We, want to show that, in TSS', Axiom (S4) is equivalent to (Sl) & (St). We first prove a number of preliminary assertions. We shall consider the theory TSS' (SL) + (St). (Note that since both (Sl) and (St) can be proved in T S the following considerations also hold for the theory T S + (Dl).) To begin with we demonstrate a certain Metatheorem on induction which is analogous to those proved at the end of Section 1, Chapter 11. We use the letters a and e as variables for semisets and the letter 2 for total supports.
+
4225. METADEFINITION. Let T be a theory stronger than TSS'. (a) A godelian term T ( X ) is Sm-ZocaZ in T if T l-
XE
T(X)
= (30 G X ) ( X ET ( a ) ) .
(b) A godelian term T(X) is Sm-iterable in T if it is Sm-local in T and has the semiset-images property in T, i.e. T 1- Sm (T (a)). We now consider an arbitrary theory T stronger than (TSS', SI, St); R is a constant for a totally universal relation. We write a A F for (3x E F ) (a = Ext,(x)). 4226. DEFINITION (TSS', S1, St). Let F c C(R) acd let X be a class. F is said to code an G-cofinal system of subsemisets o f X ( F [ E A] X) if (L) (Va a F ) (a E X) and (2) (Ve c X ) (30 A F ) (4 E a).
iff
4227. METALEMMA. Let T ( X ) be a godelian term. T ( X ) is Sm-local in T
(*)
T I- ( V F ) ( F [ G
A]
X
+
T (X) =
uT
(0))
*)
OAF
Demonstration. (a) If (*) holds then T(X) is obviously Sm-local (take for F a coding of all subsemisets of X ) . (b) Let T (X) be Sm-local.Proceeding in T, we suppose F[E A]X. Since T(X) is Sm-local we have T(o) E c T(X) whenever B A F ; hence (J T ( 0 ) c T(X). On the other hand, if x E T ( X ) then there is a
UAF
e c X such that
x E T (e); further, there is
*) The second part of this implication may be written more exactly as (Vu) (u E T(A')= (3c 0 F)(u E T(c))). Note that the last formula is normal in T.
21 7
422s
[CHAP. IV SEC. 2
BOOLEAN SUPPORTS
(r - I: such that p c U T(cT).
a
c_
CT
which implies x E T (cT).Thus we have T ( X ) G
aAF
We are now ready to state o w Metatheorem on induction (cf. 2145).
4228. METATHEOREM. Let T be a theory stronger than (TSS‘, SL, St) and let R be a constant of T for a totally universal relation. For every Sm-iterable giidelian term T ( X ) there is a giidelian term S ( X , R) such that the following is provable in T: For each X 0, S = S ( X , R) is a relation, S’{O}= X , D ( S ) is an ordinal and S”(a> = T (S”a) for every ct > 0. Demonstration. Define
+
= (VL) (0 < L < u -+ Q ” { L ) = ?r(Q”L)) , n .’. C ( X . R) = . Re1 (e)& D (e) E On & Q”(O> -c X & A (e, D (e)) *) d ( 9 , 2)
Write C for C ( X , R) and define S ( X , R)
=
U
=
PAC
S . We proceed in T.
( 1 ) D (S) i> an ordinal because it is a complete subclass of On.
e
(2) S”(0; = X for, if X E X then defining e = {(x, 0)}, we have C ( x , R) ( 3 ) S”{aj = U Q”{z) and S”GL= U @”m; hence S”{ct) = U T (@”a)and
T (S”sc)
=
T(
Pt5C
PAC
PAC
U ~”a).It remains to prove that T [ U @”a)= U T (@”a).
Pf
c
QAC
PAC
Let CT A D = (30 r k C) (CT= ~ ” c c ) ; we prove that D codes an ccofinal system of subsemisets of @”a;in other words, for every o c U @”athere
u
is a 6 A C such that 4227. (3) Hence let
(r
PAC
CT
PAC
E @‘’a.This will complete the proof by Metalemma
u ~ ” a .Define
QAC
(x, u)
E
Q
=
.
ZL E
o& (3e A C ) (e
=
= Ext,(x)& u E @”ct). Q is a relation and D(Q) is a semiset (in fact, D(Q) = 0).By Axiom (Dl) let Q , c Q be a semiset relation such that D (Q) = D (Qo); then W (Q,) is a semiset and U p”(0) is also a semiset QAWQO)
(by (C2)):denote this semiset by eo. Evidently eo E X . We claim that there i s it Q A C such that $’(O) = eo and A (Q, z). Indeed, set P E A = *) This is a slightly imprecise but short way of expressing the following idea: C(X, R) contains all codes of all semisets which satisfy the right hand side of the defining equivalcncc and contains no code of any scmiset which does not satisfy it. Obviously this definition is normal and C ( X , K) can be written as a godelian term.
21s
CHAP. IV SEC. 21
4231
BOOLEAN SUPPORTS
E 1(3g A C) ( ~ ” ( 0= ) eo & A (g, B)) (the condition defining A is seminormal; recall Metatheorem 4121). If A 0 then it has a first element p. A standard proof (using St) shows that for every y < /Ithere is exactly one Q,, A C such that g’,’(O) = eo and A (e, ,y) and that y 1 < y z < p implies g,, c Q ~ , .Hence if p is limit then setting Qs = U g y we obtain /3 4 A , a con-
+
Y
<s
tradiction (note that R is used in the definition of es; the reader may easily write down a seminormal formula equivalent to x E es). If p is isolated, say fi = y 1, then defining es = e, u T(W(e,)) x {y) we obtain B $ A , alqo a contradiction. We denote by @; thus our claim is proved. It follows by induction (using St) that for every y 5 a and every e A W ( Q o ) we have ~ “ E y $’y. Consequently e”a c @”a and so G c $ ’ x .
+
u
u
eAW(Qo)
This completes the proof that T ( e”a) PAC follows (cf. (3)).
=
U T ( ~ ” a ) ;the Metatheorem
eAc
We shall now consider the notion of well-ordering in TSS‘ with the axiom of standardnesq and S1. 4229. LEMMA (TSS’, St). If a set r is a well-ordering then every nonempty semiset G G C ( r ) has a least element (w.r.t. the ordering r). Proof. There is an isomorphismf of C ( r ) onto an ordinal a (w.r.t. r , E) (see 2141). Set e = f ” ~then ; e E a and e $. 0, hence e has a least element y , and the u E 0 such that f ’ u = y is the least element of 0. 4230. DEFI~ITION (TSS‘). A semiset Q is a Sm-well-ordering if it is an ordering such that every nonempty subsemiset of C ( g ) has a least element. Renzark. Lemma 4229 states that every well-ordering which is a set is a Sm-well-ordering. 4231. THEOREM (TSS’, St, SL). If e is a Sm-well-ordering then there exist a uniquely determined ordinal number y and a uniquely determined one-one mapping G of C (g) onto y such that o is an isomorphism of g, (E 7). Proof 1) By the metatheorem on induction 4228 a relation H can be defined by the following conditions: H”(0) = (the least element of C (4)); for a > 0, if C (Q) - Hy’a is nonempty then H”(a) = (the least element of C ( g ) - H” a ] ; otherwise H”(a} = 0. H is an isomorphism and W (H)= = C (g); hence W ( H ) is a semiset. H is one-one and thus an exact functor, hence D(H) and N are semisets. Furthermore, D(H) is a complete subsemiset of On and is therefore an ordinal. (D(H) = y where y is the first ordinal not in D ( H ) . )
r
219
4232
[CHAP. I\’ SEC. 2
BOOLEAN SUPPORTS
2) To prove the uniqueness it suffices to show that if CT is an isomorphism of E [‘ y , E 6 (where 7 and 6 are ordinals) then y = 6 and IJ is the identity. Suppose CT is not the identity; then (by St) there is a least 5 such that a’t+ 5; hence 0’5 > (. It follows that 5 has no pre-image, a contradiction.
r
In TSS’, (Sl), (St) the definition of a complcte ultrafilter can be simplitid. 4232. THEOREM (TSS’, S l , S t ) . If B is a Boolean embryo then Z E B is a complete ultrafilter if and only if it has the following properties: 1)
XEZ
=
2) a E Z
- X $ Z forail ~ E B ,
E
A U E Zforall
u EB .
Proof. Every complete ultrafilter obviously has both properties. It‘ Z has properties 1) and 2) then to show that Z is a completc ultrafilter we need only prove that
- -
J’
2
x EZ
->
y
E
z,
where is the canonical congruence. We prove that for all x, y E B ; if y then x E 2 = y E Z. We first prove this in case x - O y. a) We have 0, 6 Z; for 0 E Z so that A0 E Z whence 0, = --A0 .$ Z. Since { u , - u ) Q Q Z, we have u A - u 4 2. b) If z = U(4) then x
A Z E Zf 2 c z = ( V t E q ) ( t
G
Z)
=
= ( V f E q ) ( A ? E Z )= A ( A f ) E Z . f=l
We now suppose that 2 > 0 is the least ordinal for which there exist x and y y and 1 (x E Z = v E Z) (note that (St) is used!): such that x a) If - x - y then - x E Z = - ~ E Zand hence X E Z = ~ E Z . b) If x -<.u then x E Z = U E Z and U E Z = ~ E Z hence ; S € Z = yEZ. c) 1f.x = A \ z , y = A w a n d z - - < , , w t h e n , - ~ Z ~ w c Z ; h e n c e x c Z= =yEZ. d) If x = u A u, y = u and u A - u < z 0, then x E Z = (u, c) c 2 ; since u A - 1 7 - < a O B $ Z we have { u , u ] c Z = u E Z . e) If x = u A -0, y = 0, and u A u - < a ~ then y $ Z . We also have x I$ Z ; for, if x E Z then u. ---ti E Z whereas u A u E Z since u A u < = u, so that v , - u E Z, a contradiction.
-
-
220
CHAP. 1V SEC. 21
BOOLEAN SUPPORTS
4237
-
This completes the proof of the assertion x y + (x E Z = y E Z ) . We can now easily prove y 2 x E Z -+ y E Z ; if x E Z and x A y x then x A ~ E and Z hence (x, y > E Z, i.e. ~ E Z Hence . Z is a complete ultrafilter. Fina!ly we show that Axiom (S4) can be proved in TSS’, (Sl), (St):
-
4233. THEOREM (TSS’, S1, St). There exists a total support which is a complete ultrafilter on B,. Proof. We let
K ’ y ) = ((x,
E,
0 ) ; (x
E
Z& c
K ” { u } = ( ( z , E , a ) ; ( z , E, a ) & [ ( z E K”a&
E
E =
=
0 ) v (x $ Z &
& =
1& x
E =
l)]}
* O)} ,
B,&
0) v ( z $ K”a&
(cf. 4212; z is a total support). Let Zo = K” On. We shall prove: 1) that Z , is a complete ultrafilter on B,, and 2 ) that Z , is a total support.
ad 1): We obviously have x E Z , = - x $ Z, for all x E B, and a E Zo = h a E Z, for all a E B,. By Theorem 4232, Z , is a complete ultrafilter on Bo. E
ad 2 ) : If
D
D =
r”Z we define
= f ” Z , . Hence Z , is a total support.
E
7
= ( u , x)
E
r so that
Thus we have proved
4234. THEOREM (TSS‘). Axiom (S4) is equivalent to (Sl) & (St).
We shall now consider (locally) semiset total supports which are complete ultrafilters. 4235. DEFINITION (TSS’). Supp, ( Z ) seniiset Boolean support).
= Supp, ( Z ) & Supp,
( Z ) (locally
4236. AXIOM (SS) (The axiom of locally semiset total Boolean support) There exists a locally semiset total support which is a complete ultrafilter on a Boolean embryo. 4,737. THEOREM (TSS’). Axiom ( S 5 ) is equivalent to (S4) & (Pot). P r o o f . ( S 5 ) implies (S4) and (S2); (S2) implies (Pot). Conversely, (S4)& & (Pot) implies ( S 5 ) ; for, if Z is a support satisfying the conditions of (S4), then by (Pot) and by the proof of 4128 Z is a locally semiset support. 22 1
4238
BOOLEAN SUPPORTS
[CHAP. IV SPC. 2
4238. MI:TATHEOREM. a i r (dTSS/dEupp (Z)) is a faithful model of TSS’, S5 in TSS’, Supp, (Z), dTSS/?Gupp (Z). (Cf. the demonstration of 4222.) 4239. DEFINITION. Supp, (Z) support).
= Supp,
( Z ) & Supp, ( Z ) (semiset Boolean
4240. AXIOM (S6). There exists a total support Z which is a complete ultrafilter on some complete Boolean algebra b = ( h , c , f ) . (The axiom of total semiset Boolean support.) 4241. THEOREM (TSS’). Axiom (S6) is equivalent to (S3) & (St). Proof. It is obvious that Axiom (S6) implies (S3) and (S4) and hence (Si), (Pot) and (Rd). Conversely, if we suppose that (S3) a n i (St) hold then we may use all results concerning Boolean embryos. We let Z, E u be a total support and suppose, by 41.07, that Z , is a disjointed support. Let b = = u x { O , 1) and let Z = (Z, x (0)) u ( ( u - Z,) x {I)); Z is a disjointed support. It follows that P ( Z ) = ryZ. Let f l = Cnv ( r l ) ;f , is a mapping of a subset of P ( h ) in to b and f l /‘ P ( Z ) is a n-apping of P ( Z ) into 2.Let y o be an arbitrary clement of h - Z ; we define f ’ z = f;z for z E D ( f l ) , f ’ z = yo for L E P(b) - D(f,). Then f is a function which maps P ( b ) into b in such a way that z E Z = f ’ z E Z. For .y E u ’ we let c’ (x, 0) = = (x. 1) and c’(x, 1) = (s,0), so that x E Z = c’x 4 Z; hence b = = [b, c , f ] is a Boolean embryo and Z is a complete ultrafilter. If r is the canonical congruence on [b, c , f ] , then we let E = r ” ( x } for x E b, 6 = -_ = {Z; x E h } , 3’: = f ’ u z , C’S = c’x and 2 = (E;x E Z}; then 8 = (6, E , f ) is a complete Boolean algebra. We have x E Z = 2 E Z and hence E E z = E C’x Z ; thus z G = u ( z ) G Z = /\U(z) E z = f ’ z E z and 2 is a complete ultrafilter on (6, 2, f). Since Z = (Cnv (r))” Z , we have Dep(Z,Z) and Z is a total support.
+
z
4242. COROLLARY (TSS’) (S6)
=_
(Pot)& (Bd)& (St).
4243. METATIIEOKIIM. air (2TSS/c‘Ciipp(Z)) is a faithful model of TSS’, S6 in TSS’, Supp, (Z), c‘TSS/ZSiipp (Z). Demonstrution. 1) We proceed in TSS’, Supp, (Z), dTSS/dGupp (Z). Supp, (Z) implies that Z is a total semiset support on a real Boolean embryo in the sense of Supp (Z). It follows that the embryo in question is a set; we denote it by [b, c , f ] . On factorizing by the canonical congruence we obtain a cornplete Boolean algebra (6, C , f ) and a complete ultrafilter Z on it; Z is a total support since Dep (Z, Thus we have proved (S6)G”ppz.
z).
222
CHAP. IV SEC. 21
4249
BOOLEAN SUPPORTS
2) One can show in the usual way that the model is faithful. 4244. THEOREM (TSS’). If 5 is a coinplete subalgebra of b and Z is a complete ultrafilter on b then 2 n 6 is a complete ultrafilter on 6.
Proof. Exercise. We shall now consider semisets which correspond to a restriction of a complete ultrafilter on some complete Boolean algebra b to a base for b. 4245. DEFINITION (TSS’).If a = ( a , 5 ) is a separatively ordered set then we say that u E a is dense in if
( v X E a ) ( 3 y E u ) ( 3 z E a ) ( z5 x & z 5 y ) . 4246. LEMMA (TSS’). Let b be a coniplete Boolean algebra. If 5 is the canonical ordering and b, is a base for b, then u E b, is dense in 5 (as an ordering of b,) iff V u = 1,. Proof. Exercise.
4247. DEFINITION (TSS’).If 5 is a separative ordering of b then a semiset E b is called a complete ultrajlter on ( b , S ) if it has the followingpropcrties: B
1) ( V x , y E c ) ( 3 z E b ) ( z 5 x & z 5 y ) , 2) (Vu) (u is dense .+ u n B += 0).
s
4248. LEMMA (TSS’). Let be a separative ordering and suppose that b is a complete ultrafilter on ( b , 5 ) . If ti E 0,u E h and u 5 u then
B E 0
E 6.
Prooj. We let wEq-\v=uv
1(3tEb)(t 5uatsw).
The set q is a dense subset of b; for, if x E b then either (ill E b) ( t 5 x & t 5 u ) or x E q. Hence there is some x E q n B . We have x = r ; for, if x =t= u then -I (3t E b) ( t 5 u & t 5 x) and 1 (3)( t 5 u & t 2 x), a contradiction. Hence u E 0. 4249. LEMMA (TSS’). Let b be a complete Boolean algebra, let 5 be the canonical ordering and suppose that b, is a base. If B is a complete ultrafilter on b then B n b, is a complete ultrafilter on (b,, (and obviously h P .( 7-( b,, .)>.
s)
223
4250
BOOLEAN SUPPORTS
[CHAP. I V SEC. 2
Proof. If x, y E a n b,, then x A y E a and there exists z 5 x A y such that z E b,; this proves 1).If u is dense in (b,, 5 ) then Vu = 1, by Lemma 4246; hence there exists z E u such that z E a and 2) is proved. 4250. THEOREM (TSS’). Let 6 be a complete Boolean algebra, let 5 be the canonical ordering and suppose that b, is a base. If 2, is a complete ultrafilter on (b,, I ) then the semiset 2 = {x E b; ( 3 y E 2,) ( y 5 x)} is a complete ultrafilter on b, Z , = 2 n b, and Dep (2,2,). Proof. 1) We shall prove x E 2 3 - x 4 2 for x E b. Suppose that x E b . If x E Z & --x E 2 then there exist u E Z , and u E 2, such that u 5 x and v 5 y; hence there exists z E bo such that z 5 u and z 5 v; this is a contradiction sjncc O,# b,. If x # 2 and -x # Z then we let U E u = v~ b,& & (I,? 5 x v v 5 -x); u is obviously dense in b, since v u = 1,. There exists 1: E u n 2, such that either u 5 x or u 5 -x. In the first case we obtain x E Z, a contradiction; the second case is treated similarly. We shall prove u c Z E Au E 2 for u c b. It suffices to prove u n 2 $. 0 = Vu E Z . The implication from left to right is obvious. To prove the converse we suppose v u E Z . Let u E ii E u E b, & ( u 5 - v u v (3w E u) ( u 5 w)). Since ii is dense in (b,, 5 ) there is some u E E n Z,. It is obvious that r $ -Vu, since V u E 2 ; hence there exists w E u such that u 5 w , so that u n Z 0. 3) We obviously have 0, I# 2, since there is no element in the base which is 5 0,; hence 2 is a complete ultrafilter on 6 . By Lemma 4248 we have 2, = Z n h,. Set r = ((x, y ) ; y 5 x}. Then 2 = r”Z,, hence Dep(2, 2,).
+
+
4,951. THEOREM (TSS’). If 5 is a separative ordering of b and 2, is a complete ultrafilter on ( b , 5 ) then 2, is a support. Proof. We may suppose that b is a base for a complete Boolean algebra (6, i;, 1)and that 2, = 2 n b where 2 is a complete ultrafilter on (6, Z,J ) . Hence 2 is a (disjointed) support. Since Dep (2, Z), and Dep (Z,, Z), Z, is also a support (not necessarily disjointed).
4252. DEFINITION (TSS’). A relation I is antimonotone on an ordered set a if x 5 y .+ r”(x} 2 ~ ” { y }
for all x, y
E a.
4253. THEOREM (TSS‘). Let a be a separatively ordered set and suppose that 2, is a complete ultrafilter on a. If Dep (a,2,) then there is a relation I antimonotone on a such that c = r”Z,,. 224
CHAP. IV SEC. 21
4257
BOOLEAN SUPPORTS
Proof. There exists a complete Boolean algebra 3 = (6, E , f ) such that a is a base for 3 and the canonical ordering of 5 coincides on a with Let 2 be a complete ultrafilter on 8 which is an extension of 2,. We have c = = 7”Z where F is a disjointed relation. If we let
s.
then r is the required relation.
4254. THEOREM (TSS’, Supp, (Z,), dTSS/aGupp (Z,)). (st)4 (St)-Z,). Proof. If every nonernpty semiset of ordinal numbcrs has a least element then in particular every nonernpty semiset which is dependent on Z, has a least element. 4255. COROLLARY (TSS‘, Suppl (Z,), dTSS/dGupp (Z,)).
(a) (b) (c)
( ~ 44 ) ( ~ 4 ) ~ ” 0 ~ ;( ~ 1 ) (S5) 3 (S5)GUPP‘Z1) ; (S6) -+ (S6)Gupp(z” .
4256. METATHEOREM. The model air [dTSS/dGupp (Z)) js a faithful model of TSS’, (Si) in TSS’, ( S i ) , Supp, (Z), dTSS/dGupp (Z) ( i = 4, 5, 6 ) . We conclude with a theorem concerning (S6) which is analogous to the last theorem of the preceding Section. 4257. THEOREM (TSS‘, Supp(Z,), dTSS/aGupp(Z,)). If 2 is a total Boolean support on 6 then there exists a complete subalgebra 6 of b such that Sim (Z,, 2 n 5). Hence 2 n 6 is a total support in lhesensc of Girpp (Z,). Proof. By the last theorem of Section 1 there exists a c b such that a n Z is a total support in the sense of Gupp (Z,); if we let 8 be the least subalgebra of b such that a c 6 then we have Dcp ((1 n Z , 6 n Z). It suffices to prove Dep (6 n 2, Zl). To prove this we show that 6 n Z is a class in the sense of Gupp (Z,) or, to speak loosely, we construct 6 n Z inside Gupp (Z,). Since (St) implies (St)GuPO(Z1), we may use the Metatheorem on Induction in Gupp (Z,). Since 5 is a set, it is also a set in Gupp (z,).We construct F as foIIows:
lf”(0)
=
H”{E}
= {-X;
a, x E H ” E ) u {Az;z 5 H ” E }
(E
> O),
6=H”o~. 225
425 7
BOOLEAN SUPPORTS
[CHAP.
IV SEC. 2
Further we let
We can prove 6 n 2 = K”On by induction; we also have CIsGupp(zl) (K” On). This completes thc proof.
226
CHAP.
IV SEC. 31
4304
COMPLETE ULTRAFILTERS
SECTION 3
Properties of complete ultrafilters
In the present section we shall imestigate certain properties of complete ultrafilters; in particu!ar we shall be interested in the relationship between similar complete ultrafilters. We also consider propxties of semisets determined by the properties of a complete Boolean algebra with Boolean support; more details will be given in Chapt. V, Sect. 3. Throughout this section 9(6, 2 ) means " Z is a Boolean support on b", i.e. "b is a complete Boolean algebra and Z is a complete ultrafilter on b". 4301. LEMMA (TSS'). Let a be some set, let Z c a and suppose that 9(b, Z). Then there exists w E 2 such that u E a for every n o n ~ c r ou 5 MI. Proof. Wc have V ( b - a ) $ Z and - V ( b - a) = A(-c, u $ a } E Z. If we let w = - V ( b - a) then ( b w)' G a ; for, if there exists u E b' such that v 5 w and u $ a then u 5 w 5 -u, so that u = O,, a contradiction.
I
4302. METATHEOREM. The following is provable in TSS' for any SF q(b, x, 0 ) : If Z is a Boolean support on b and if q(b, x, y , 0 ) holds for every x E Z. then there exists w E 2 such that q(b, u, y, 0 ) holds whenever u 5 w and 0 =k 0,. 4'1
4303. DEFINITION (TSS'). Let b be a complete Boolean algebra. If u E b and u 4 O,, we define c u = {u A u, u E cr>.
Ib
CJ
c b,
4304. DEFINITION (TSS'). Let b be a complete Boolean algebra and let c
be a subalgebra of b. We denote by cIb u the algebra whose field is clb u and
whose operations are defined as follows: the meet of a nonempty subset of clb u is the same as its meet in 6, while the meet of the empty set is u; if u E c then the complement of u A u is - u A u where - and A are the operations of b. 227
4305
[CHAP. IV SEC. 3
COMPLETE ULTRAFILTFRS
,1
u is the same as bl u (see 2413), and
clb
u, with the operations as defined above,
4305. Remark. In particular b c ( , 1, = c.
4306. LEMMA (TSS’). The set is a complete Boolean algebra.
Proof. The operations are obviously uniquely defined. If we set clb u = d thenod = 0,and 1, = u . F o r x ~ c w e h a v e ( xA u) A (-x A u) = 0, = 0,. Similarly, (x A u ) A ( y A u) = x A u i f f x A u A y = x A u i f f x A u A A - y = 0, iff (x A u) A ( - y A u) = 0,. The verification of associativity is left to the reader.
4307. LEmfA (TSS’). Let c bc a subalgebra of 6, let u E Z and suppose that 9(b, Z). If Z , = Z n ( c u ) then I) 2 ,
=
(zn c ) ,1
,1
11,
2) Z , is a complete ultrafilter on c
,1
ii.
Proof. 1) If x E Z,then x E Z and x = y A u for some y E c. It follows that J‘ E Z and so we have y E Z n c and x = y A u. Conversely, if x E ( Z n c) u then x = y A u for some y E 2 n c, i.e. x E Z and y E c.
,1
2) Suppose that q c c ,1 u. We claim that q c Z , iff A4 E Z , . This is clear if q is empty; suppose it is nonempty. If q G Z, then Aq E Z and Aq E c u; hence Aq EZ,. If Aq E Z , then q c Z ; since q c c u, it follows that q c 2,. Finally it is clear that for any u E c we have u A u E Z , iff --: A ~ $ 2 ~ .
,1
,1
4308. COROLLARY (TSS’). Let Z be a Boolean support on b. 1) If c is a subalgebra of b then Z n c is a Boolean support on c.
2) If u E 2 then 2
I, u is a Boolean support on 6 I
11.
4309. LEMMA (TSS’). Lct Z be a Boolean support on b and let c be a subalgcbra of 6.If zi E Z then 2 n c and ( Z n c) u are similar. Proof. We have (Z n c) u = r”(Z n c ) where, for u,, u2 E b, (u,, u z ) E E r 3 u1 = u z A u. Hence (Z n c) u is dependent on (Z n c). Conversely, we have (Z n c) = s”((Z n c) u), where, for u,, uz E b, (u,, u z ) E s = E t’, E c & u2 = v1 A u ; hence 2 n c is dependent on (2 n c) u.
,1
,1
Ib
,1
,1
4310. COROLLARY (TSS’). Let Z be a Boolean support on b. If U E Z then Z and Z u are similar.
,1
128
CHAP. I V SEC. 31
4316
COMPLETE ULTRAFILTERS
4311. DEFINITION (TSS’). 1) If b is a complete Boolean algebra then a disjointed relation r is said to be a b-set if D (r) E b. 2 ) If r is a b-set thcn for x E W (r) we denote by r, the unique element of D ( r ) such that ( x , r,) E r . If x 6 W ( r ) then we let r, = 0,. 3) If r is a b-set and if u E b then we let r Ib u = {(x, v A t i ) ; (x, u ) E 4) If r, s are b-sets then we write r G s for (Vx) (r, = s,).
1.1.
4312. LEMMA (TSS’). Let Z be a Boolean support on b. If I’ is a 6-set and u E z then r ” = ~ ( r u)’’ ( Z Ib u). Proof. If < x , v > ~r and U E Z then (x, v A u ) ~ Ibu r and v A U E Ez u, so that r”Z E ( r u)’’(Z u). Conversely, let (x, u> E r Ib u and let u E 2 u. Then we have v E Z and there exists w such that (x, w> E r and u = w A u. Thus we have w E Z and x E r”Z.
I,
Ib
,1
,1
,1
4313. THEOREM (TSS’), Let Z be a Boolean support on b. If r and s are b-sets then r”Z = s”Z iff r u G s u for some u E Z . Prouf. The implication from right to left follows from the preceding lemma. Suppose that r”Z = s”Z, i.e. that r, E Z = s, E Z for every x E W ( r ) . clearly u E Z . If v = A [ ( r , A Let u = A [(1‘,A s,) v ( - r , A -s,)];
Ib
,1
A
sy) v
r,
xeW(r)
(- ry A A ZL =
-sy)]
u
A
r,
Y*X
then A [(I, A
sx) v
(-r,
A
and similarly s, A u = u A r, A s,; hence r, and so we obtain r u G s jL u.
l6
-sJ] A
= t’ A I’, A
u = s,
A 14
s,
for every x
4314. DEFINITION (TSS’). Let b be a complete Boolean algebra, let u E b and let a G b. We say that u is layered by a (Thru, (u, a)) if for each x E a either u 6 x or u 6 -x. Thru,(a) denotes the join of all u which are layered by a. (This join itself need not be layered by a.) 4315. LEMMA (TSS’). Let b be a complete Boolean algebra and suppose that Z is a Boolean support on 6. If a c b then a n Z is a set iff Thru, (a) E Z . Proof. Suppose that a n Z = a , is a set. Then a2 = a - Z is also a set and a3 = { -x: x E a2> is a subset of Z . Hence u = A(al u u3) belongs to Z and is layered by a; i.e. u $ Thru, (a) and so Thru, (a) E Z. Conversely, if Thru, (a) E Z thcn some u E Z is layered by a . Then a n Z = = {x E a; x 2 u ) so that a n Z is a set.
4316. THEOREM (TSS’). Let Z be a Boolean support on b and let CT where r is a b-set. Then CT is a set iff Thrub (D (r))E Z .
=
r”Z
229
4317
[CHAP. IV SEC. 3
COMPLETE ULTRAFILTERS
Proof. By 4315, Thru,(D(r)) E Z iff D(r) n 2 is a set. But since r”Z is similar to D (r) n Z, D (r) n Z is a set iff r”Z is a set. 431 7. DEFINITION (TSS’).Semisets a and e are called strongly similar (sSim (a, Q ) ) if there exists a relation r such that a = r ” and ~ Q = Cnv (r)”a.
4318. DEFINITION (TSS’). Let 6 be a complete Boolean algebra and let b are said to be b-strongly similar on u
u E b. Subsets a,, a, of ( d i m , (al, a,, u ) ) if
(b
-
{Ob)
= a,
Ib
-
lob)
i.e. if the set of all nonzero meets of elements of a , with u coincides with the set of all nonzero meets of elements of a2 with ti. The join of all u such that sSim, (a,. a,, u) is denoted by sSim, (a,, a,). 4319. LEMMA(TSS‘). Let Z be a Boolean support on b. If a,, a, c b then ( I , n Z i s strongly similar to a2 n Z iff sSimb (a,, a,) E 2. Proof. If sSimb ( a l , a 2 )E Z then a , is 6-strongly similar to a, on some 1c E 2. If, for s E a , and y E a 2 , we let (x, y) E r = x A u = y A u, then a , n Z = r”(a, n Z) and a, n Z = Cnv(r)”(a, n 2); hence a , n Z and 0, n Z arc strongly similar. Conversely, if a, n 2 is strongly similar to a, n Z then we may suppose that the corresponding relation r is a subset of a , x a,. We have (a, - W ( r ) )n Z = ( a , - D(r)) n Z = 0 and o # Z where o = V[(a, - W ( r ) ) LJ ( u Z - D(r))], Set w = A [(x A y) v (-x A - y ) ] ; since <x.y>er
r implies x E Z = y E Z, we have w E Z . Consequently u = w - c E Z. If s E a , - W (r) then x A u = 0,; similarly for y E a, - D (r). If I E w (r) and <.s, y> E r then x A u = A u and so a, Ib u - (4) = - a, Ib u - {ob], Hence a , and a, are 6-strongly similar on u and u is in (x. y )
E
Z . Thus we have proved that sSim, (al, a,)
E
Z.
4320. THEOREM (TSS’).Let 2 be a Boolean support on b and let r and s be b-sets. The semisets r”Z and s”Z are strongly similar iff sSim, (D (r), D (s)) E 2. Proof. We have sSim (r”Z, D (r) n Z) and sSim (s”Z, D (s) n Z). Hence sSim ( ~ ” 2s”Z) , iff sSim (D ( r ) n Z. D (s) n Z); however, the latter is equivalent to sSim, (D (r), D (s)) E Z.
4321. LEMMA (TSS‘). Let Z be a Boolean support on 6. Iff is a mapping of b into itself such that f ” Z E Z then thcre exists w E Z such that u f’u. whenever u 2 w . 230
CHAP. I V SEC. 31
4323
COMPLETE ULTRAFILTERS
Proof. For every u E b we have ( u - f ’ u )4 2 ; hence We set w = - V (u - f’u); if u
s
ueb
- w ; hence u - f ’ u
=
ob,
5 w then (u - f’u) s
so that u
V (u - f ’ u )$ 2 . usb w and (u - f ’ u ) 5
g f’u.
4322. LEMMA (TSS’). Let Z be a Boolean support on b and let f be a mapping of b into itself such that f”Z E 2. If f ’ u u whenever u E 2, then there exists tv E Z such that f’u = u whenever u 5 w, u .f. 0,. Proof. By Metatheorem 4302 there exists w , E Z such that f’u u for every u 5 w , , u 4 0,. By the preceding lemma there exists w 2 E Z such that u g f ’ u for every u w2, u =!= 0,. If we let w = w, A w2 then we have w E Z and f ’u = u holds for every u S w, u 4 0,. We shall now investigate undcr what conditions two Boolean supports are similar.
s
s
4323. THEOREM (TSS’). A Boolean support 2, on b , and a Boolean support 2, on 6, are similar iff there exist u, E Z , , u2 E 2, and an isomorphism f between b, u, and b2 uz such that Z , lb2 u2 = f”(Z1 ]bl u l ) . Proof. If the condition holds then Z , lbl u1 and Z , u2 are similar; by Lemma 4309, 2, u, is similar to 2, and Z , u, is similar to 2,; hence 2, is similar to 2,. To prove the converse, suppose that 2, and 2, are similar; let r i (i = 1,2) be disjointed relations and suppose that D ( r i )E hi, W ( r , ) = b,, W (r,) = = b,, 2, = rYZ, and Z , = r y Z , . We use indices 1 and 2 to distinguish the operations in 6, and b,. For u E bi we set (b,)” = (u E b,, u 2 t i ) ; for x E b , we define
I
1
Ib,
la,
, ,I
,
}Z;X
=
A2(rY(biy) A z A2(Cnv
(r2)”
(bi)”)
2
and for x E b, we define
,
,
If u 5 u then obviously h;u 5 h;u and, since r, is disjointed, u E 2, implies h;u E 2,; similarly for h,. Wc let g;x = h;(h;x) for x E b , and g;x = = k;(hix) for x E bZ. If x E Zithen gix E 2,( i = 1’2); we shall prove that g;x s i x. If x E b , and (x, y ) E r, then h;x 5, y, so that h;h;x = g;x 5 s 1h;v g , Alr;{y] 5 , x; similarly for x E b, and g2. By Lemma 4322 there exists w iE Z isuch that g;u = u whenever u 5 wi.If we let u , = = w1 A k;w, and u2 = h;ul then u1 E 2, and u2 E 2,; we shall prove that 23 1
4324
[CHAP. IV SEC. 3
COMPLETE ULTRAFILTERS
r
I
1
I s2
= h, ( b , u , ) is an isomorphism between b , u1 and b2 u2. We have u2 = h;u, h;(k;w,) 5 g;w2 = w2, so that u2 S 2 w2 and therefore g;u = u for any u u2. If x 5, y 5 , u1 then h ; x 5 , h i y S2u2 and hencefis an order-preserving mapping of b , u , into b2 u2.The functionj is one-to-one; for, if x’, y E ( b , u,) and h;x = h;y then x = g;x = = It;h;x = g ; y = y. The range off is b, u,; for, if y S 2 u2 = hiu, then h;y 5 , g i u , = t i , and, setting s = h;y we have x E b , 1 u1 and h;x =
j
s2
I
= h;(h;y)
I
I
I
= g;y = J’.
I
I
Since f is an order-preserving one-one mapping of b , u1 onto b, ii,, we can easily check that f preserves meets and complements and hence f is an isomorphism between b, u , and b2 u 2 . Since f is a restriction of h,, we have Z 2 u2 2 f ” ( Z 1 u I ) and since g 1 is the identity on b , u1 we finally obtain the equality 2, u2 = f ” ( Z 1 u , ) .
I
l2
I
1,
1
1,
4324. DEFINITION (TSS’). Let p be an isomorphism between complete Boolean algebras b, and b2. If r is a &,-set then we let fi’r = ((x, p’u); (x, v) E r). 4325. LEMMA (TSS’). Let p be an isomorphism between complete Boolean algebras b , and b,. If 2, is a complete ultrafilter on 6 , then Z2 = p”Zl is a complete ultrafilter on b2 and we have PZ, = (i’r)’’Z,
For any 6,-sct I’. Proof. Exercise.
4326. THEOREM (TSS‘). Let Z , and Z , be complete ultrafilters on b, and 6 , respectively and suppose that Z , and Z 2 are similar. If ri is a &-set ( i = 1,2) thcn rYZ, = r y Z 2 iff there exist u , E Z,, u2 E 2, and an isomorphism p bctwcen 6 , u , and b, u 2 such that r2 u2 & i ’ . ( r , u,) and P”(Z1 Ib, = z2 la, 1‘2. Proof. If the condition holds then
I
4
I
la,
Ib2
Conversely, suppose that r Y Z , = 4 ‘ Z 2 ; since Z , is similar to Z, there exist u l 0 E Z,, uZ0E 2, and an isomorphism p o between b, ul0and b, [ uz0 such that pt(Z, ulO) = Z , uZ0. If we let Fl = 7 , ulo, f 2 = r2 [ uz0, Z = Z , ulO,Z, = Z, u , ~and s2 = fio’Tl then we have fyZ2 = SYZ,
I
232
I
I
I
I
I
CHAP. IV SEC. 31
4328
COMPLETE ULTRAFILTERS
I
I
I
and there exists u, E 2, such that s, u, & 1, u, (= r2 u,). Finally, choose u, such that p i u , = u, and let p = p o ( b , [ u,). Clearlj‘ u I E Z , and p is an isomorphism between b, u1 and bz u,. We also have
I
I
1
3 ( r 1 u,) = jb((r1 .lo)
I
r
I4 = [ i b h I 4 1 I
u2
= s2
I
142 =%
rz
4327. THEOREM (TSS‘). Two Boolean supports Z, and Z , on similar iff 2, = f”Z, for some automorphismf of b.
Proof. If 2, = f ” Z 1 and iff is an automorphism then 2, is (st; similar to 2,. Conversely, suppose that Z , is similar to Z2 and suppose that 2, =k Z,. Choose v , E Z, - Z,, and set v2 = -0,; thus v, E Z , and Zi v i is similar to Z i(i = 1,2). By Theorem 4323 there exist u i E Zi (ui5 ui) and an isomorphism g between 6 ul and 6 24, such that Z2 t i 2 = = g”(Z, ul); clearly, u1 A u, = 0 ,. To prove the theorem, it is enough to extend the mapping g to an automorphism of b. For x E b we let f’x = = [x - (u, v u2)] v g’(x A u,) v Cnv(g)’(x A u,). The function f is one-to-one and maps b onto itself; for any x, y E b we have x 5 y = = f’x f’y; hence f is an automosphism. Clearly x 5 ul implies f’x = = g’x. We shall prove 2, =f”Z1. If X E Z , then x A u1 E Z so ~ that g’(x A u,) E 2,; since f’x 2 g’(x A u,) we have f’x E Z,. If y E Z 2 then y A u, E 2, so that Cnv (9)’ ( y A u,) E 2,; if x E b and y = f’x then x A u , = Cnv (9)’ ( y A u,) and hence x A u1E Z , . Since s 2 x A u , we have x E Z , and we are done.
I
I
I
I
I
4328. THEOREM (TSS’). Let Z be a Boolean support on b and suppose that b , and b2 are subalgebras of 6. Z n b , is similar to Z n b, iff b , ii = = b, u for some u E Z.
Ib
Ib
Proof. Let u E 2 be such that b , 16 u = b2 Ib u. By Lemmas 4307 and 4309, Z n bi is similar to Z n (bi u) (i = 1,2) and hence Z n b, is similar to Z n b,. Conversely, suppose that Z n b2 and Z n b , are similar. There are disjointed relations r l , r2 such that Z n bl = ry(Z n b,) and Z n b2 = = ry(Z n bl). It follows that Z n b , and Z n b, are strongly similar (consider rl u Cnv(r,)!). By Lemma 4319 we obtain a u E Z such that bl IbU = b2 [ b u s We shall now investigate properties of complete ultrafilters on products of separatively ordered sets (in particular, on products of two separatively ordered sets). We first prove a theorem which concerns complete ultrafilters on products
Ib
233
4329
[CHAP. I V SEC. 3
COMPLETE ULTRAFILTERS
of any number of factors; the products are taken over a cut I having the singleton property.
4329. THEOREM (TSS’). Let a
=
n’a , be a product of nonempty separaXES
tively ordered sets (without greatest element) and let 2 be a complete ultrafilter on a. If we set Z , = { u E a,; (u, x) E Z } for each x E s then 1 ) Z , is a complete ultrafilter on ax, 2) f E Z = (Vs E D (f))(f’x E Z,) for any f
E a.
Proof. 1) If ti, u E 2, then { ( u , x)}, { ( u , x)} E 2, and so there existsfe a such that f 5 { ( u , x)} and f 5 ( ( u , x)}. It follows that f ’ x E a,, f ’ x u and f’x 5 , u ; hence we have (Vu, IJ E Z,) ( 3 w E a), ( w 5 , u& w 5 , 0). If q IS dense in a, we shall prove that q x {x) i s dense in a. If f~ a and x E D ( f ) then f ’ x E a, and there exist u E 4 and u E a, such that u sxu and t 5 , f ’ x . Similarly for x 4 D (f). For y E D (f) ( y x) we define g’x = = I and g’y = f ’ y ; we have g =< f and g 5 { ( u , x)}, so that q x {XI is dense in a. Consequently, if q is dense in a then there exists an element [ ( u . Y)) of q x {x) which is in 2 and hence u E 2,. Hence Z, intersects every dense subset of a,, so that Z , is a coinplete ultrafilter on a, as asserted. 2) The implication from left to right is obvious. If the right-hand side holds and if b is the complete Boolean algebra with base a then f = A i ( f ’ x , x)) (where the meet istaken in b); since f ’ x E Z , we have
ss
+
XED(/)
{ ( f ‘ x . Y)) E Z and hence f E 2.
=f
E U(2) for every f~ a. Analogous considerations can be carried out for the operation o (cf. 2521): If aiis separatively ordered with greatest element lui ( i = 0, l), and 2 is a complete ultrafilter on a, o a,, then we let 2, = { u E ao; ( u , lu,) E Z } and Z1 = { u E a , ; (la, u ) E 2). It follows that 2 = Z , x 2, (the ordinary cartesian product) and that Zi is a complete ultrafilter on ai. (Prove !) 4230. COROLLARY. We have f
234
E
2
CHAP. V SEC. I ]
FULL UPWARD EXTENSION
5101
CHAPTER V SECTION 1
The full upward extension of the universe
a ) Parallels between TSS“ and TSS“, S5
Our first aim in the present section i s to generake certain constructions already made in TSS” or in weaker theories. We consider TSS” and TSS”, S5 simultaneously, observing certain analogies between the results in these two theories. The second theory is stronger than the first, and in it we can present the promised “upward” method for constructing models. By this method we obtain a model of a theory of sets (TS”) in a theory of semisets (TSS”, S5). The results obtained for TSS” will be used in the next section. We already have certain parallcls between TSS’ and TSS”, S5. We recall Metatheorem 2145 on recursion for iterable terms and Metatheorem 4228 on recursion for Sm-iterable terms. The first result concerns sets and real classes while the second concerns semisets and arbitrary classes. Actually the first result refers to TSS’ and the second to TSS’, S4; however TSS“ is stronger than TSS‘ and TSS”, S5 is stronger than TSS‘, S4. The reason for dealing with these stronger theories is that we shall need the notion of rank and, for stronger results, not only the existence of a total support and the Standardness Axiom, but also the axiom (Pot) which gives for each set a “semiset coding of all subsemisets”. There are other analogies. In TSS“ we have Theorem 3109 which states the existence of a (real) regulator for every real equivalence relation. In TSS”, S5 we have the following result: 5102. THEOREM (TSS”, S5). For every equivalence relation S there is a class P -C C (S) such that for each x E C (S) the class { y E P; ( y , x) E S } is a nonempty semiset. Proof. Define x E P = [x E C ( S ) & (Vy E C (S)) ((x, y) E S -+ ~ ( x )5 235
5102
FULL UPWARD LXTLNSION
[CHAP. V SEC. I
2 ~(y))]; thus for each equivalence c!ass P contains all elements of smallest rank. If x is an arbitrary element of C ( S ) then ( y E P ; ( y , x) E S} is a semiset (since all elements have the same rank); by (St), this semiset is non-empty. 5102. DEFINITION (TSS). R is an economicul functor (or a n economical relation, Econ ( R ) ) if, for each x E D ( R ) , ( y E D ( R ) ;EXt, (x) = Ext, ( v ) > is a semiset (i.e. each R-extension has only a seiniset of R-codes). Note that T S S b Ncon ( R ) + Econ ( R ) , i.e. every nowhere constant functor is economical. We can prove a certain generalization of the axiom (C2) concerning economical functors. 5103. THEOREM. (1) (TSS”) If R is a rea economical functor and if W
is a set then D ( R ) is a set.
(K)
(2) (TSS”, S 5 ) If R is an economical functor and W ( R ) is a semiset then D ( R ) is a semiset.
Proof. (1) Each Ext, (x) is a set: if we define F = {(x, 4); x E D (R)&y = = Ext, (x)} then F is a real relation. W (1;) = D ( R ) and D (1;) c P (W ( R ) )
is a set; since R is cconomical, F is regular; hencc W ( F ) is a set by t414, 1454, 1403(c). (2) Set cr = W ( R ) and let a be a set which includes cr. By (Pot) there is a coding Q of all subsemisets of a : let S
=
( ( ? ’ , s ) ; s ~ D ( e ) & ? . ~ n ( R ) & E x t ,=( ~Ext,(y)). )
W e h a v e D ( S ) ~ D ( e ) a n d s oD(S) isasemiset;foreach x ~ D ( S ) , E x t ~ ( . x ) is the semiset of all R-codes of Ext, (x), (The fact that R is economical IS used here.) Hence, by (C2), W (S) is a semiset. Obviously D ( R ) = W ( S ) . 5104. LEMMA. (1) (TSS”) Every real extensional relation is niorphic to a real strongly extensional (a fortiori, economical) relation.
(2) (TSS”, S 5 ) Every extensional relation is morphic to an extensioiial economical relation. Proof. (for both (1) and (2)). Let R be a relation with the assumed pIoperties. Set S = ( ( t i , 1.); ti, r E C ( R )& Ext, (u) = EXtR ( u ) ) . S is an equivalence; take a regulator P of S (in case (1) P is chosen to be real). Set R = R n P 2 , EI = S P . Evidently H is a morphism of R , R and thercfore R is extensional. For . Y E P we have Ext,(x) = ExtR(s) n P : [ y E C ( R ) ;Ext, (y) = Ext, (x)) = { y E P ; ( y , x) E S) is a semiset and so K is economical.
r
236
CHAP. V SEC. I]
FULL UPWARD EXTENSION
The proof of (2) is ready; to prove (1.) we continue as follows: Put is a real regular equivalence. Define
= S n P2;
s
5106
s=
Evidently, R is the required relation and A is the required morphism. 5105. DEFINITION (TSS). R is an almost regular relation (AReg(R)) if for every x E C ( R ) there is a semiset 0 G Ext, (x) such that Ext, (x) = = SAT,(o).
Evidently every regular extensional relation is almost regular. 5106. THEOREM. (I) (TSS”, Elk(R), dTSS/d%(R)). If the relation R is real, almost regular and almost universal then Axiom (C2) holds in the
sense of %(R).
(2) (TSS”, S5, Elk (R), JTSS/d% (R)). If the relation R is almost regular and almost universal then Axiom (C2) holds in the sense of % (R). Proof. ( 1 ) By 5104 (I), we can define a relation R and a morphism H of R, R such that R is strongly extensional. Evidently, R is pairing, regular and almost universal provided R is pairing, almost regular and almost universal. Hence the theorem follows by 1313 and 1416.
(2) Since R is E-like, the axioms of TC hold in the sense of %(R). By 5104 (2), we can construct a relation E and a morphism H of R,R such that R is extensional and economical. Moreover, is pairing, regular and almost universal provided R is pairing, almost regular and almost universal. Hence using 1313 we can suppose without loss of generality that R is regular and economical. Under this assumption we prove (C2). First we have Sm*(X*) = = Sm (X*). Indeed, if Sm*(X*) thenX* G EdR(U) for some u ; consequently, Sm(X) by the regularity of R, while conversely if X* is a semiset then X * E ExtR( u ) for some u by the almost-universality of R. Now suppose X* E C (R) and Exct* (X*) and set X = Dec, (X*). We have D* (X*) = D ( X ) , Extx (u ) = Ed:. (EdR(u)) etc., hence we obtain Reg (X) from Reg* ( X * ) and Econ (X) from Ncon* (X*). Thus Sm (D (X)) = = Sm (W (X)) (+ by 1414 and t by 5103 (2)) and therefore
Sm* (D* (X*))
3
Sm (D (X))
= Sm (W (X)) = Sm* (W* (X*)) . 237
5107
[CHAP. \‘ SEC. 1
FULL U P W A R D EXTENSION
5107. DEFINITION (TSS). A relation R is saturated-universal (denotation: Satunvr ( R ) ) if for every 0 E C ( R ) there is a x E C ( R ) such that Ext, (x) = = SAT,(a). Note that every saturated-universal relation is almost universal. We shall prove in TSS”, S5 that there are extensional almost regular saturated-universa1 relations. This is important because of the following 5108. METATHEOREM. The F-definition % (R) with the specification Extl (H)& AReg (R) & Satunvr (R) determines a model of TS” in TSS”, S5. Dernonstration. We proceed in the theory TSS”, S5, (Extl(R) & & AReg (R)& Satunvr (R)), aTS/aYt (R). It follows immediately from Satunvr (R)that R is pairing. Hence by Theorem 5106 all axioms of TC and also (C2) hold in the swse of % (R). We prove (Cl) and (C3) in the sense of % (R). First R can again be supposed to be regular and economical (cf. the proof of Theorem 5106); if R is saturated-universal then IS also saturated-universal). Thus every saturated subsemiset 0 of R is the extension of some element of C (R), and so wt‘ have (C3)*. Define f f y }
H”{n
+
=
I’.
1) = (X
E
c (K);ExtH(x)
E
C(R); ExtK(x) =
=
0) , If”(iI)).
(Using Metatheorem 4228.) Since I1 is regular. Ii”o is a semiset; obvioudy I f ” o i s saturated. Hence there is an x* such that x* = H ” o and we have
o* € * x * & (Vn*
€*
.*) ( ( n * ) * € * .Y*) .
Iiidecd. if i~ E x* then ZI E N “ { n ) for some t i . We have (Vu‘ E ff”(ri}) (ExtR( 1 1 ’ ) = Ext, ( u ) ) and for each 2: E W ’ { n + 1) we have E d R ( u ) = ZZ”{n}; hence setting ti* = Ext,(u), u* = Ext,(u) we have u* = (u*}*.Thus Axiom (Cl)* is proved. The proof of (D2)* is easy. It remains to prove the existence of extensional almost regular saturateduniversal relations i n TSS”, S5. This will be done in a quite general way in an analogy with the construction of a real upward extension (of a real regular strongly extensional relation) describcd in Chant. 111 Sect. 2. (Cf. Metatheorem 3223.) 5109. MIXATHEOREM. There are godelian terms Fupw (Q, S ) and Imb (Q) such that (1) the following is provable in TSS”, S 5 : If Q is a regular extensional economical relation and if S is an arbitrary totally universal relation 238
CHAP. V SEC. 11
5109
FULL UPWARD EXTENSION
then Fupw ( Q , S) is a regular extensional economical and saturated-universa1 relation. Moreover,
( 2 ) in TSS”, S5 we can fix constants Q for a regular extensional economical relation, S for a totally universal relation and R for Fupw (Q, S). Adding i?TS/J% (R), we obtain a theory in which the following is provable: There is a complete subrelation Q of R such that Imb (Q) is an isomorphism of Q and C (9)is a class in the sense of % (R) and V* = Ker* (C (0))holds.
e;
Fupw (Q, S) is called the full upward extension of Q (constructed with the aid of S ) . Denionstration. We proceed in TSS”, S5. Given Q define F’x = ( 0 , x ) for x E C (Q) and set Q = ( ( ( 0 , x), (0, y ) ) ; (x, y ) E Q). Evidently F can be defined by a godelian term Imb (Q). Given S define another totally universal relation S such that Q c c S and with the following additional propcrties: (a) semisets which are Q-extensions have the same codes in S and in Q,(b) S is economical and regular. Indeed, define
S = Q u ( ( u , (1, s)); 14 E Ext, ( x ) & Sm (Ext, ( x ) ) & 8~ 1(3jj E C (Q)) (Exts ( x ) = E x t ~ (y)) & 8~-I ( 3 j E~ D ( S ) ) (Ed, ( x ) = E d , ( Y ) 8~Z(Y) < ~ ( x ) ). Thus Q extensions are coded by their @-codes and each other semiset is coded by all elements of the form (1, x) where x is an S-code for this semiset of last rank. By (St) S is a totally universal relation; obviously S is economical. We now define H by recursion asfollows: H”(O} = C( c1 > 0 + H”(ct} = = ( y E D (S);Ext, (y) c H”c1 & Sat, (Ed, (y))), The definition is correct by Metatheoreni 4228 (note that we use (Pot) to have the Sm-itcrability). Further set R = S (If” On). Evidently R can be defined by a godelian term Fupw ( Q , S). It follows from the condition defining II”(c1) that x E C ( R )implies Ed, (x) = Ext, (x) and therefore Ext, (x) is R-saturated, hence R is extensional, economical and regular and 9 _C c R E c S. We prove that R is saturated-universal. Take a subseiniset a of C(R); since R is economical SAT,(a) = SATs(a) is a semiset and therefore there is an c1 and an x E H”{a} such that SAT, (a) = Ext, (x). Thus R is saturated-universal. Let us proceed now in the theory indicated in (2). Note that this theory is a conservative extension of TSS”, S5 and that all axioms of TS” hold
e),
r
239
5110
FULL UPWARD EXTENSION
[CHAP. V SEC. 1
in the sense of 92 (R). Denote Imb (Q) by F . F is evidently an isomorphism of Q and Q, and Q is a complete subrelation of R. Moreover, by our construction, C (Q)is a saturated subclass of C (R)and therefore CIS*(C and Comp* (C We prove V* = Ker* (C ( Q ) ) .Let Z* be a class in the sense of 92 (R) such that C ( Q ) c Z* and P* ( Z * ) E Z * . Then it follows by induction (using (St)) that H"a E Z* for every a; hence V* = H" On E E Z*. This proves V* = Ker* (C (Q)).
(a))
(a)).
5110. Remark. Note that if S , , S , are two totally universal relations and Q is a regular extensional economical relation then Fupw ( Q , S,) and Fupw ( Q , S , ) are morphic. The proof of this fact is left to the reader as an exercise.
b) F u l l upward extension of E; the inodel Up An extremally important case of the full upward extension is Fupw (E, S ) . This contra$ts with the real upward extensions because Rupw (E) is evidently isomorphic to E and therefore of no interest. 51 2 1 . DEFINITION (TSS", S5). A relation R is a fuZl upward extension
(Fupw(R)) if there is a totally universal relation S such that R = Fupw (E, S). We denote Imb (E) by Imb and Imb" V by Tor (the torso of the full upward extension). 5112. METADEFINIIION AND
DIAGRAM.
TSS' is the theory
TSS", S5, F U ~(Fup), W i?TS/J%(Fup) (the theory of semisets in which the upward extending model will be studied). Thus Fup is a constant fixed for an arbitrary full upward extension. It follows by Metatheorem 5109 that nit (aTs/i?%(F~p))is a direct model of T S in TSS'; hence TSS' is a conservative extension of TSS", S5.
240
5113
FULL UPWARD EXTENSION
CHAP. V SEC. I1
Consider the following diagram:
Arrow 1 is the identity and arrow 2 is air (aTS/diTt(F~p)).(Translations of formulas in the last model will be denoted by asterisk.) All arrows in this diagram denote some direct translations (and we shall show that they are models); double arrows denote identities. The reader is recommended to draw this diagram step by step as it is being studied. We defined the constant Tor in TSS“, S5 by Tor = Imb” V and we know that TSS” b Cls* (Tor)& Comp* (Tor)& V* = Ker* (Tor). We shall establish other properties of the class Tor. First, the following lemma follows immediately from the properties of the isomorphism Imb: 5113. LEMMA (TSS’). (a) (Vx) (3! x*
E*
Tor) (x*
=
Imb’ x) ,
(Vx* E* Tor) (3! x) (x* = Imb’ x) ;
c Tor) (x*
=
Imb’ o),
c Tor) (3! o) (x*
=
Imb” c) ;
(b) (Vo) (I! x* (Vx*
(c) (VX) (3! X * E Tor) ( X * (VX*
=
Imb’ X ) ,
c Tor) ( 3 ! X ) ( X * = Imb” X ) ;
(d) (x* = Imb” x & Y*
=
Imb’ Y ) -+ [x
E
Y
E
x* E* Y*]
. 24 1
5114
[CHAP. V
FULL UPWARD EXTENSION
SEC. 1
This has the following 5114. METACOROLLARY. If q ( x , TSS’ t (x* = Imb” x & ---t
=
[cp(x,
0,
o, 0 , X ,
0)
is a R F then
& y* = Tmb’ o &
& X * = Imb’ X & 0 ) .+
x,
y*,
0 , 0,0 ,
0)
= ‘P*(X*,
0,
0,X*,
In particular, let cpi(x, y , z ) be RF’s equivalent to z I , . .., 7) where the Fi are godelian operations. Then TSS’ k (x* = Imb” x &
0) +
[cpi(x, y , z )
o)] . =
Fi(x, y ) ( i =
= cpT(x*, y*, z*)]
,
Hence we can proceed as follows in TSS‘. Let x*, y* E* Tor and let x * = = I m b ”x, y* = Imb’ y . Set z = Fi(x, y) and z* = Imb” z . Then z* E* Tor and cp?(x*, y*, z*). Hence (VX*, y * ) ( 3 z * ) (z* = FT(x*, y*)), i.e. we have proved Clos* (Tor) in TSS‘. Further we prove AUncl* (Tor). Recall that all axioms of T S hold in the sense of % (Fup). Hence we must prove that in the sense of %(Fup) every subset of Tor is a subset of an element of Tor. Let x* E Tor. There is a CJ such that x * = Imb” o and there is an a such that o G u . For a* = Imb’ a we obtain x* E a*. Hence we have proved 5115. LEMMA(TSS‘). Mcl* (Tor). 51 26. DEFINITION (TSS’). (a) Mcl,, ( X ) = . Mcl ( X ) & (32 E X ) (Vo ( 3 r E X ) (o = r”Z);
cX)
.
(b) MclD, ( X ) = Mcl ( X ) & ( 3 F ) (Un ( F , On, X ) & U(W ( F ) ) = = X & (Vu E X ) ( F n u E X ) ) . Mcl,, ( X ) is rcad “ X is a model-class w i t h Sl”, Mcl,, ( X ) is read “ X is a model-class witlz D2”. These names are justified by the following obvious 5217. MFTALEMMA. (a) 6 t (M) with the specification Mcl,, (M) determines a faithful model of TSS’, S1 in TSS’. (b) Gt (M) with the specification Mcl,, (M) determines a faithful model of TSS, D2 in TSS’. 51 18. LEMMA (TSS’) In the sense of 91 (Fup) Tor is a model-class with S1 and D2 and, moreover, On* c Tor.
Proof. There is a total support Z (which is a local semiset Boolean support, although we do not use this fact). Denote Imb” Z by Z*. We have Z* G Tor. 242
CHAP. V SEC. 11
5121
FULL UPWARD EXTENSION
If u* E Tor then there is a r~ such that u* = Imb” u and there is an r such that u = r”Z. Denoting Imb” r by r* we have r* E* Tor. Evidently the formula u = r”Z is a RF; thus we obtain u* = r*”*Z* by Metacorollary 5114. We have thus proved Mcl,: (Tor); we now prove x* = Imb” x -+ -+ [Ord (x) = Ord* (.*)I. Suppose x* = Imb’ x; then Comp (x) = E Comp*(x*) since Comp(x) is a RF. Similarly if Trich(x) means (Vu, u E x) (u E u v u = u v u E u ) then we have Trich (x) E Trich* (x*). We prove Ord* (x*) = Ord (x). Let First (x, y ) be the formula y E x & & [y 0 -+ (3u E y ) (u n y = O)]. First (x, y ) is a RF (in TSS). Suppose Ord(x). Let y* c x* and y* = 1mb”o. Then u G x and if r~ 0 then by (St) there is a u E u such that u n u = 0, and so we obtain First* (x*, y*). Conversely, suppose Ord* (x*). Then Comp (x) and Trich (x). If y c x then for y* = Imb” y we have y* c x* so that First* (x*, y*). First (x, y ) follows and we proved Ord (x). We now prove On* c Tor. Suppose not; then there is a least a* in On* - Tor and there is a u c On such that a* = Imb” u. By (St) On - CJ has a minimal element a ; since u is complete it follows that u = a. Then a* = Imb” a, hence a* E* Tor, a contradiction. We now prove MclD,* (Tor). By D2 there is a real function F such that D ( F ) = On and U(W ( F ) ) = V. Denote Imb” F by F*. Evidently (x, a ) E E F = (x*, a * ) E* F* for x* = Imb” x, a* = Imb’ a. It follows that D* ( F * ) = On* and W* ( F * ) c Tor. Moreover, if x* E* Tor, x* = Imb’ x then, for some a, x ~ F ’ aand hence x* E* F*’*a*. Thus we obtain U*(W* (F*)) = Tor and MclD, (Tor) follows.
+
+
This leads us to the following definition: 5119. DEFINITION (TS”). F M d ( X ) . Mcl,, ( X ) & MCl,, ( X )& On E = Ker ( X ) . ( X is a f u l l model-class if it is a model-class with S1 and D2 such that On E X and V = Ker ( X ) . )
cX& V
5120. THEOREM (TSS”). Tor is a full model-class in the sense of ‘3 (Fup). In theories with stronger regularity axioms the notion of a full model-class can be simplified.
5121. LEMMA.(1) (TS, U3) M is a full model-class iff it is a model-class with S1 which contains all urelements.
(2) (TS, D3) M is a full model-class iff it is a model-class with S1. Proof. (1.) Every full model-class must contain all urelements since Ur $ M implies Ker ( M ) =+ V. Conversely if Ur c M then V = Ker ( M ) . 243
5122
FULL UPWARD EXTENSION
[CHAP. V
SEC. 1
We proved in TSS, U3 that every model-class contains all ordinal numbers. Further it follows easily in TSS, U3 that every model-class is a model-class with D2 (moreover, in TSS, U3, dTSS/aGt (M) we have (U3) '*'"I). (2) is an immediate consequence of (I). 5122. METADEFINITION. (1) TS' is the theory TS", FMcl(M). (2) Up is the extension of air (dTs/d%(F~p)) interpreting M as Tor. FMcl (M) is a fixing in TS" (cf. arrow 3 of Diagram 5112); by 5120, the direct translation Up is a model of TS' in TSS' (arrow 4). The model Up is of great importance; we shall prove in the course of the subsequent discussion that Up is faithful. 5223. METALEMMA. 'Dir (dTSS/dGt (M)) is a model of TSS", S5 in TS", dTSS/dGt (M).
Demonstration. (See arrows 5 and 6.) D)ir (aTSS/dGt (M)) is a model of TSS; (Sl) and (D2) hold in this model. Hence it suffices to show that (St) and (Pot) hold. We proceed in TS', dTSS/dGt (M). Denote the notions in the sense of Gt (M) by means of the superscript 0. Recall that On E M. It is obvious from the definition of Ord (x) that Ord (x") + Ord" (xo); hence On E E On". Suppose Onn - On $. 0 and let EO~O - On. Then Po E M and so fir-]is a set. On the other hand, for every CL E On we have a E fi' becai.ise c1 E 8' v fi" E CY holds and 8" E a implies 8" E On. But this means that On c fi" which is a contradiction. Thus wc have proved that On = = On". Now let on c Ona. Then a" is a set of ordinal numbers (we are working in a set theory!) and so if it is non-empty then it has a first element. This proves (St)l'. We now prove (Pot)". If S" is a totally universal rclatjon in the sense of Gt (M) and a is an element of M then, for each u c a , let F'u be the set of all S"-codes of x of the lest rank. (All semisets are sets?). Then F is a function, D ( F ) is a set and hence u ( W ( F ) ) is a set. Hence if we define So = So u ( W ( F ) ) then So is a set (since So is regular and D (So) is a set) and So E M, hence So is a semiset in the sense of G t (M); let 0" = So. Evidently or: is a "semiset coding" of all subsemisets of a in the sense of Et (M). This proves (Pot)U and hence (S5)O follows.
r
5124. The model Up (arrow 4) can be extended to a direct model of TS', aTSS/aGt (M) in TSS', dTSS/dGt* (Tor) (arrows 5, 7, 8) where Gt* (Tor) is the following F-definition: (VX*) ([Ma () Xg
(vx~?, Y o ) (X" 244
=
EO
X*)
Yo
3
X * E Tor] & sec
= X"
E*
Y E )& sec .
CHAP. V SEC. 11
5127
FULL UPWARD EXTENSION
(This definition is induced by the model U p and by the F-definition Gt (M) added to TS‘.) We prove that the composition of arrows 6 and 8 is equivalent to the identity 1 * 7. Lemma 51 13 which was proved in TSS’, has the following 5125. COROLLARY (TSS’, dTSS/aGt* (Tor)) (VX)(3!XU)(Imb”X = X”)&(VX3)(3!X)(Imb”X =Xu);
Xo = I m b ” X & Yo
=
Imb’ Y . -+ [ X E Y = X u +
YO].
Our result follows by 1269. Consequently 6 * 8 is faithful and therefore 6 is faithful. We have also the following useful 5126. METACOROLLARY. Let q ( X , 0) be an arbitrary TSS-formula. Then TSS’, dTSS/dGt* (Tor) t (X”
=
Imb” X & 0) -+ q(X, 0)
= qL(X“, 0) .
Similarly, the model describcd by arrow 6 can be extended to a direct model of TSS”in TS’, dTSS’/dGt (M) (arrows 1, 9, 1.0). In this latter theory 0-variables, and all 0-notions correspondingtoTSS-notions,are first defined and we know that all axioms of TSS”, S5 in the 0-sense are provable in the terminal theory of arrow 6. In particular, we have the constants FupU, Toru and Imbu and we can give the definition of *-variables as saturated parts of C(Fupo) (nore that it does not matter if we say “in the sense of Gt(M)” or not because the notions of a relation and a saturated part are absolute). Furthermore, we can derive the definitions of all *-notions corresponding to TS-notions. We know that FMcl* (TorU) is provable in TS’, dTSS‘/dGt (M). We have the following 5127. LEMMA(TS’, aTSS/dGt (M)). There is a morphism K of E and Fupu such that K ” M = Torn. Proof. We start with hubo which is an isomorphism of E f M and FupO Toru. Define H ” ( 0 ) = Imb’;
r
c1
> 0 + H”(c1)
= ( ( y , x);
x c D (H”a)& Ext,,,u ( y )
=
(H”a)” x } .
It follows from Ker (M) = V, by the construction of Fup” and by the absoluteness of ordinal numbers, that if we set K = H”On then K is a morphism with the required properties. 245
5128
FULL UPWARD EXTENSION
[CHAP. V SEC. 1
5128. COROLLARY (TS', dTSS'/dGt (M))
(VX) (3! X * ) ( X * X*
=
=
K"X& Y*
K"X)& (VX*) (3! X ) ( X * =
K'Y, + [ X
E
Y
X*
=
E*
K'X) ; Y*] ;
K"M = Tor" .
By Lemma 1269 and the remark following it, the composition of arrows 4, 10 is a faithful model. Hence the model Up (arrow 4) is also faithful. We summari7e our discussion into the folowing 5129. METATHEOREM. All arrows in the diagram 5112 are direct models and the diagram commutes. Hence (1) the model Up (arrow 4), i.e. the extension of lOir((iTS/d92(Fup)) interpreting M as Tor, is a faithful model of TS' in TSS'; (2) the model a i r (dTSS'/dGt (M)) (arrow 6) is a faithful model of TSS", S5 in (TS", dTSS/aGt (M)).
c ) Kelution.~,cardinalities, regularity and choice i n the sense of the model
UP As we have diown, Up is a model of TS", FMcl (M) in TSS'. Recall that *-classes (clarses in the sense of Up) were defined in TSS' as saturated subclasses of the relation Fup (full upward extension of the relation E) and E* was defined as usual by ill(Fup). We also defined the class Tor and proved FMcl" (Tor) and defined the isomorphism Imb of V and Tor. We proved Lemma 51 13 which can be interpreted as follows: Imb makes sets into *-elements of Tor and semisets into *-subsets of Tor. Using this lemma and Metatheorem 51 14 it is easy to characterize various notions of the model (*-notions) in terms containing no asterisked symbols; in particular, we can characterize some important *-notions concerning *-elements of Tor in terms of simple notions concerning sets and semisets. The following lemmas should be understood as the most important examples of such characterizations. 5130. LEMMA (TSS"). If x* (a) Rel* (x*)
= Re1 ( 0 ) ;
(b) Un*(x*)
= Un(a);
(c) D* (x*)
=
246
D(a).
=
Imb"
0
then
FULL UPWARD EXTENSION
CHAP. V SEC. 11
5136
Proof. (a) and (b) follow by Metatheorem 5114. By the same Metatheorem, we have y E D (c) = Imb” y E* D* (x*) and (c) follows. Furthermore, in the course of the discussion 511.8 we proved the following
5232. LEMMA(TSS”). On* = Imb” On; i.e. x E On iff (Imb’x) E* On* and if x* E* On* then there is a n x E On such that x* = Imb” x. We now consider equivalence of sets. Suppose x* = Imb” x and y* = = Imb” y. If x* x * y* then there is a n f* c x* x * y* such that Unf (f*,x*, y*). From x*, y* E* Tor we deduce f* c Tor and hence by Lemma 51.1.3 and Metatheorem 51 14 we obtain (la) Un, ( 0 , x, y) i.e. x G y (x is absolutely equivalent to y , cf. the definition 2201). Conversely, if x x y then obviously x* w * y*. This proves the following. A
5232. LEMMA (TSS“). If x* = Imb” x and y* = Imb” y then x* w * y* = & y ; similarly, x* y* = x y. We are thus led to definitions which differ from the customary definitions in that certain bound set variables are replaced by bound serniset variables.
<*
=x
5133. DEFINITION (TSS’).
card (a) = &nf(a, (a i s
1 (3p < a ) ( p & a) ( a is an absolute cardinal) ,
8) = ( a 5 p&
(36) (Un (a, a, p)& U(W ( 0 ) ) = p)
absolutely cofinaI with
5234. LEMMA.(1) -,eonf(a, p). (2) (TSS‘) Let a*
=
(TSS’)
p).
e a r d (a) -+ Card (a) and Conf (a, p) -+
Imb’ a, B*
=
Imb’ p. Then
(a) Card* (a*) = e a r d ( a ) , (b) Conf* (a*, p*) = eonf (a, p) . 5135. LEMMA (TSS’, S4). Every natural number is an absolute cardinal; o is an absolute cardinal. Proof. We prove that if n is a natural number and c is a mapping with domain n then c is a set. We denote this statement by q ( n ) and let A = = ( n ;q ( n ) } (this class exists by Metatheorem 4121). It follows by induction, using (St), that A = o.The lemma follows. 5136. LEMMA(TSS‘). The following are equivalent: (1) Cardinals are absolute (i.e. every cardinal is an absolute cardinal), (2) for every a, if a* = = Imb’ a then K,*,= Imb’ (KJ. 247
5137
[CHAP. V SEC. I
FULL UPWARD EXTENSION
Proof. Obviously Imb” K O is the first limit ordinal number in the sense of Up. (Note that /? = u + 1 is a restricted formula.) Hence K,*, = Imb” (KO). Suppose that all cardinals are absolute. If K,*,= Imb’ ( H a ) holds for all a < A and if A* = Imb’ 1- then K;* is some Imb’fi; certainly fi 2 K, (because e a r d (fi) implies Card (b)); the converse inequality holds by the absoluteness; Imb” Kj, is the *-least cardinal number *-greater than all K:, (a* < * I.*) and (2) is proved. The implication (2) + (1) is obvious. 5137. LEMMA(TSS‘). Suppose x * following are equivalent: (1) x*
=
Imb’x and y*
=
Imb”.
The
<*P* ( y * ) - (o*)* ,
(2) ( 3 ~(Exct ) (Q) & D (Q) Proof. In TSS‘, x
c y*
y).
< P ( y ) - (0) = ( 3 f ) (Un ( f , x, P ( y ) )& Un, (f))=
= (3r)(Exct ( r ) & D(r) (3r*
= x & W (Q) G
=
x & W(r) E y).
Hence (1) is equivalent to
x * x*) ( E x c ~ (*r * ) & D* ( r * ) = x * & W* ( r * ) c y*)
which is equivalent to (2) by Metatheorem 51 14. We wish now t o characterize some notions concerning cardinalities of set9. To do this we need (El)*,so we ask what must be added to TSS‘ in order that (El)* be provable. In the following metatheorem we answer this question and we also give conditions for certain other axioms of regularity and choice to hold in the sense of Up. 5138. METATHEOREM. Up is a faithful model of
(a) TS, U3, FMcl (M) in TSS‘, U3 ; (b) TS, D3, FMcl (M) in TSS”, D3 :
(c) TS”, E l , FMcl (M)in TSS‘, ElO,where (E10) is
(q (30) ( 3 4 (Un,
a))
( 0 , x>
(consequently, U p is a model of TS”, E l , FMcl (M) in TSS‘, El); (d) TS, E2,FMcl (M) in TSS’, E20, where (E20) is ( 3 F ) (Un, ( F , V, On) (consequently, U p is a model of TS, E2,FMcl (M) in TSS’, E2); (e) TS, D3, (M = Cstr) in TSS‘, D3, (Constr) ; (f) TS, D3, FMcl (M) & M =k V in TSS’, D3, 1 (C3) (consequently, U p is a model of TS, D3, 1 (Constr) in TSS‘, D3, 1 (C3)). 248
CHAP. V SEC. I ]
FULL UPWARD EXTENSION
5138
Demonstration. (a) It suffices to prove (U3) E (U3)'n(FUP) in TSS'. By Metacorollary 5126 we have (U3) E [(U3)Gt(M)]W(Fup) in TSS', dTSS/i?Gt* (Tor))(arrows, 6 , s ) ;hence if we prove (U3) = (U3)Gi(M)in(TS", dTSS/i?Gt (M)) (the terminal theory of arrow 6) we obtain (U3)R(Fup) = = [(U3)Gt(m)]R(Fup) in (TSS', aTSS/dGt* (Tor)) (by arrow 8) and hence (U3) = (U3)R(Fup) in the last mentioned theory. Since the formula (U3) = E (U3)R(Fup) is a TSS'-formula, and the idcntity arrow 7 is conservative, we obtain TSS' t (U3) = (U3)"F"p). We proceed therefore in (TS', aTSS/dGt (M)). Obviously (U3) ( U3)"'('). Conversely, suppose (U3)u holds. Since M is full, V = Ker (M) from which it follows that Ur c M and Urn = Ur; thus Mn(Urn) implies M(Ur). Furthermore M = = Ker" (Urn) c Ker (UrF) = Ker (Ur); hence if 2 z Ur and P (2) c 2 then Z z Ker (Ur) 2 Ker" (Ur") = M, hence Z z Ker (M) = V. Thus V = Ker (Ur) and so (U3) holds. (b) is demonstrated in the same way. = (E10) in TSS'. Suppose and let x be (c) We prove an arbitrary set. Set x* = Imb" x. There is a y* G Tor and an x* such that Unz (y*, x*, a*). Setting Imb' a = y* and Imb" a = x* we obtain Un, (a,x, a) by 5130. Conversely, suppose (E10) and let y* be a set in the sense of %(Fup). Since Fup is regular, y* is a semiset, CT say, and it follows from (E10) by (St) that there is a semiset e and an ordinal number a such that Un, (Q, a, a). Hence a can be Sm-well ordered and so by (St) there is a semiset a. G a such that distinct elements of a. have distinct extensions (in Fup) and such that for each u € 0 there is a E E C T with ~ the same extension. There exist p o , cxo such that Un, (eo,ao, ao). Now define xEz
= ( 3 u E a) ( 3 y E ao) (EdFup(x) 82 (79
=
(Imb' y , ExtFup (u))* &
u>E Qo) .
z is saturated and there is a q E C (Fup) such that z = EXfFup(4). Heme z is a set in the sense of 111 (Fup); let q* = z; it is easy to see that Unz (q*,y*,a,*) where txx = Imb' txo. (d) is proved analogously. (e) It suffices to prove (M = C ~ t r ) ' ( ~= ~ ~(V ) = Cstr) in (TSS', D3). But (V = Cstr) E [(V = C ~ t r ) ~ ~ ( ~ is) ] provable ~ ( ~ ~ ~ )in (TSS', D3, dTSS/dGt* (Tor)) and (V = Cstr)Gt(M)= M = Cstr is provable in (TS', D3, aTSS/aGt (M)). The assertion follows. 249
5/39
[CHAP. V SEC. I
FULL UPWARD EXTENSION
+
( f ) We know that TS, D3 k (V Cstr) =_ (3X)(Mcl ( X ) & X $: V). The identity model of TS, D3, (V =I= Cstr) is a model in TS, D3, (FMcl (M) & & M 4 V); hence it suffices to demonstrate that Up is a faithful model of TS, D3, (FMcl (M)& M V) in TSS', D3, 1 (C3). To d o this we prove (V =k M)"(F"p'= 1 (C3) in TSS'. Suppose (C3), i.e. every semiset is a set. Then it follows from the construction of Fup that C(Fup) = Tor; hence (V = M)W(Fup'. Conversely, if there is a semiset u which is not a set, then 0 x (0) is a subsemiset of 'Tor which is not a set (and which is obviously Fup-saturated), hence there is a JJ E C (Fup) such that ExtFup (y) = 0 x (0). Obviously y Tor and hence Tor C (Fup), i.e. (V M)W(Fup).
+
+
+
We now consider TSS', E l . Suppose x* = I m b ' x and a* = Imb" a. * - ===;* I hen LY - x (i.e. u* is the cardinality of x* in the sense of Up) iff u is the smallest ordinal number y such that y 2 x. As a consequence it follows (in TSS'. E l ) that for every x there is the smallest y such that y & x. The reader can easily show that the last statement is provable in TSS', S4, E l (recall that S4 = , S l & St), i.e. we need not use D2 and (Pot). Hence we introduce the f(d10wi1ig I .
5139. DEFINITION (TSS', S4, El). For each x, 2 is the smallest y such that s (2 is called the absolute cardinality of x). 7 =
5140. LEMMA (TSS", El). If x* x** i f f a = 2.
= 1mb"x
and a*
= Imb'a
then CY*=
I n TSS', S 5 , El we can prove that for any x there exists a smallest a such that all subsemisets of x can be one-one coded by elements of a, i.e. such that ( 3 ~(Exct ) (Q) & D ( Q ) =
CY
& (Vu
E X)
(31) (u = Ext,
(L))
Iiidced, by (Pot) there is a coding eo of all subsemisets of x. By EL and St, D (eo)can be Sm-well ordered and by St we can obtain a Q, which is a nowhere constant coding of all subsemisets of x (Note that el can be defined by a seminormal formula.) If the absolute cardinality of D (el) is u we obtain from Q, a nowhere constant coding of all subsemisets of x by all elements of IX. If there were a coding Q2 with D ( Q , ) E O and ~ D(e2) < a then we shodd have a one-one-mapping of o! onto D(e2) which contradicts the fact that a is the absolute cardinality of some set and hence an absolute cardinal. 250
FULL UPWARD EXTENSION
CHAP. V SEC. I 1
514.5
This justifies the following definition: 5741. DEFINITION (TSS‘, S5, El). (a) For every a, 5(2,a) is the smallest y such that (3q) (Exct (Q) & D (e) = y & (Va E: a ) (3) (a = Ext, (1))). (i.e. such that all subsemisets of a can be one-one coded by y); (b) For any
X,
/?, 3(fl,a ) is the smallest y such that
(i.e. such that all semiset mappings of a into
fl can be one-one coded by y).
5142. LEMMA (TSS’, El). Let a, /? be infinite absolute cardinals, set a* =
/?* = Imb’ /?.Then
= Imb“ a and
(a) X* (2*, =*)
=
Imb’ 3 2 , a) ,
(b) 2* (fl*, x * ) = Imb” i(fl, a) . Proof. RccalI that 2 ( K p , K,) is the same as
Kr’.
( a ) If y* = Imb’ y then the following are equivalent: (1) y* w * P* (a*I? ( 2 ) there is an exact functor 4 with D(Q) = y which codes all subsemisets of a. Therefore the smallest y* which is *-equivalent to P* (a*) equals to Imb*3 2 , a).
(b) is proved analogously. In the same way the reader may show that the following definition is correct and prove the lemma following it. 5 143. DEFINITION (1 ) (TSS’, S4).
Gf ( E ) = Min {fl; Conf (fl, a)} , ( 3 ) (TSS’, S5, E l ) i{a)= 4(a, cf ( x ) ) . 5144. LEMMA (TSS’, El). Let a be an absolute cardinal and set a* Emb” a. Then
=
=
(a) cf* (a*)= Imb’ @(a)) ,
(b) I* (a*)
=
Imb’ i ( a ) .
S145. LEMMA (TSS’, S4). cf (cf (K,)= i f (N,).
(The proof is analogous to the proof of 2236.)
251
5201
ULTKAPROOUCT
[CHAP. V SEC. 2
SECTION 2
The ultraproduct model
In the preceding secrion we studied the extending model Up which, roughly speaking, “extends” the universe of the theory of semiscts. The theory TSS‘ in which we studied this model has the general assumption of the existence of a locally semiset total Boolean support as an axiom. But, because of various relative consistency proofs, it is necessary to study the model Up as a model in theories stronger than TSS‘, in which it is assunied that there exists a total Boolean support on an algebra with some particular properties. Naturally, the question arises whether such theories are consistent (relative to TSS). The task of the present section is to obtain a positive answer to this question. (See Metatheorem 5215). The well-known ultraproduct method is used is used to construct appropriate models of TSS’. 5201. DEFINITION (TSS’). If U is a real ultrafilter on a Boolean algebra B and P is a partition filter 011 B then we define
U l C (B,P )
=
If’; Un ( f ) & D ( f ) E P } ,
(the ultraproduct class),
(the ultraproduct relation). 5202. Remark. If P is the partition filter containing all partitions ofB then we write Ulc ( B ) and Ulr ( B ,U ) instead of Ulc ( B , P ) and Ulr (B, P , U ) ; 252
CHAP. V SEC. 21
5204
ULTRAPRODUCT
if it is clear which B, P and U are concerned we write Ulc and Ulr. In what follows Bool (B, P, U ) means “ B is a complete Boolean algebra, P is a partition filter on B and U is a real ultrafilter on B”. Note that Ulc and Ulr are real classes.
-5203. DEFINITION (TSS, E2, Bool (B, P, U)). For any x we let k, be the functions which assigns x to the unit element: k,
= {<x,
I,>}
.
For any X we let =
{ f U~I C ; V { U E D(f);f‘u E X } E U} .
Note that Ext,,, (k,)
=
2.
5204. LEMMA (TSS, E2, Bool (B, P, U)). Ulr is a nonempty extensional relation; for any f , g E Ulc we have Ext,,, ( f ) = Ed,, (g) iff
Proof. Clearly Ulr is nonempty; if x E J’ then ( k , , k y ) E Ulr. We write f instead of ExtUlr( f ) . We prove the equivalence
the extensionality then follows immediately. We suppose first that u ; f ’ u = g’u} E U (for brevity we omit the clause u E D ( f ) etc.). If h ~ f t h e nV{w A u ;h’w ~ f ’ u E} U; since
V(u A
v { U A U;f’U =
= v{u
g’L’3 A
A U A W;f’U =
v { W A U ;h ’ W E f ’ U }
g’U& h’W
h’w E g ’ U & f ‘ u = g’U}
=
Ef’U} = v { W A U A U;
v{w
A 0; h’W E
g’u} ,
it follows that the last join is in U and so h E 4. We now suppose that V { u A u ; .f’u = g’u} 4 U. It follows that V { u A u; f ’ u =Ig’u} = E U so that e.g. V ( u A u ; f ’ u - g’u $: 0} E U. Using a selector we ca? define a function h on D (,f) A D (9) such that h’w E f ’ u - g’u whenever the conditions w 5 u E D (,f), w 5 u E D (g) and f ’ u - g’u =k 0 are fulfilled. Thus we have h ~ f a 7 dh $ 6 . 253
5205
ICHAP. V SEC. 2
ULTRAPRODUCT
x
5205. LEMMA(TSS, E2, Boo1 (B, P, U)). For any X , is a saturated part of Ulc. (Exercise.) We shall consider the model air (dTSS/a,JI (R)) in TSS, E2, Boo1 (B, P,U) & R = Ulr (B, P, U), (iTSS/d% (R). We may introduce setvariab!es in the sense of % (R). We have the following important metatheorem, i n which f again denotes Ext,,, (f).
5206. MFTA~HFORLM. For a n y normal TC-formula (p(x, 0 , X , 0 ) wrthout constants the following is provable in TSS, E2, Boo1 (B, P, U). dTSS/(i% (Ulr (B, Y,U)):
Demonstration. By induction. If cp is atomic then cp is either s E X ; we have f ~ .y^*= (f, y) E Ulr = V ( u A v ; f ' u E y'o) E U and x
j
or
E
fE*X
r . f E X r V { l t ; f ' u E X ) E U .
If cp is cp, & cp2 then we have
Suppose finally that cp is (3y) $ ( y , xl, 0 ) . If cp*(f,, 0 ) then there exists y such that $*(ij,fl,0 ) ;we have V { v A u I A 0 ; $ ( y ' u , f ; u l , 0 ) ) E U and so V ( u , A 0 ; ( 3 ~ 1i//(y,, ) f i u , , o)} E U. Conversely, if V { u , A 0 ; (3p) $ ( j , f ; u , , 0 ) ) E U then we let j = D ( f l ) A 0 ; clearly j E P. w e let ( y , it) E R
= u Ej & 5
254
U18L
(3u,, 0 ) ( u , E D ( f l ) & .& $(Y,.f;ul,
0)).
0&
u
5
CHAP. V SEC. 21
5209
ULTRAPRODUCT
Let F , c R be a function such that D(F,) = D ( R ) . Since VD(F,)EU and F , is a set, we may letf = F , u { (OB. -VD ( F , ) ) } . C l e a r l y f ~Ulc and we have
v{u
A
ui
A 0;
$(f’u.f’;ui,
0))EU,
hence $* (fl, 0 ) . Remark. In the course of the above proof we could use T S S instead of T S S + E2.
+ (D2 & El)
5207. COROLLARY. For any closed SF formula cp the following is provable in TSS, E2, Bool (B, P, U), c?TSS/d% ( U r (B, P, U)):
5208. METACOROLLARY. Axioms ( A l ) to (A7), (Bl) to (B7) and (C1) hold in Dir (dTSS/d% (Ulr (B, P, U)) as a model in TSS, E2, Bool (B, P, U), dTSS/8% ( U r (B, P, U)). Demonstration. Axioms A1 to A7 and CL can be expressed equivalently (in TE, d l , d2) by normal set formulas. Therefore, by the preceding corollary, they hold in the model. In particular, A1 holds in the model and therefore TSS, E2, Bool (B, P, U) k ( U r is a pairing relation). Consequently, TSS, E2, Bool (B, P, U) k ( U r is an E-like re1atio.n) and so Axioms B1 to B7 hold in the model. In the sequel we shall be interested in TSS, E2, Bool (b, p, z). The fixing of constants b, p, z by Bool (b,p, z) is permissible in TSS, E2 since E2 implies the existence of an ultrafilter which is a set on any complete Boolean algebra which is a set.
5209. METATHEOREM. air (8TSS/d% (Ulr (b, p, z)) is a model of TSS, E2 in TSS, E2, Bool (b,p, z), dTSS/d% (Ulr (b, p, z)) . Demonstration. By Metacorollary 5208 it suffices to show that C2 and E2 hold in the model. To prove that C2 holds in the model it suffices (by 5106(1)) to prove in TSS, E2, Bool (b, p, z) that Ulr (b, p, z) is almost regular and almost universal. We proceed in TSS, E2, Bool (b, p, z). Let f~Ulc,letq=U(W(f))andletg~= a .g~UIc&W(g)cq&g~*f; we prove that SAT,,, ( a ) = ExfUlr(f). Let g E Ext,, (f) and suppose x E q ; set u = -V{u; g’u c q } . Clearly u I$ z. If u =?= 0, let h’u = x and let h’u = g’u whenever u E D (9) and u A u = 0,. It follows that h E a and 255
5210
cj
=
/I E
[CHAP. V SEC. 2
ULTRAPRODUCT
f i ; hence g E SAT,,, ( u ) . Conversely, if g E SAT,,, ( u ) then we choose h = 4; it follows that 11 E f and so g €1.
u such that
We have proved that Ulr is almost regular. If a E Ulc then we let x = = W ( f ) ; clearly a _c X and so Ulr is almost universal.
u
fEa
We prove (E2)*. Let A be a mapping of V into On; i.e. suppose that x) E A ) . It follows by Metatheorem 5206 that
(vx) ( 3! y ) (Ord ( y ) & ( y ,
hence A is a mapping of V into On in the sense of % (Ulr). By the following lemma, A is a real class in the sense of the model. LEMMA(TSS, E2, Bool (b, p, z), dTSS/Z% (Ulr (b, p, z)). For any real class X we have Real* Proof. l r f E Ulr then we let g be such that D (g) = D ( f ) and g'u = = f'. n X for all u E D (f). Clearly g E Ulc and 8 = j; n 1;hence Real*
(x).
(x).
5210. Remurk (TSS, E2, Boo1 (b, p, z), dTSS/d% (Ulr (b, p, z)). I n the sense of % (Ulc (b, p, z)) b is a complete Boolean algebra (by Metatheorem 5206). 5211. Wc shall now restrict ourselves to the case where p is the partition filter consisting of all partitions. Bool (b, z) means "b is a complete Boolean algebra and z is an ultrafilter on b".
5212. THI:OKEM (TSS, E2, Bool (b, z), dTSS/8% (Ulr (b, z)). In the sense of 91 (Ulr (b, z)) there exists a complete ultrafilter Z on 6 ; moreover, for all u ~ b k,, c Z i f T u E z . Proof. For f~ Ulc we let f~ Z = V{u A f " u ; u E D ( f ) & f ' u E b} E z. We write [ l ( u , f ) instead of u E D ( f ) & f ' u E b and y(u, v,f, g) instead of 11 E D ( f ) & v E D ( g ) & f ' u = g ' ~ . 1) Z is a saturated part of Ulc. Supposc f~ I and = Ext,,,
( f ) . Since V { u
= v { U A f'u
A 0;
= v { U A 0 A g'v;
f ' u ; ~ ( u . f ) }A
V{U A
0;~
=
f, where f =
( uv , f , g ) }
s
B(u,f)& Y(u, V , f , Y)} =
b(V,Y)&
< v{V A g ' v ; S(u7 9 ) )
-
A
4
Y(U, v,f, 9 ) )
3
it follows that g E Z. (The first inequality follows from the fact that the elements of D ( f ) are mutually disjoint.) Hence CIS*(Z). 256
CHAP. V SEC. 21
5214
ULTRAPRODUCT
2) Clearly Z G b and k l bE 2. 3) Let f~ b and let W (f) E b. We let g be such that D ( 9 ) = D (f) and g’u = - f ’ u for all u E D (f). Clearly, 8 is the complement of f in the algebra b in the sense of Ulr. For any u E D (f) we have u = (u A f’u) v v (u A g’u) and so =
hence f~ 2
v{u
A
f ’ u ; u E D (f)) v v { u
A
g’u; u E D (9)) ;
= g # 2.
4) Let h c b and let V*hE* 2; we prove h n Z + 0. Since h c 6 we have V { u ;h’u c b) E z and we may suppose V { u ;h’u c b) = 1,. Defining f by D (f) = D ( h ) and (Vu E D (h)) (f’u = V h ’ u ) we have = V*8. Since 2 wc have V { V h ’ u A u;u E D ( h ) ) E 2. For every u E D ( h ) let be a well-ordering of the elements of h’u. Let (w:; u E C,}, be a sequence of pairwise disjoint elements such that w, 5 u, and V w: =
!E*
{u:; u E ).5 =
V 0: (i.e. L‘
w;f = u;f and w:
5u
=
uz - V w; for u > 0). The set containing B
all elements w,“ A u (u E D (h),a E [.) together with the element - V ( V h ’ u A u ) is a partition of b. We define a function g whose domain is this partition by setting g’(w: A u) = u: for each a, u and g’(-V(h’u A u ) ) = 1,. For this g we have the following: (a) g E Z because V { g ’ u A u ; u E D ( 9 ) ) 2 2 A W: A U)) = A 24) = Vh’U A U E Z;
v(v(Uz urn
(b) g
E
v(vW: u a
h because V{u A
u ; g’u E
Y
h’u) 2
v (w:
u*u
A
u) = V ( v h ’ u
A
u ) E z.
U
HencegEZnh. In view of 1-4,Z is a complete ultrafilter o n 6 in the sense of W (Ulr(b,z)). As for the additional property of 2, we have V { u A klu; u E D (k,)}
=
V {1,
A
u} = u ;
hence u E z = k, E 2. Using the above theorem we obtain some important consistency results. 5213. METADEFINITION. We say that a set-formula /3(x) describes a complete Boolean algebra in a theory T (stronger than TSS’, say) if T F 1 (3x) /3(x)& (Vx) (/3(x) + x is a complete Boolean algebra with at least two elements).
5214. METALEMMA. Let r be a sequence of closed set formulas. If p(x) describes a complete Boolean algebra in (TSS, E2, T‘), and if 9(6,Z) is 251
5215
ULTRAPRODUCT
[CHAP. V SEC. 2
a formula which asserts in TSS’ that b is a complete Boolean algebra and 2 is a complete ultrafilter on b, then the axiom
is consistent with TSS, E2, r. Indeed, by Metatheorems 5209 and 5206 and by the preceding theorem, a i r (dTSS/a!R (Ulr (b, z)) is a model in the following diagram:
5215. METATHEOREM (Support Principle). Let r be a sequence of closed set formulas. If p(x) describes a complete Boolean algebra in TSS, E2, r and if 9(b, Z) has the same meaning as in the preceding metalemma then the axiom
is consistent with TSS, E2, r. Indeed, Dir ( d T S S / d G u p p (Z)) is a model in the following diagram:
5216. Remarks. (1) Axiom ( S p ) is the strengthening of (S6) promised above; whereas by (S6) there exists a total support which is a complete ultrafilter on some complete Boolean algebra b, here we assert that the algebra in question satisfies P(b). (2) If T is a term such that T I- (T is a complete Boolean algebra which is a set) then (ST) has the same meaning as (S(b = T)) i.e. (3Z) (9(T, 2) & & Z is a total support).
258
CHAP. V SEC. 21
ULTRAPRODUCT
5217
(3) Consider the following diagram:
The theory TSS, E2, (Sg) is stronger than TSS”, S5 (arrow 1); Up is a model of TS”,FMcl (M) in TSS‘ (arrow 3) and TSS‘ is a conservative extension of TSS”, S5 (arrow 2). The theory TSS, E2, ( S g ) , Fupw (Fup), dTS/d%(Fup) is a conservative extension of TSS, E2, (Sg) (arrow 4) and is stronger than TSS” (arrow 5). Consequently Up is a model of TS”,FMcl (M) in the terminal theory of 5 (composition of arrows 3 * 5). As we know from Section 1, E2 holds in the last model. For the sake of brevity we denote the sequence Fupw (Fup), dTS/d%(Fup) by (+ upw), so that the terminal theory of 5 is denoted by TSS, E2, (SQ), (+upw). The last theory is a conservative extension of TSS, E2, (SQ). If I‘ is a sequence of set formulas and j? describes a complete Boolean algebra in TSS,E2, r, we know by the support principle the (Sg) is consistent with the last theory and therefore TSS, E2, (SQ) upw is consistent relative to TSS, E2, r. This is a very important corollary of the support principle. Let us give an example to show how this principle is used.
+
5217. Example (Independence of the axiom of constructibility). Let Noat (6) be a set formula which asserts in TSS’ that 6 is a n algebra with no atoms. By 2536 (36) Noat (6) is provable in TSS, E2 and so TSS, E2, (S Noat) upw is consistent (relative to TSS). In this theory (3.) (c is not a set) (i.e. 1 C3) is provable (the support on b cannot be a set) and is equivalent to (3x* E Tor) (x* #* Tor). Hence Up is a model in the following diagram.
+
L--
I
But the former theory is stronger than TS, D3, E2, i (Constr), so that 1(Constr) is consistent with TS, D3, E2. 259
5218
ULTRAPRODUCT
[CHAP. V SEC. 2
For more details and other consistency results see Chapter VI. The following metatheorem is a very important provability result. 5218. M E r A T t I E o R E M . Let f be a sequence of closed set formulas and let
cp be a closed set formula. Suppose that P(x) describes a complete Boolean algebra in TSS, E2, The formula cp is provable in TSS, E2, if and only if cp is provable in TSS, E2, f ,(Sg); in other words, (SP) extends TSS, E2, f
r
r.
conservatively w.r.t. set formulas. Demonstration. TSS, E2, f,( S P ) is a n extension of TSS, E2, r;hence if cp is provable in the latter theory then it is provable in the former. Conversely, suppose that cp is provable in TSS, E2, f,( S P ) and consider TSS, E2, f. The theory TSS, = (TSS, E2,
r, P(b), Boo1 (b, z), aTss/;%
( U r (b, z)))
is a conservative extension of TSS, E2, f ; we denote the notions of %(Ulr(b,z)) by the means of *. We construct a support model in 9t (Ulr (b, z)) i.e. consider TSS, = (TSS,, Supp* (Z"), aTSS/dGupp* (Z*)). By 5212, TSS, is again a conservative extension of TSS, E2, f ;we denote the notions of Gupp* (Z*) by the means of 0. By 5214 and 5215 Gwpp* (Z") is a model of TSS, E2, f,(Sp) and so TSS, k cp5. By the absoluteness of set formulas in support models we have TSS, t cp* and so TSS, t cp*. By 5207 we obtain TSS, I- cp and so TSS, E2, r t cp.
260
CHAP. V SEC. 31
CHARACTERISTTCS OF ALGEBRAS
5302
SECTION 3
Characteristics of complete Boolean algebras and properties of relations and functions
We shall investigate the way in which the properties of semisets, and especially the properties of semiset relations and functions, depend on the properties of complete Boolean algebras with total Boolean support. Before doing this we shall consider the important notion of an approximable mapping. We shall investigate this notion fmt in the theory of semisets without axioms of supports. In the presence of axioms of support the notion of an approximable function can be characterized in terms of the properties of complete Boolean algebras with total Boolean support, and it closely related to the problem of cardinality in the model Up. We begin with some important notions concerning b-sets. 5301. Remark. According to 4311, each b-set r determincs, for any x, a unique element rx of b; we shall write r,,, for rCx,+.
5302. DEFINITION (TSS'). A b-set r is called a b-relation (Relb(r)) if r E V 2 x b; a b-relation r is called a b-relation on x and y if W(r) E c x x y; a b-relation r is called a b-mapping or a b-function (Unb (r)) if
a b-mapping r is called a b-1-1-mapping (Un: (r)) if moreover
a 6-mapping r is called a b-mapping of x info y (Unb ( r , x, y)) if r E ( y x x) x b and if (rKr,K E y} is a partition of b for every L E x. 261
5303
CHARACTERISTICS OF ALGEBRAS
[CHAP. V SEC. 3
5303. LEMMA (TSS’). If Z is a total Boolean support on b then
(a) Re1 (0)
= (3.)
(b) Un ( B )
E
(c) Un,
(Relb ( r ) & B
=
r”Z) ;
(3) (Un’ ( r )& CT = r”Z) ;
= ( j r ) (Unf ( r ) & B
(0)
(d) Un (0, x, y)
=
r ” ~ ;)
= ( 3 r ) (Unb ( r , x, y ) & B
=
r”Z) .
Proof. The implications from right to left are obvious; we shall prove the converse. Suppose s is a disjointed relation such that v = s”Z. (a) If CT is a relation then u = r”Z where r = s n (Vz x b). (b) If Un (0)then we may assume that Relb ( s ) and s”Z = v . Let a c Vz be such that B c a. Since sKI,A sK21$ Z for every ( K ~ , L ) ,( K ~ i), E a (K, K ~ we ) have
+
v
b K , ,
KI*KZ
*
SK,J
6z
< K * , I > . EQ
and, denoting the above join by u, we have - u E Z , so that s”Z
=
r”Z
where r
= s
I b (- u) ;
we also have Unb ( r ) . The proof of (c) is analogous. (d) Un (0,x, y ) implies Un (u) and hence there exists a b-mapping r such that r”Z = B. For each L E x the system { r K LK; E y> is disjointed in 6, its join belongs to Z and so that the complement of this join does not belong to Z . We choose a fixed element K~ in y independently of L E X . Define f K o l = rKolv - V r K L ,and FKL = rkr (K =I= K ~ ) .Clearly P”Z = r”Z and liEY Unb (T.,x, y). Hemurk. Thus 6-relations may be regarded as “codes” for those semisets which are relations. The elements of the range of a b-relation can be considered as “indices” and the 6-relation itself can be considered as a “two parameter system of elements of b”; i.e. a 6-relation r can be considered not as assigning ordered pairs to elements of a complete Boolean algebra but as assigning elements rKIto ordered pairs ( K , L ) . In the same sense, b-functions may be considered as “parametric systems of disjointed systems i n b”; for, if i E x then the rKr( K E y) form a disjointed system in b. We shall now investigate semiset mappings. 262
CHAP. V SEC. 31
5309
CHARACTERISTICS OF ALGEBRAS
5304. DEFINITION (TSS’). We say that a mapping r~ of x into y is z-approximable if there exists a relation q E (y x x ) such that 1)uEq,
(4 4
2) (vu E D (4))(Ed,
(u can be embedded into a cylinder of gauge z). We say that mappings of x into y are z-approximable if every(semiset) mapping of x into y is z-approximable. (Notation: Appr (x, y , z ) . )
5305. LEMMA (TSS’). If Appr ( x , y , z ) and x l Appr ( x , , y l , z l ) . (Obvious.) In particular, we have
< x , y , < y , z1 b z then
5306. LEMMA (TSS’). Appr (x, y , 2) holds iff every mapping of x into y is a set; in particular, Appr (x, 2,2) holds iff every subclass of x is a set. Proof. The fact that every mapping of x into y is 2-approximable means that every such mapping can be embedded in a set relation having only singleton extensions. Such a relation must coincide with the original function and so this function is a set. Every subclass (r of x is similar to its characteristic function; i.e. to the function g such that, for any u E x, g’u = = 1 = u E t~ and Q’U = 0 = u $ 0 ; g is a mapping of x into 2.
<x
5307. LEMMA(TSS’). If Appr ( x , 2, 2) and ( y x x l ) APPr (XI, p ( Y ) , 2). Proof. Let 0 be a mapping of x 1 into P ( y ) ; define ( u , u) E e Clearly g 5 y x x , ; hence g is a set and so q is a set.
then
= u E du.
5308. LEMMA (TSS’). If Appr (K,,2, 2) then Appr (N,, P (a,),2). Proof. The assertion follows from the preceding lemma since apx x
0,
w ap.
Recall the definition 5133 of the predicate e o n f (a, 8). In TSS‘ we have Eonf(cf(K,), K,) for every b. In addition we have 5309. LEMMA (TSS’). If K, S cf(K,) then
Proof. Let f be a (set) monotone mapping of cf(K,) into K, such that
U(W(f))
= 0,.
263
5310
CHARACTEXISTICS OF ALGEBRAS
[CHAP. V SEC. 3
1) If a is a mapping of K, into cf (K,)such that u ( W ( 0 ) ) = cf (K,) then the composition off and a is a (semiset) mapping e of K, into K, such that
U(W(e))
=
K,.
2) Conversely, if e i s a mapping of K, into K, such that U(W(e)) = K,, then we let a be a mapping which assigns to each t < K, the least ( < cf(K,) such that f’( > e’t. We prove U(W(a)) = cf(t-4,). If q < cf (K,) then there exists 5 < o, such that a’t 2 q; for, if we let qo = = U(f”q) then qo < o, and there exists 5 such that e’5 > qo whence =
0’5 2 r].
5310. LEMMA (TSS’). I f Ka x
K, and
LX
5 p then e o n f ( X u , K,).
Proof. If a is a I-1-mapping of o, onto o, then u(W(o)) = a, and hence we have e o n f ( X u , K,). 5311. LEMMA (TSS‘). If K, < K, and 1Appr (K,,K,, K,) then e o n f (s, , K,). Proof. Let Un (a,K,, K,) and suppose that a is not K,- approximable. We shall prove that U(W(a)) = K,. If y < os and y + u ( W ( o ) ) then U(W (a)) c y and y < K,, i.e. a E y x a,, which contradicts the fact that a is not N,-approximable.
N, be regular and suppose that S, < K,. If e o n f (Nu,8,) then 1 Appr (Nu,N,, K,). Proof. Let a be such that Un (a,Nu,K,) and U(W ( 0 ) ) = 0,; we shall 5312. LEMMA(TSS’). Let
show that a is not K,-approximable. Let a c Y, let r E o, x o,and suppose that Ext, ( L ) < o, for every L E w,; then U(W (Y)) = o, since 0 c r ; by regularity we have Sup ( r ” { ~ )< ) o,for any L E 0,. Hence Sup (r”oa) = = U(W ( r ) ) < w,, a contradiction. Thus 7 Appr (K,,K,, K,). 5313. THEOREM (TSS’). If K, < cf (K,) then
e o n f (Nu,N,)
=
1 Appr (K,, cf (K,),cf (K,)).
This follows from lemmas 5309,5311 and 5312.
< D and K, 2 K, then 1 A p p r (Km, 2,2). Proof. If Un, (a,Xu, K,) then there exists a relation e c o, x oawhich
5314. LEMMA (TSS’). I f
LX
-
well-orders o,and which is isomorphic, by means of a semiset isomorphism, to E 0,. This semiset is not a set. Since o, x o, o, there exists a subclass of w, which is not a set; hence 7 Appr (Nu, 2 , 2).
r
264
CHAP. V SEC. 31
5318
CHARACTERISTICS OF ALGEBRAS
5315. THEOREM (TSS‘). If Appr (K,,2,2) holds for every a < p then every cardinal K, K, is an absolute cardinal. 5316. LEMMA (TSS’, S4). Appr (n, 2,2) holds for every n E o. This follows by 5135. We also know (in TSS’, S4) that o is an absolute cardinal. Using the notion of approximability we shall now find a condition equivalent to u < p & K a & K,.
In what follows we use Axioms St, Pot and E l ; thus the results hold in
TSS‘, El for example.
5317. METATHEOREM. If q ( x , Y) is a N F and if a is a semiset variable then TSS’, Pot, St, E l k (Vx -+
(3g
cb
E u)
x u ) (vx
(3a E b) ( q ( x , a)
E u)
-+
q ( x , @”{XI).
Demonstration. We give a proof in the theory TSS’, Pot, St, E l as follows. Let z E b x c be a coding of all subclasses of b; such a coding exists in view of Axiom Pot. Let f be a well-ordering of c ; such a mapping exists in view of El. In view of (St) we may define eo as follows: ( u , u> E go = = D E u & u E c & u is the least element in the ordering f of c such that V(V’
Ed,(4).
Clearly eo is a function. If we define
then
e has the required properties.
5318. LEMMA (TSS’, St, Pot, El). Let CI < p and suppose that eonf(K,,K,,). If K, 2 K, for every y such that CI < y < p, then tc, 2 K,.
Pruuf. Let a be a mapping of K, into K, such that U(W (a)) = og. For every 1 < o, there exists el such that Un, (el,a,,A): hence, by Metatheorem 5317, there exists e G (0, x om)x o, such that, for every A < o,, @”{A} is a one-one mapping of o,onto 1.For L, K < o,we define ~ ’ ( 1 ,K) = = (e”{o’~>)’ K. Clearly z”(o, x 0,) = as;hence there exists a one-one mapping of o, into w, x 0,. Since o, x w, x ma, there exists a one-one mapping of cop into 0,; since o, o, this completes the proof.
<
265
5319
[CHAP. V SEC. 3
CIIARACTERISTICS OF ALGEBRAS
S319. THEOREM (TSS’, S5, El). If Nu < N, then for Nu 2 N, to hold it is necessary and sufficient that 1 Appr (Nu, N,, Ny)for every regular K, such that a < y 5 j?.
Proof. If K, &. N, then, by 2204, N u & N, for every y such that a < y
5
I j?; hence by Lemma 5310 eonf (Nu, N,,).Conversely, suppose the condition holds. Let a < 1 5 p and suppose that Nu &. N, for all y such that a < ;! < 1.If Ni is regular then 1 Appr (Nu, K,, Nn) and by 5311. we have
conf (N,,KA). If cf (N,) 5 X u then Conf (Nu,N,), hence e o n f (Nu, KI); if Nu < cf(h’,) then using iAppr (Nu,cf(K,), cf(N,)) we obtain emf (Nu,NJ. Thus we have proved eonf (‘Nu,K,) and by Lemma 5318 we have h’, & Nn.Now, using SL and St, we obtain Nu & N,. 5320. THEOREM (TSS’, S5, EL). ( 1 ) In order that
and sufficient that
N, and all smaller cardinals be absolute, it is necessary for every a < j? .
Appr (Nu.Nu+I , Nu+
( 3 ) In order that N, and all larger cardinals be absolute, it is necessary and sufficient that Appr (Nu,N u + ,, Nz+ for every a 2
fi
Proof. (1) There is a non-absolute cardinal 5 K, iff Nu & Nu+lholds for some a < /?; but this latter condition holds iff 1 Appr (Nu,Nu+1, Nu+,) for some a < j?. The proof of (2) is analogous. We observe that (L) also holds for KO in view of 5135. Under the assumption that Nu M KO for some a > 0 we can prove the existence of complete ultrafilters on certain Boolean algebras. 5321. THEOREM (TSS’, S4, El). Let
N u 2 No and suppose that b is
a complete Boolean algebra such that the cardinality of the set q of all partitions of b I S Xu. Then there exists a complete ultrafilter on 6. Proof. Let g be a one-one mapping of ooonto q and let A be a selector for b. We define H ” { O } = { u } , where u $1 0, is some chosen element of Q’O, and H ” { n + 1) = { u } , where u 0, is some chosen element of ~’(+ n 1) which has non-7ero meet with A H ” ( n + 1). The meet AH”(n + 1) always exists. since If”(n + 1) is a set. We let Z, = H ” o and Z = = { w ; (jn) (w 2 f ~ ” ( n ) ) }z . is a complete ultrafilter on b.
+
266
CHAP. V SEC. 31
5325
CHARACTERISTICS OF ALGEBRAS
We shall now find out how the approximability of mappings depends on the properties of a Boolean algebra with total support.
5322. DEFINITION (TSS’). If b is a complete Boolean algebra and r is a b-mapping of x into y then we say that an element u of b is z-layered by r (w.r.t. b; Thru, (u, r, z)), if for every p E x,
( u intersects less than z elements in each “level”). We say that 6 is x - y - z-distributive (6,(x, y, z)) is, for every b-mapping r of x into y , the join of all elements which are z-layered by r is 1,. 5323. Remark. 1) In TSS’ + E l the notion of K, - K, - 2 -distributivity coincides with the usual notion of N, - XB-distributivity for complete Boolean algebras. 2 ) In TSS’ + E l , K, - K, - K,-distributivity is equivalent to the following property: For every collection of N, partitions, each of power 5 K,, there exists a partition consisting of K,-layered elements. 5324. LEMMA(TSS’). If there is a total Boolean support Z on 6 and if b is x - y - z-distributive then Appr (x, y , z). Proof. If C-J is a mapping of x into y then there exists a 6-mapping r of x into y such that r”Z = C-J. Since the join of all z-layered elements is l,, there exists a z-layered element u E 2. If we let u E q = u E b & u A u =/= 0,, then s = r”q is a set and 0 c s; for, if p E C-J then r p E 2 and rp intersects u. Clearly the extension of a n y element of D (s) is strictly subvalent to z.
.
5325. LEMMA(TSS’). If Z is a total Boolean support on b and Appr (x, y, z) then there exists u E Z such that b u is x - y - z-distributive. Proof. For any 6-mapping r of x into y there exists a n s such that r”Z E E s 5 y x x and every extension in s is strictly subvalent to z. Since r”Z E s we have
I
A PEX
V rqpE Z
qEExt,(p)
for any r a n d s. Thus, if we set u = A V A Vr,, where r ranges over b-mapr
s
P Q
pings of x into y , s ranges over subrelations of y x x which satisfy the 261
5326
[CHAP. V SEC. 3
CHARACTERISTICS OF ALGEBRAS
condition ( V p E x) (Ext, ( p ) < z), p ranges over elements of x and q ranges over elements of Ext, ( p ) , then we have u E 2. For every ( b u)-mapping r of x into y there is a b-mapping P of x into y such that P,, A u = rp4. We have V A V FP4 2 u and hence V A V r p q = I+. Since A rm is
I
S
P
s
Q
I
P
P
4
v 4
a z-layered set for any s. the algebra b u is x - y - z-distributive.
I
5326. LEMMA (TSS‘). If b is x - y - z-distributive then b u is x - y -
- z-distributive for each u
1
> Ob.
Proof. For every b tr-mapping r of x into y there is a b-mapping ? of x into y such that Ppq A u = rp4; if u E b is z-layered by r then u A u is z-layered by r. 5327. THI:OREM (TSS’). If Z is a total Boolean support on 6, then Appr (x, y , 2 ) iff b u is s - y - zdistributive for some u E 2. We shall give two sufficient conditions for absoluteness of cardinals in the theory with Boolean total supports.
I
5328. TriEoKmt (TSS’). If p(b) 5 K, then b is K, - K, - K,-distribut!ve for all 2. Proof. Let r be a 6-mapping of K, into K,. Since { r x t ;K E K,} < K, [or every ~ E K , ,it follows that every element of b is N,-laqered by r . 3.329. COROLLARY ( 1 ) (TSS’). If there is a total Boolean support on b and if K, 2 p(b) then Appr (Nu,K,, K,) for every a. ( 2 ) (TSS’). Let b be a complete Boolean algebra with total Boolean w p port. If cf (K,)2 p ( b ) and a < jl then
(3) (TSS’, El). Let b be a complete Boolean algebra with total Boolean support. If K, 2 p(b) then K, is an absolute cardinal. Proof. Suppose KF & K, for some u < /I; then we have Conf (Km,N#). (a) If X, is regular then we have Conf (Ha, K,) by Corollary (2), a conrradiction. (b) If X, is singular, then X, > p(b), since p(b) is regular (cf. 2510); hence there exists Ka, 2 K, such that p(b) Nu,< K, and K,, K,,+ We now apply (a) to the regular cardinal K,, = K,,,
,.
,.
5330. THEOREM (TSS’, El). If b is a complete Boolean algebra and if Kn < v(b) then b is K z - KB - 2-distributive for every p. 268
CHAP. V SEC. 31
5332
CHARACTERISTICS OF ALGEBRAS
Proof. Let r be a 6-mapping of w, into a, and let a be an K,-multiplicative base for b. We have V r S y= 1, for every y E a,. For each v E a we shall 6 4
construct an element u E a such that u 2 u and u is 2-layered by r. We define a decreasing sequence {f’t},<w, of elements of a as follows. Let f’0 be an element of the base which is 2 v A rso0,where 6, is least 6 such that rno A v $; 0,. Suppose f’t has been defined for all { < i, (A < a,) in such a way that for each t there exists some 6, such that f’t 5 r,<{; then we let f’l. be an element of the base which is 5 (Af’t) A r,,,, where 6, is least 6 such that
(Af’l) A rdL$; ob. Such an element exists since Af’r A
=+ 0, by
Xu-rnultiplicativity.Finally, we let u bz some element of the base which is - A f’t. It follows that the join of all elements which are 2-layered by r I 6c-U.
is 1, and hence b is K, - K, - 2-distributive. 5331. COROLLARY ( I ) (TSS’, El). If b has a total Boolean support and if K, < v(b) then
APPr (% K,, 2) holds for every K,. (2) (TSS’,EL). If b has a total Boolean support and if Nu < v(b) then every semiset 0 z K, is a set. A
(3) (TSS’, El). If b has a total Boolean support and if K, 5 v(6) then K, is an absolute cardinal. (Otherwise, we would have K, & K, for some
a < B, contradicting Corollary (2).)
(4) (TSS’, El). If b has a total Boolean support and cf (K,) 2 v(b) then Conf (K,,K,)
= eonf ( K ~K,) ,
for each a < p. (By (1) we have Appr (Nu,cf (K,), cf (K,)) for every Nu < < cf(%).) We now use the characteristics p and v to get more information about supports on the product of two algebras. Recall that if 6 , and b , are algebras then b, o b , denotes the algebra with base 8, 08,. Every complete ultrafilter on h, o 8, is a product 2, x 2, of complete ultrafilters on 6, and 6, respectively (cf. 2532). 5332. THEOREM (TSS‘, El). Let bo and b, be complete Boolean algebras and suppose that 2 = 2, x 2, is a total Boolean support on h, o 8,. 269
5332
[CHAP. V SEC. 3
CHARACTERISTICS OF ALGEBRAS
If every exclusive system on 8, is of power 4 Nu,i.e. p(bo) 5 Nu+,,and if 8, is Nu-rnultiplicative, i.e. v(b,) 2 Nu+1,then every semiset CJ K, depends on Z,. Proof. Let q be a partition of b, o 6,. We say that u E 6 , stratij5e.s q if for every w E q such that w A ( I b o , u> ObO@blthere exists u E do such that w A (la,, u ) = ( u , v ) ( A is the meet in b, 0 b,). Clearly if v stratifies q and V 5 v, ij E &, then ij also stratifies q.
<
+
,
1) We prove that if q is a partition of bo 0 b, then for every u E 6, there exists ij E d l , ij v such that V stratifies q. There exist u, E 6, and u, E 6 , such that (u,, u,) w A v > for some w E q. Suppose that we already defined a n exclusive system ( u s } s < Ain 6, and a decreasing system f ~ ~ in 6 , such that for every p < A we have (us, u p ) 5 w for some w E q; it follows that A < O,+1. IfVoUp < o I,, then We let (UA,UA) be such that Ua 50 I,, - 0
s
s
-,Vous,oa 5 Alva and (u]., u A ) 5 w for some w E q. There exists .I < a A
A
,
< mu+ such that Vous = l,,, since p(b,) 5 Nu+ Let 6 E d , be such that
V
A
i lAlus; we prove that
ij
+ Oboqbl then we let 1
Clearly (E,, V ) = w
A
stratifies q. If w E q and if w
A
(tho, (.>
+
< l b o , 5).
2 ) We prove that if t is a set of partitions of b, G b , and if t 4 m, then for every v E 6 , there exists t; E d,, V u such that 0 stratifies each q E t ; we say that ij unijormly stratifies t . Consider an arbitrary well-ordering o f t , t = {qs, p < mu), and let 6, 2 u be some element of b , which stratifies 4., If {cB, [I < A} is a decreasing sequence of elements of 6, such that every fib stratifies qa < A) then we let V A be some element 5 A, iTa which strati-
s
(i
fies qA. Finally we let V
5 ,A,
B<m.
8
fiP; V uniformly stratifies t .
a,, then there exists 3) It t is a set of partitions of b, o 6, and f a partition q , of b, such that every element of q , uniformly stratifies t . This partition q , can be defined by transfinite induction using 2). 4) We finally prove that if CJ is a 1-I-mapping on mu then c = s”Zo for some 7. Let r be a (b, o b,)-mapping such that CJ = r”Z and let
270
> ~ <
CHAP. V SEC. 31
CHARACTERISTICS OF ALGEBRAS
5336
Since t is a system of partitions of b, 0 b , and has power 5 Nu,there exists a partition q 1 of b, whose elements uniformly stratify t . Let u be the unique element of q 1 n 2,. For every w ~ u ( t define ) U , E b, as follows: if w A =I= =k Oaocb, then
In a n analogous way we can prove the following generalization: 5333. THEOREM (TSS’, El). Let a, and a , be separatively ordered sets and suppose that 2 = 2, x Z , is a total Boolean support on uo 0 a,. If every exclusive system on a, has power at most K, and if u, is Nu-multiplicative then every (T 3 o,depends on 2,. (Exercise.) 5334. COROLLARY (TSS’, El). Let Z = Z, 0 Z, be a total Boolean support on 8, o 8, and suppose that p(b,) 6 v(b,) and K, 5 v(b,). If N, is an absolute cardinal in the sense of Gupp(Z,) then K, is an absolute cardinal. 5335. THEOREM (TSS’, St). If K, is the least cardinal such that 1 Appr (Nu, 2,2), i.e. such that some part of Ku is not a set, then
Proof. For regular K, the statement follows trivially from Lemma 5305. Hence we suppose that cf(K,) = K, < K, and that CT c w, is not a set. Let e be a mapping of o, into a, such that ~ ( W ( Q )= ) o,. For each [ < a,,(T n is a set, and so the function T defined by 7’5 = cr n (4”) is a semiset and maps o,into P (0,);since u ( W (T)) = 0, T is not a set. Hence we have
($0
1 APPr (cf(K,), p (a,),2 ) .
5336. THEOREM (TSS’, St). If K, is the least cardinal for which there is a y such that 1 Appr (Nu,y , 2 ) then K, is an absolutely regular cardinal number. Proof. We have Appr (N,,2 , 2 ) for every y < a; hence e a r d (Nu)by Lemma 5314. Suppose K, is absolutely singular. It follows that thcre exists a mapping CT from some K, into K, such that u(W ( 0 ) ) = K,. Since Appr (K,,K,, 2), the mapping D is a set; hence K, is singular. If e is a mapping on o,and if is not a set then the mapping T on o,defined by 27 1
5337
[CHAP. V SEC. 3
CHARACTERISTICS OF ALGEBRAS
r
r
e a’{ is not a set; every e a’( is a set but u(W(z)) is not. Hence for some y we have 7 Appr (N,,y , 2), a contradiction. Relations were considered as functors, i.e. as “mappings” between semisets and sets. A similar approach can be adopted in connection with b-relations:
z’l =
5337. DEFINITION (TSS’). Let b be a complete Boolean algebra. If r is a b-relation such that r c (y x x) x b, then we define
for all
i
E x.
4L-J
=
{
u>, ( ( K ,
L>
u> E r>
9
5338. LEMMA (TSS’). Ler 2 be a Boolean support on b. If r is a b-relation and a = r”2, then Ext, ( 1 ) = ( r [ ~ ] ) ”2 for all 1. 5339. DEFINITION (TSS’). A b-relation r is called a b-exact functor on z if r is a b-mapping on z and if the following condition holds for all c1 $. i 2 in z : sSimb ( D (r[c2]) = Ob. (r[L1])7
5340. LEMMA (TSS’). If 2 is a Boolean support on b and r is a b-exact functor on 2 , then r”Z is an exact functor and D ( r ” 2 ) E z. Proof. Let i l , i 2 be distinct elements of z and let a = r ” 2 . We have Extu(il) = ( r [ i l ] ) ” Z and similarly for c2. By 4320, we have Ext,(~,)$. .I.Ext, ( i 2 ) ; the rest is obvious. 5341. DEFINITION (TSS’). We shall write Beth, (2, K,, K,) if there exists a b-exact functor r on my such that r c (a, x a,) x b. We shall write Beth, (Nu, K,, KY)if there exists a b-exact functor r on a,such that for each i E a, r [ i ] is a b-mapping of a , into a,. 5342. THEOREM (TSS’, S5, EL). If there is a total Boolean support on b then
I) Beth, (2, s,, K,) implies K, I 3 2 , K,) , 2) Beth, (K,,K,, K,) implies K, 5 f i ( ~ , K,) , . Proof. 1) Suppose Beth, (2, K,, K,). Then there exists a n exact functor on w, such that every extension is included in ag.Hence K, 5 q(2, No). 2) Similarly. 5343. LEMMA (TSS’, S5, El). If K, and K, are absolute cardinals then K F 5 i ( K , , K,) and 2’p 5 3(2, Kg). (Obvious.) 272
CHAP. V SEC. 31
5344
CHARACTERISTICS OF ALGEBRAS
5344. LEMMA (TSS’, S5, El). If there is a total Boolean support on b
and if b
%
K, then
1) If K, is an absolute cardinal and then i ( 2 , K,) 5 K?. 2) If K, and K, are absolute cardinals then
Proof. 1) K y is the number of all possible b-sets r
o, x b. 2) Kf;lax(Wa3Kg) is the number of all possible b-relations r 5 (o,x a,) x b. G
273
6101
AXIOM OF CONTINUUM
[CHAP. VI SCC. 1
CHAPTER VI SECTION 1
Independence of the Axiom of Continuurn
We shall now illustrate the general theory by means of some examples. We shall demonstrate among other things the independence of the Axiom of Continuum.
u ) Generalized Cantor' ulgebras.
6101. DI.I.INITION (TSS'). Let K, 5 KB be cardinal numbers; for any s E wg let b, = at (2) (see 2520) and let I be the ideal of all subsets of cop of
power less than K,. We denote by Cant (Ka, KO)(or more briefly Cane) the complete Boolean algebra with base b! = 'b,.
I1
,COB#
6102. In the sequel we assume that K, is regular and that the axiom (Cont) holds. 6103. LEMMA (TSS'. E l , Cont).
(a) v(cant!)
s X, ,
(b) p(Cantt) 5 X,+l . Proof. (a) by Theorem 2523. (b) by Theorems 2524, 2525, 2526 and the axiom Cont.
6104. THEOREM (TSS', E l , Cont). Let Z be a total Boolean support on Cant!, where SI 5 p and K, is regular.
(a) If y < c1 then Appr (K?,Xd, 2) for all 6 , (b) If y > x then Appr (Kd,K,,X?) for all 6 214
.
CHAP. VI SEC. I ]
6106
AXIOM OF CONTINUUM
(c) Every K, is an absolute cardinal. (d) For every y cf (K,)= Gf (Ky). Proof. (a) See 533 1 (1). (b) See 5329 (1). (c) See 5329 (3), 5331 (3). (d) See 5329 (2), 5331 (4). First we shall examine the case K,
=
K, for regular
S,.
I
6205. LEMMA(TSS’, El, Cont). (1) If Nu is regular then Cant: u is not K, - X u - Nu-distributive for any u OCnnt. (2) Cane has power X u + ,. Proof. (1) Letf be a 1-1-mapping of o, x o,onto 0,;for <.q < o,let uF,, be the function which assumes the value one atf’(5,rl) and the value 7ero at f ’ ( 6 , q) (6 < t), and is undefined otherwise. Clearly E bz. We let r = q), uSq);5, q < a,}; r js a Cant:-relation and for every L‘ E bz there exists q such thatf’(5, q) i s not a fixing point of L for any 5. For u c h a n q we have D A ut,, Ocnnt.a for every 5. Hence no basic element is S,-layered by r, and consequently no non7ero element is K,-layered. (2) The power of b: is S,; each element of Cant: is a JOIII of not more than K, disjoint basic elements, so that Cant: Nu+1. We h a w found a disjointed system of power K, of b:; distinct subsets of this system determine distinct elements of Cane and hence Cane w
+
(((e.
+
<
6206. THEOREM (TSS’, El, Cont). If K, is regular and Z is a total Boolean support on Canc then 1) y
=
a E 1 Appr (K,,K,,KY)for any y ;
2) 3 2 , K,) = s,+ for any y. Proof. 1) By the preceding lemma and 5327 we have 7 Appr (Km,Ha,X u ) ; if y a then we have Appr (K,,K,, K,) by Theorem 6104.
+
2) By Theorem 6104, all cardinals are absolute; since \?(Cant:) 2 K,, every semiset of power < K, is a set, and so 2(2, X t ) = 1(2, X,) = S.,+for each y < a. If y 2 a then 5(2,KP)2 X,+,; on the other hand, every part of K, is determined by a Cant:-set whose range is included in my. Since Cant: M and = K,+l,there are only S f + , such Cant:-sets. Consequently 3 2 , K,) 5 H ~ + ~ . Suppose now that h’, < cf (Xb) and that K, is regular.
K2,
275
6107
[CHAP. VI SEC. 1
AXIOM OF CONTINUUM
6107. LEMMA (TSS’, El, Cont). If K, < cf(K,) and if K, is regular then (a) K, i K, < cf(K,) (b) cf (K,) 5 K, 5 K,
--f
-+
Beth,,,.0(2, K,,H,); BethCant=p(2, K,,Kp+1).
Proof. (a) Let f be a 1-1-mapping of o,x a, onto o, and for each < o, and q < a,, let urq be a function which assumes the value 1 a t f’([, q ) and is undefined elsewhere. Let r = (((c, q), urs);5 < o,.q < a,}. It suffices to prove that if q1 + q2 then sSimCantuB ( D (r[ql]), D ( r [ q 2 ] ) )= - OCaotUa. If u E b! then for some 5 neither f’(5ql) nor f’(t1q2> is a fixing
r
+
pcini of v ; it follows that u A uCq1 u A urs2, since f’(5, q l ) is a fixing point of the left-hand element but not of the right-hand element. Hence u distinguishes r[ql] and r [ q 2 ] , and so sSimCant=B (D ( r [ q l ] ) ,D ( r [ q 2 ] ) )= - OCant.6.
-
(b) Let c be the set of all subsets of w, of powzr K y ; clearly c K,+l. For u , u E c let ( u , u ) E t = (u - u) u (v - u ) < Ha;t is an equivalence and for every u E c we have Ext, (u) M K,. Hence there exists a set d of pairwise nonequivalent elements of c such that d w K,+ 1. We choose a wellordering of d of type o,+ and for every u E d we chose a well-ordering of u of type o,;let f ’(c, q ) be the 5-th element of the r]-th set in d. If we let u5,, = = ((l,f’(t,q ) ) } and r = ( ( ( 5 , q ) , ucJ; 5 < a?, q < then for q l + q2 and u E 6; there exists 5 such that f’(5, q l ) f’(5, q z ) and such that none of these elements is a fixing point of v. The rest of argument is similar to that sub (a).
,
+
6108. THEOKEM(TSS’, El, Cont). Let K, be a regular cardinal such that
Ha< cf (K,).If Z is a total Boolean support on Cant! then (a) xY< K, 3 2 , sy)= K,+, , (b) K, 5 N, < Cf(K,) -+ i ( 2 , K,) = K p , (c) cf (K,) 5 N, 5 N, 3 2 , K,) = K,+ 1 , (d) S, 5 K, 3 2 , K,) = N,+l. -+
-+
-+
Proof. (a) is proved in the same way as (2) in Theorem 6106. (b) Since b{ M K, and since every element of C a n e is a join of at most K, disjoint basic elements we have Cant; M K,. By the lemma 6107(a) 3(2, N,) 2 2 H,. Since K F y = K, it follows from Cant! M K, that there are K, (Cant!)-sets with range c a,,; hence 5(2, K y ) 5 N,. ( c ) By Lemma 6107(b) we have i ( 2 , K y ) 2 K,, Since K p = K,+l it follows from Cant! z K, that 3 2 , K,) = N,+l.
,.
216
CHAP. V I SEC. 11
6110
AXIOM OF CONTINUUM
(d) If K, 5 K, then j ( 2 , K,) 2 2', 3 2 , K,) = K ,+ 1'
N,+,; since N;,
=
6109. THEOREM (TSS', El, Cont). If N, and of the preceding theorem then
=
wc have
K, satisfy the hypothesis
(a) cf (K,)< K = + I{N,) = I(K,) = K,+ I , (b) K, 5 cf (K,)< cf (K,) -+ n(K,) = Max (K,,K,+ ,) , (c) Cf(K,) Cf W,) qH,) = Max 1, K,+ 1) . Proof. (a) is obvious, since every semiset of power < H a is a set. j(N,) 2 i ( 2 , cf (K,))= K,. Hence i(K,) 2 (b) j ( K , ) 2 I&) = K,, 2 Max (K,,K,, Conversely, for every semiset CT which maps cf (K,) into K, there exists a Cant!-functicn r c (a,x cf (a,))x Cant! such that cr = r"Z. Since p(Cant!) 5 Nu+ we have D (I) Max (Ku,cf (K,)) = Each u E D (r) is a join of at most K, disjoint basic elements, = cf [K,,). each having less than K, fixing points. If we choose a set q,, of at most K, basic elements for every u E D (r) such that u = Vq,,,and if we let t be the qu, then t cf (K,)and r is a b(')-function, set of all fixing points in
s
w,+
+
<
u
<
uaD(r)
where b(') is the complete Boolean algebra generated by the base
n'/tat (2).
X€mp
The cardinality of b(') is at most cf (a,,)so , that the number of all b(')-functions P c (a,x cf(a,)) x b(') is at most cf(K,)Hy = X,,,; the number of all subsets of asof power at most cf (Ky)is K$(KY)= K ,. Hence we have found a set of cane-functions of power at most Max (N,, K,+ 1) such that every semiset-mapping of cf (a,)into a, is coded by at least one of these Cane-funct ion s. Hence
i(K7) 5 Max (K,,K,+ . (c) Supposing cf (K,) 2 cf (N,) we have
i(K,) i(K,)
2 I(N,) 2
K,+, , 3 2 , cf (K,)) 2 N,+ 1 , =
and i(K,) 5 j(K,, K,) = Max(K,+,, KY+l). Consequently = Max (Kp+1, K,+ 1). 61 10. DEFINITION (TSS').
i(K,,) =
217
6111
[CHAP. VI SEC. 1
AXIOM OF CONTINUUM
The reader may Icrify as an exercise that the following is provable in
T S , D3, Mcl,, (&I),i?TSS/?st(M):
Metariiutlirinarica I
ccii sequences.
Let i , ,u be good definitions of cardinal numbers 111 TSS, D3. E2: denote the cardinal numbers satirfyiiig ; Iand it by K, and K, respectively. (1) SLIPPOW PSS, D3, I 2 I- (S, is regular). Then the axiom 6111.
MrTAiHFORLbi
Mcl,, (31) & (Xx is the only cardinal such that 1 Ap (M, K,))
(Appr,)
is consistent diagram :
tviilh
TS. D3. E2, Cont; in fact, Up is a model in the following
I
I
In particular the following is consistent with TS,D3, E2, Cont: MCI,,( f-)
& (S,is the only cardinal such that 1 AP (L, Nu))
(2) Suppose TSS. D3. E2 k ( K m is regular and cf(KP) > Nu). Then the axiom
(G i m
I
1
Sa -+ I(SJ
(A)
cf(X.;) <
(b)
s, 5 cf (S,)< cf (K,) + I(K,) =
= S,,,
Max (Kb, K7+,) ,
( c ) cf(Kp) 5 cf(N,) = M a x (So+
+ X(HJ ==
K,,
, =
I I J
for
is consistent wilh TS, D3, E2; in fact, Up is a model in the following diagram:
?s03,€2, 178
1 _I
TSS/03,E2,Conf ( S Conf,P 1+ upw
CHAP. VI SEC. 11
6112
AXIOM OF CONTINUUM
Demonstration. (I) We proceed in TSS,D3, E2, Cont, (S Cant:) + upw. By 6104 all cardinals are absolute and therefore Mcl,*,(Tor). By 5142 and 5136 we have 1*(2*,K;*) = Imb” (i(2, Ny)) = K,*,+,.for every y and y* such that y* = Imb” y ; i.e. (Cont)* holds. Set Imb’ K, = K:*. We have A(N& hence by 5126 we have AD(N$) (we use dTSS/dGt*(Tor) for the moment) but since A i s a good definition we have (Mcl,*, (Tor) & lbo(K,*,)) + -.+ ,l*(N:*), i.e. L*(K:*) holds. Finally, for any y and y* such that y* = Imb” y, Ap* (Tor*, X;*) is equivalent to (1)
(Vf*
c Tor) (Un* (f*,Ky*r, K;*) + (3r* E Tor) (f* E r* & & ( V L * ) (Ext:
(L*)
<* K;*)))
and (L) is obviously equivalent to (2) (Yo) (Un (0, X,, Ny)--+ (3r)
(0
E
r & (VL) (Ext, (6)
? N,)) .
In (2) we may equivalently replace by < because cardinals are absolute; hence (2) is equivalent to Appr (Ny, K,,N,) and hence by 6106 to K7 =k X u . Rut this is equivalent to K;* ; . : K this proves (Appr:*). We have thus demonstrated that U p is a model in the diagram sub (1).
+
+
(2) We proceed in TSS, D3, E2, Cont, (S Cant:) upw. By 5138 and 6104 (D3)* and (E2)* hold and cardinals and cofinalities are absolute; hence if we set.:N = Imb” K, and K& = Imb’ X, we have ,l*(K$) and p*(‘Kt,) (cf. the proof of (1)).Let y be an arbitrary ordinal number and set K,*.=
Imb” K,. Then we have cf* (N;*) < * Kz* + cf (K,) < K, 3 i ( K y ) = K,, --f l*(N:,) = Ny*.+l*rand similarly for (b), (c) in (Gim,*!*) (cf. 5144 and 6109). Thus we obtain (Gim:!*). This completes the demonstration. (For the corollary i n (I) see 5138; L denotes the class Cstr.) = =
It is of some interest that using Up as a model in TSS,D3, E2, (S Cant:) + + upw we can prove the consistency of 2O’ = K, without knowing anything about the model-class Cstr (0). 6112. METATHEOREM. Up is a model in the following diagram:
TS,D3,E2,
TSS, 03,E2,
279
6113
AXIOM OF CONTINUUM
[CHAP. Vl SEC. 1
Demonstration. We proceed in the latter theory. Denote Cant: by 6 . Obviously v(b) K,; hence every subsemiset of wo is a set and ti, is absolute. It suffices to prove the following: If e E oo x a, is an exact functor then K, N,.First we prove 2K0 2 ti,. For this purpose let i be a one-one mapping of oox o1 onto o1 and write in, instcad of i’(n, 5). For I E e x p ( 2 , o ) we define ugr as the element of Cant: such that D (ticr) = = i“(o x ( 5 ) ) and u& = r’n. Further define a b-set f such that D (f) = (urr; 5 E o1& r E exp ( 2 , o)} and f ’ugr = ( r , (>. If Z is a total Boolean support on b thenf”Z is a semiset mapping of o, onto exp (2, o)(exercise). Now if CT is an exact functor which is a subsemiset of o x o, such that D (c) = a,,then, for every 1, Ext, ( 1 ) is a subset of o;thus the mapping e which associates with every i the set Ext, ( b ) is a one-to-one mapping of o,into P(o); hence X u 2’O N l , i.e. 2*(2*, K:) = X;., q.e.d.
3
4
3
6113. Remark. In the same way one can prove the following in TSS, 113, E2: if there is a total Boolean support on Cane:: then every subsemiset
of o, is a set and 5(2, Nu)= X u + l .To derive metamathematical consequences, one must be careful about the definition of K,; e.g. it suffices that K, be defined by a definition il such that TS, D3, E2, Mcl(M), dTSS/dGt (M) 1(Vy) [(V6 < Xy) (Card (6) = Cardn (S))] + 1(K,) = iln(Ky)]. 6114. Remark. The reader may establish the following as an exercise: if Nuis regular and K, = cf (ti,) < K, then 3 and 5 satisfy the same equalities if we assume the existence of a total support on Cant!” as if we assume the existence of a total support on Cane. However, if there is a total support on Cane” then there is no total support on any algebra with a base of power less than NEB+,,whereas if there is a total support on Cant! then there is a Boolean support on a n algebra with a base of power N,, since Cant! has a base of power K,.
b) Siinultaneous ziiolation of the continuum hypothesis. 6115. DEFINITION. Let
f be a function whose domain consists of all less than a fixed ordinal A and such that K, is regular. If, for all ct, fi E D (f), we have
ordinals 1) f ’ c t 2)
c1
f’P
whenever a < fi ,
Nu < cf (s.c/.,),
then we call f a continuum function on A. 280
CHAP. V I SEC. 11
6117
AXIOM OF CONTINUUM
The algebra PCant ( f ) corresponding to the continuum function f is given by the base where I is the ideal of all subsets of D ( f ) such that x n j3 < K, for every regular cardinal K, c 1. Clearly 1 is an ideal; the condition x n /3 < K, is nontrivial only if B = K,.*). For the definition of b,f"see 6101. 6116. We shall show that in TSS', El, Cont, the existence of a total support on PCant ( f ) implies 3 2 , K,) = K,., for all a E D ( f ) .
6117. LEMMA (TSS', El, Cont). Let f be a continuum-function on , Iand let a E D (f).If
and b2 ( f , a)
then
=
n
I/(D(f)-(,+
x>a
x-D(f 1
1))
f'x
bx
-
a) b) c) d)
b ( f ) is isomorphic to b, ( f , a) 0 b, ( f , a) ; every exclusive system in b, ( f , a) has power at most Nu; b, ( f , a) is Ka-multiplicative; the algebra with base b, ( f , a) has power at most Kf,,. Proof. a) Obvious - see 2529. b) Since b c is a product of K,,, copies of the algebra at(?), it follows by 2533 that b, ( f , a) is a base of the same algebra as nJ at (2), where XBS
s = { ( y , 6 ) ; 6 ~ a & G ~ D ( f ) & y < K / ' and ~ } X E ~ E, x ~ s & D ( x ) ~ E I / ( . + 1) & (V6 E D (x)) Ext, (6) < Kd). If a = [ 1 then (b) follows from Theorem 2524, since Norm ( j ) = K,, and since every exclusive system in at (2) has power at most 2; if a is a limit number then (b) follows from Theorem 2526, since Norm(j) = K, and since every exclusive system in at (2) has power at most 2. c) This follows from Theorem 2523, since the ideal l/(A - (a + I)) is &-additive and since is Ka-multiplicative.
+
*) We mention that the existence of regular cardinals such that K, = /I is not provable in the theory of sets; such cardinals are called weakly inaccessible and we shall not treat them in this book.
281
611s
AXIOM OF CONTINUUM
[CHAP. VI SEC. 1
d) By easy calculations, b,(f, a) z Kya. Hence the assertion follows by b). 6115. THEOREM (TSS’, E l , Cont). Let f be a continuum function and let
Z be a total Boolean support on YCant ( f ) . Then a) all cardinals are absolute, b) 3 2 , Ka) = K,.= for all
SI E
D (f).
Proof. If S,is regular then, using the same argument as sub b) in the preceding lemma, we can prove that in b ( f ) - and hence in PCant(f) - thc number of disjoint elements is at most K]., so that the cardinals greater than K, are absolute. If K, is singular then, using the estimate 2524. bve see that every exclusive system has power at most KA+,, so that all cardinals greater than are absolute. Hence it is sufficient to prove that S, G Sa., cannot hold for Y. 5 i.; the case a = A is to be investigated only if S Ais singular. I f Y < 1- and if K, is regular then p[b, (f, a)) 5 Ka.,.l 5 v(b, ( f , a)), so t h a t by 5333 Un, (a,Ka. K,+,) implies Dep (6,Z , ) where Z = Z , 0 Z 2 and %, i s a support on b, ( f , E). On the other hand, Ka+l is absolute in the sense of S u p p (Z,) (because p(b, (f, x ) ) = K a +,), a contradiction. If Y 5 1. and if K a is singular then let S, = c?‘ (S,.+,). If K,+ & Ka then S; < S, by 5145 and 5134(a). Since p(b,(.f, c)) 5 K,+ 5 v(b, ( f , t)),we have eonf (S5.Kz+,) i n the sense of Giipp (Z,). Hence b, ( f , t) is not S: - Kn+,-- Kz ,-distributive contradicting p(b, ( f , 5)) 2 Sc+l(see 5313 and 5328). This completes the proof of a). h) If r E D ( , f ) thcn every subsemiset of o, depends on Z , and we have i ( 2 . s,) 2 s,.,in the sense of ~ u p (z~); p for, if e is a support on b,/’“ then i ( 2 . K,) 2 KJ,z holds in the sense of S u p p (Q). On the other hand, as in 6108 sub (b), Lemma 6117(d) yields 5(2, Na)5 K,,, in the sense of 6 u p p (Z,) and, by absoluteness of cardinals, we have 3(2, K,) 5 K,,,.
,
+
61 19. METATIIEOREM. Let q ( x ) be a set formula such that
TSS”
t- (3s)(&)& ( V x ) (cp(x)) -+ x is a continuum function) ,
TSS”, FMcl (M), c?TSS/dSt (M) t (VX E M) (q(.)E ~ ” ( x ) ) .
Then Up is a model in the following diagram:
C H A P . VI SEC. I 1
6123
AXIOM OF CONTINUUM
where (Powers,) is the axiom (Vcc E D (f)) (2”a consistent with TS, D3, E2.
=
XI.,).
Hence (Powers,) is
c ) Collapsing algebras.
6/20, DEFINITION (TSS‘). Let X a 5 K, be cardinal numbers, let c, = (H,)for every x E o,and let I be the ideal of all subsets of w, of power less than K,. We denote by Coll (K,, KB)or simply Collt, the complete Hnolean algebra with base ct = n‘c,. = at
X€l&
We shall investigate the case when X u is regular and less than or equal to cf (X,). 6121. LEMMA (TSS’, E l , Cont).
(a) v(cOl1;) 2 h’, ;
(h) p(C01lt) I K, + . Proof. (a) foilows from Theorem 2523; (b) follows from the fact that cf hiis power X,, so t h a t every disjointed systcm has power at most X,. 6122. LI’MMA (TSS’, E l . Cont). Collt is not
K,
- K,
- K,-distributive.
Proof. For i < o, and ’1 < og let uqc’r = v] and lel u , , ~be undefined Furthermore, we let r = {((q, c), u , , ~ ) ; E w,, clsewherc; clearly, u,< E rl E W ~ } . Since us,5 A u , , ~= Ocoll,~ whenever tiL =t= t i 2 , it follows that r is a Collt-mapping. Let u E c t be a basic element and let 5 be such that t iv 5 is not a fixing point of u for any v ] . Then u A uqc( = u u us<) is nonzero for each v] E op.Hence no element is K,-layered.
2.
6123. THEOREM (TSS’, E l , Cont). Assuming the existence of a total Boolean support on Collt we have K, & K,; moreover, if y 5 SL or y 2 2 p + I then K, is a n absolute cardinal. Prooj. Let Z be a total Boolean support on Coll!. Let r have the same meaning as in the previous proof. Put 0 = r”Z. Then 0 is a mapping of a,into op;we prove that is a mapping onto cog.Let q < o,,u E ct. Then there exists 5 < o, such that u A u , , ~+ 0 (see abovc). In other words, u,; = lCo,,,b. Hencc there is a 5 such that u,,~E Z . Consequently ( v ] , 5) E
v
&CJ,
E a.
The second part follows from Lemma 6121 and Coro!laries 5329 and
532 I .
283
6124
AXIOM OF CONTINUUM
[CHAP. VI SEC. 1
6124. METATHEOREM. Let I., p be definitions of infinite cardinal numbers in TSS, D3, E2 and suppose that TSS, D3, E2 I- (Va, p) (A(Na) & p(NB) . + -+ . X u regular & X u 5 cf (Np)). Let (Collapst) be the following axiom (formulated in TSS, D3, E2, Mcl (M), aTSS/dGt (M)):
Then U p is a model in the following diagram:
I TSS, D3,€2,
I
I
I
(In fact, the model in question is the extension of U p which interprctes the symbols defined by aTSS/dGt(M)by the symbols defined by dTSS/dGt* (Tor) (cf. diagram 5112, arrow 8).
284
CHAP. V I SEC. 21
AXIOM OF CHOICE
6201
SECTION 2
Independence of the Axioms of Choice
The aim of the present section is to establish the consistency of the negation of the weakest axiom of choice E l (and of certain stronger statements) w.r.t. the theory TSS + D3. Earlier we proved ihe consistency of 7 E l with TSS + U3 using the model 6 t (HS (z)) with a suitable specification of the group-filter z; this model - and each specification of it - will be referred to as the permutation model. We also proved the consistency of the axiom of regularity D3 with TSS using the model 6t(Ker), where Ker is the constant denoting the regular kernel; this model will be referred to as the nuclear model. Since the latter model is an essentially faithful model of TSS + D3 in TSS, aTSS/aGt (Ker), the consistency of l E l & D3 with T S S is equivalent to the consistency of l(El)G*(Ke')with TSS. To prove this consistency we cannot make use of permutation models, since TSS, U3, Sym (z), dTSS/aGt (HS(z)) t Ker = KerQ*(HS(a)); in order that the sentence 1(El)~t(Ker)*6t(HS(a)) should be provable in some extension T of TSS, U3, Sym (z), aTSS/aGt (HS (z)) it is necessary that T !- l ( E 1 ) W K e r ) . Fortunately we have another method at hand. We shall start with some extension of TSS + 1 E l which can be shown to be consistent using the permutation model and consider the extending model in this theory. We shall first investigate how to extend TSS" so that l(EL)6t(Ker)is provable in the sense of the model Up. 6201. DEFINITION (TSS'). Let 6 be a complete Boolean algebra. A nonempty set j of subalgebras of b is called an ideal (of subalgebras) if the following hold:
1) if c1 ~j and if c2 is a subalgebra of c1 then c2 ~ j , 2) if cl, c2 E j and if c is the subalgebra of b generated by c1 u c2 then cEj, 285
6202
[CIIAP. VI SBC. 2
AXIOM OF CHOICE
3) if c E j and f is an automorphism of b then f ” c E j . 6202. DEFINITION (TSS’). If b is a complete Boolean algebra then the set Ki ( b ) of all subalgebras c of b such that (3d E Ker) ( c M d ) j s called the ideal of nuclear subalgebras. 6203. LEMMA(TSS‘). Ki(b) i s an ideal of subalgebras (in the sense of Definition 6201). Proof. Ki(b) is noneinpty since it contains the algebra whose field is (4, lb>. Condjtios 1) and 3) obviously hold; let us prove 2). It is suficient to show that if u c b can be mapped 1-1 into Ker then c can also be mapped 1-1 into Ker, where c is the field of the subalgebra of b generated by a. Let f o be a 1-1 mapping of some do E Ker onto a. Obviously, we can assume that all the elements of do have the same rank. We define mappnigs f , with domains d, as follows:
[(-fkL (1, x>> ; x 4,)uf;, {(Af2+iz, (0, z > > ; z E d , a + i } u f 2 a + i
fi,., I
=
.fZz+2
=
fl
=
1
Uf,
a
3
(lalimit).
There exists a least r such that W ( f a ) = c; we define an equivalence e on D ( f a ) by setting ( u , u ) E e = f i u = f i l l . If d is the set of all maximal sets of equivalent elements and if we define f ’ x = = (3u E x) ( f i u = y) for each x E d then f is a 1-1 mapping of d onto c. In addition to the assertion of the lemma we have proved the following result: 6204. LEMMA(TSS’). Let b be a complete Boolean algebra and let c be the subalgebra of b generated by a set a c b. If a M q for some q E Ker then c E Ki (6).
6205. LEMMA(TSS’). If Z is a total Boolean support on a complete Boolean algebra b then for every c c Ker there exists c E Ki (b) such that c depends on Z n c. Proof. We have (T = r-”Z for some 6-set r such that W ( r - ) E Ker. D ( r ) can be 1-1 mapped into Ker by assigning r”{x} to each x E D ( r ) . Let c be the subalgebra of b generated by D(r); we have c ~ K i ( 6 )and since Dcp (0,D ( r ) n Z ) and Dep (D ( r ) n Z, c n Z), we have Dep (6,c n Z ) . 61’06. LEMMA (TSS”, (S6), dTSS/JGt (Ker)) (a) (st)G*(“er) , (b) (Bd)Et(Ker) . 286
CHAP. VI SEC. 21
6210
AXIOM OF CHOICE
Proof. Each CJ c On is a semiset in the scnse of G t (Ker) and ordinal numbers are ordinal numbers in the sense of Gt (Ker); hence ( a ) follows easily. We now prove (b). For each c E Ki (6) let a(c) be the least ordinal number such that c pmC,,;let a = Sup (a(c);c E Ki ( 6 ) ) and set a = pa. We prove that a is a bounding set. If CJ c Ker then CJ = r”(Z n c) for some r a n d some c E Ki (6). We may suppose that W ( r ) E Ker and D ( r ) E: c ; if we let q = {r”{x}, x E D ( r ) ) then q M D ( r ) c a ; if we Ict g = = { r ” ( x ) ; x E Z n c> then e E q and Ue = CJ.
<
< <
6207. THEOREM (TSS‘, S6, c?TSS/ZGt (Ker), dTSS/dGt* (Ker*)).
Proof. By the preceding lemma it suffices to prove (Pot)E*(Ker) (cf. 41 34, 4241). We have Imb” Ker E Ker* (where * is used to rcprcsent the notions
in the sense of Up); this follows from the fact that Imb’ ( 0 ) E Ker* and that for every a, Imb’ p a c Ker* implies Imb’p a + c Ker*. Suppose a E Ker and ( b , a ) E Imb. Clearly y* = Ext,,,,,, ( b ) E* Ker* and there exists x* E* Ker* such that, in the sense of U p , x* is a n exact functor whose domain is an ordinal number and which codes all subclasses of y*. We have x* E Tor; if we let e = [Cnv(Imb)]”x* then e is a n exact functor whose domain is an ordinal number and e codes all semisets included in a . This prows (Pot)‘*(Icer) and hence also (S6)G*(Ker). 6208. Remark. It follows that if we want (El) to fail in the sense of the model Gt (Ker) constructed within the model U p it is sufficient to prove that there exists no total support in the sense of Gt (Ker). 6209. DEFINITION (TSS’). Let j be an ideal of subalgebras of b. We call 2 Ob there exists U ; it follows that d Ib U iS a subalge11 2 ob such that U 5 U and d 16 U 5 C bra of c Ib u. An ideal j is called primary if it has a principal algebra. c B j a principal algebra in j if for every d E j and every u
6210. THEOREM (TSS”, aTSS/dGt (Ker)). If Z is a total Boolean support on b then the following conditions are equivalent:
1
(a) there exists u E 2 such that Ki (b u) is primary; (b) in the sense of Gt (Ker) there exists a total (Boolean semiset) support. Proof. Assume that (a) holds. We may suppose that Ki(b) is primary; for otherwise we may consider b [ u and 2 u. Let c be a principal algebra in Ki (6) and let c’ E Ker be a complete Boolean algebra isomorphic to c.
la
287
6210
[CHAP. V I SEC. 2
AXIOM OF CHOICE
Let f be an isomorphism of c‘ and c and let Z’ = f”(Z n c). Since Z’ is a complete ultrafilter on c’, Z’ is a support in the sense of Gt (Ker). We prove that Z‘ is a total support in the sense of Gt (Ker); thus we prove that Dep (a,Z’) for each IJ c Ker. We have Dep (a,Z n d) for some d E Ki (6); for every u E b there exists u E b such that u 5 u and d u E c u ; hence there exists u E Z such that d u E c u. It follows that
Ib
,1
Ib
lb
r
(here we use the relation r = I ( d Ib u)). w e also have Sim (Z n c, z n n ( c u)), Sim (Z n d, Z n ( d u)) and Sim (Z n c, Z’), so that Dep (a,Z’) which proves (b). Conversely, let Z, be a total Boolean support on 6, in the sense of Gt (Ker). By 4257, there exists a subalgebra 6’ of b such that 2 , is similar to b’ n Z; hence, by 4323, there exist uo E Z, and uo E b‘ n Z such that 6 , uo is isomorphic to 6‘ 0., It follows that 6‘ uo E Ki (6 uo) and that 6’ w o E E Ki ( 6 wo) for each w o 5 u”. Let c E Ki.(b); the subalgebra 6” of 6 generated by (b’ u c ) also belongs to Ki (6). We shall prove that Z n b’ is similar to Z n b“. Since b’ 5 b“ we have Dep (Z n b’, Z n b”); since 6” E Ki (6) there exists an algebra in Ker isomorphic to 6’’. Let f be the isomorphism in question and let a = f”(Z n b”). We have a E Ker, Dep (a,Z,), Sim (a,Z n b”) and Sim (Z,, Z n b’), so that Dep (Z n b”, Z n b’). By Theorem 4328, there exists u E Z such that b’ u = b“ u, and since c c b” we have c u E 6’ u, and c w c b‘ w for each w 5 u. Hence we have
Ib
Ib
I
I
Ib
Ib
Ib
I I
1
I
lb
,1
Ib
By 4302, there exists u E Z such that
We may suppose u j uo, so that 6‘
I
,1
I
u E Ki (6 u) and
for, if c E Ki ( b u ) then the subalgebra d of 6 generated by c belongs to Ki (b) and c E d u. Hence Ki(6 u) is primary and 6’ ,1 u is the principal algebra.
Ib
I
CHAP.
VI SEC.
21
AXIOM OF CHOICE
6214
6211. Remark. (TSS’, El, WSym (z), aTSS/aGt (HS(z))). Let b, c E Suppose b is a complete Boolean algebra in the sense of Gt (HS(z)) and c is a subalgebra of b in the sense of Gt (HS(z)). Then c E Ki* (6) iff PInv (c) E z. For, if x E HS (z) then x can be 1-1 mapped in the sense of 6t (HS (z)) into Ker iff PInv (x) E z. E HS (z).
6212. We shall now introduce some notations in TSS’. Let a = (a, g ) be a separatively ordered set fruitful w.r.t. a symmetric g-filter z and let 9’= HS(z). L2t b be the complete Boolean algebra with base a and let b, be the complete Boolean algebra in the sense of Gt (HS (z)) with base a, = (a n 8, S n 9). Let f be the embedding of b, into b described in 3352 and let Z be a complete ultrafilter on b. It follows that [Cnv (f))” Z is a complete ultrafilter on b, in the sense of Gt(8). This ultrafilter will be denoted by Z , . In the sequel we shall suppose a, b, z etc. to have the meanings given to them above. We shall also assume that the algebras b and b, are constructed in the canonical manner from the saturated subsets of the corresponding bases (cf. 2442). Under these assumptions we shall investigate the model Gupp ( Z , ) * Gt (HS (z)), whose sets are the elements of HS (z) and whose semisets are those subclasses c of HS (z) which are, in the sense of Gt (HS (z)), dependent on Z , (i.e. such that (r = r”Z, for some r E EHS(Z)). We shall sometimes be interested in whether elements of V - HS (z), i.e. nonsymmetric subsets of HS (z), can also be semisets in the sense of the above-mentioned model. The following theorem provides the answer in case a is hereditarily symmetric (and hence a, = a). 6223. THEOREM (TSS’, a, b, Z , z etc. as above). Suppose a E 9 , so that a, = a. If r E 8 and x = r”Z, then x E 8. Proof. We may suppose that r is a disjointed relation, since Z, is, in the sense of Gt (P),a disjointed support. If we let y = (Cnv (r))” x then 4’ c Z,, i.e. f”y E Z. Let u E a n Z , u 2 Af”y. If we let by = ( u E b,, v 2 u} then we have by E 8, since u E a = a, and Inv (br) 3 Inv ( u ) n Inv (a). We shall prove that x = r”(b;); this implies x E 9’which completes the proof of the theorem. The inclusion x c r”b; follows from y E by; since by c Z , we have r”by E r”Z, = x and we are finished. We now consider products of complete Boolean algebras. 6214. THEOREM (TSS’). For each x E Ur let b, E Ker be a complete atomless Boolean algebra. Let 1 be a cut on Ur having the singleton property. If Z is a total Boolean support on a = b, then there exists an exact functor
e such that
n’
reUr
D(e) = Ur and W (e) E Ker. 289
6215
[CHAP. VI SEC. 2
AXIOM OF CHOICE
Proof. We define Q = { ( u , x); x E Ur& ( ( u , x)] E Z}. For each x E Ur we have 0 += ~ ” { x }c b,; it suffices to prove that Q is nowhere constant. If x i E Ur then @”(xi) = ri’Z, where ri = { ( u , {(u, xi)>), {(u, xi)) E u ) . By 4320, it suffices to prove that x1 =I= x2 implies sSim, ((D(r,), D (rZ)) = 0, where b is the complete Boolean algebra with base a.Letfbe a n element of a ; we wiFh to find g _I f such that D (rl) g += D ( r 2 ) g ; we shall show that g = f has the required property.
,1
Ib
(a) If x, r$ D (f)then for every u E b,,, f A { ( u , x,)} is a basic element not defined at x1 or 0, whereas for every u E b,,, f A { ( u , x,)) is a basic element defined at xl. (b) If x, E D (f)then, for every u E bxt,f A { ( u , x2)} is a basic element which assumes the valuef’x, at x l , or Ob whereas for every u E b,, such that u < f’xl, f A ( ( u , x,)} is a basic element which has the value v at x1 (here we use the fact that b,, is atomless).
6225. EXAMPLE. Suppose Ur M K, where K, is a regular cardinal number, and let 1 be the ideal of all subsets of Ur which have power less than K,. We shall use the ideal 1 in two different ways. First we define a g-filter z on g (Ur) by lctting z = z,; z determines the class 8 = HS (z). Secondly, we define a separatively ordered set a by letting a = b,, where b, = xEUr
Cant (KpKp) for every x. Clearly a E 8;let b be the complete Boolean algebra with base a and let b, be the complete Boolean algebra with base a in the sense of 9’; we suppose both algebras to be constructed in the canonical manner. Suppose there is a Boolean total support on b and let Z, be its restriction to b,. We know that 2, is a Boolean support on b, in the sense of Gt (9’)(see 6212). Also suppose (E2). For the time being, TSS” will denote TSS’ extended by the addition of the above-mentioned assumptions; the relative consistency of TSS9 is given by 5215. It follows from the results of Chapter 111, Sect. 3 that, in the sense of Gt (P),N, < Ur holds for every CI < flbut neither K, 4 Ur nor Ur K,: moreover, a subset of Ur either has power less than Ur or its comnplement has power less than Ur. Every exclusive system q of elements of b, which is in 9’ is in the algebra with base n”‘b,, where e e l and =
<
<
Ur
PInv(e) E Inv(q) (sec 3357). This algebra is i n 8, since Ker(e) We now prove
c 9.
6216. THI!ORI:M (TSSp, aTSSl2Gt (P), aTSS/8Guppa (Z,)). The following holds in the sense of Gupp (Z,) * Gt (P): 290
(a) (b) (c) (d)
6217
AXIOM OF CHOICE
CHAP. V1 SEC. 21
All cardinals are absolute. If a < p then K, ? Ur. Nejther K, 4 Ur nor Ur K,. There exists an exact functor e such that D (Q) = Ur and W (Q) E o,.
4
Proof. As in the proof of 6117 (b) we may prove that b is isomorphic to Cant (No, K,). It follows by 6104 that if there is a Boolean support on b then all cardinals are absolute; hence all cardinals are absolute in the sense of Gupp (z,)* Gt (9).
(b) If a c p then there exists a 1-1 mapping f E 9’of N u into Ur; hence K a >< Ur holds in the sense of Gupp (Z,) * Gt (9),i t . Nu & Ur. Since Nu z K, is impossible by absoluteness of cardinals, it follows that K, & Ur
cannot hold in the sense of the model. (c) Let CT be a semiset of the model which is a 1-1 mapping of K, into Ur. There exists a relation r E B which is a b,-mapping of K, into Ur and such that = r”Z,. Suppose that Inv (r) 2 PInv ( e ) for some e E 1, e x Xu. For L E Ur let h’r be the least y < o,such that rtY=+ Oh (if such a y exists). There exists L E Ur - e such that h’r is defined; for otherwise W ( 0 ) E e and hence K, & K,, a contradiction. Since r $ e we have K E D(h) and h ’ = ~ h’r whenever K E Ur - e. If wc set h’i = y then {rx7;ti E D ( h ) - e} is a disjointed family of b , which belongs to 9 and which cannot be well-ordered in the sense of 9. This is a contradiction, since by 3357 the family {rx,,;K E E D ( h ) - e } belongs to the subalgebra with basen’leb, and conscquently CJ
Ur
to the class Ker‘*(”) (e), which can be well-ordered in the sense of Gt (9). Similarly, let CT be a semiset of the model and suppose that CT is a 1-1mapping of Ur into K,; as before, h’r is defined for every L E Ur so that there exists L E D ( h ) - e (where r a n d e arc defined as above). The rest is obvious. (d) For each y < o, let u,,be the basic element of Cant (K,,K,) which assumes the value 1 at y and is undefincd elsewhere, and define r by r = = ((7, r ) , ( u ~1,) ) . We let e = r”Z; as in the proof of Theorem 6107 (a) the reader may verify that e has the required property. 6217. COROLLARY. Let K, be a constant for a regular cardinal introduced by a good definition. The theory TSS‘ extended by adding (a) - (d) of the preceding theorem is consistent relative to TSS’; moreover, the following holds in the sense of the model Up as a model in this theory; There exists a family c of subscts of wp such that K, < c for every K, < N, and neither K, c nor c K,.
<
<
29 1
6218
AXIOM OF CHOICE
[CHAP. V l SEC. 2
In particular, for fl= 0, we obtain the consistency of the following statement: There exists a family of subsets of oowhich is infinite (in the sense of 2216) but does not contain any countable subset. 6218. We shall now continue from Chapter 111, Section 3 our investigations concerning a-boundable formulas and their permutation models. Suppose that q ( x ) is a-boundable where u is a constant introduced by a good definition and that q ( x ) has a permutation model given by a formula o(x, z) where o(x, z) well describes a symmetric g-filter on x (everything w.r.t. TSS, U3). We proved in 3348 that the theory TSS, U3, E2, (col (q. a)& o(q, z))
is consistent. Let Xp be a constant for the least regular cardinal number >j:; recall that col (q, u) implies t h a t (Vu E q) ( u M X,,). Let b be the algebra with the base a = h!(cf. 6101), where 1 is the ideal of all subsets of Ur of pow-
11’
er
Ur
< S,. It follows from 5215 that the theory TSS. U3, E2, col (q, a) & w(q, z) 9(b, Z) & TSupp (Z)
is consktcnt (relative to TSS). In this theory we can define the filter z 1 (see 3349) and the model class 9’ = HS (z 1); as usual, a, denotes the ordered set (a n 9,5 n S)arid b, is the complete Boolean algebra in the Sense of Gt (9) with base a,. Z, denotes a Boolean support on b, in the sense of Gt (9) which i\ a restriction of %. The latter theory together with all defined notations will be denoted by TSS” as beforc. The following thcorem is crucial for the dcrnonstration of the consistency of T S + D3 + (3.x) cp(x): 6219. THEOREM (TSS9, i?TSS/i?Gt(Y), JTSS/c?Gupp” (Z,)). In the sense , subclass of p: is a set. of Gupp (Z,) * G t (91every Proof. We shall say “in the sense of the model” instead of “in the sense of Gupp ( Z , ) * G t (9’)”.Let us recall that the sets in the sense of the model arc thc elements of 9’ and the semisets in thc sense of the model are thc semisets CJ such that CJ = r ” Z , for some r E 9.The power of jj: is less than Xp, so by 533 1 (2) every part of is a set (and, consequently, every part of p: n H S (zl) is a set). It remains to prove rhat every semiset in the model which i s incltided in jj: n HS (zl) is a syminctric set. We shall prove the 292
CHAP. VI SEC. 21
6221
AXIOM OF CHOICE
following stronger statement: if x E HS’ (z, q) = HS (z 1) n Ker’ (9) and if x = r”Z, for some r E B then x E 8.Suppose k E Inv,, ( r ) n Invur (zJ), k = hJ n PInv,, (e), h E z, e E P&Jr); we prove that k E Inv,, (x). We may suppose that r is disjointed; iff is the canonical embedding ofb, into b and if we let ( t , f’u) E F = ( t , u ) E r then hvUr(f) 2 1nvcr (z.1) and li E c Invur (F). Since F”Z is a set there exists an exclusive system ui of elements of b with the following properties: 1) V m
=
Thru (D (F))
E Z.
2) If u E m then u is a maximal element such that
E Inv,, (m),so that k c Invu,(m). Let k such that p , and p have the same extension onto q (see 3350) such that @;tr A u $. Ob. Since rn is an exclusive system it follows that 8;. = u and hence h;(b”) = b”, where b” = { v E b, v 2 u}. On the other hand we have f”Z = ?”(b”), since (Cnv (F))” x c b” c Z, i.e. x E F”b” E r”Z, = x. Hence
Clearly Inv, (F)
c Inv,, (D (F))
r
r
u E rn n Z and for each p E k let p1 be an element of
Since fi Ker’ (q) = p, Ker’ (q), we have p’x = x and hence p The following theorem can be proved as 6216 (d):
E
Inv
”, (x).
6220. THEOREM (TSS”, dTSS/aGt (S), aTSS/aGuppZ (Z,)). The following statement holds in the sense of Gupp (Z,) * Gt (9): There exists an exact functor e such that D (e) = Ur and W (e) c q,; moreover, if L E Ur then U(e”{t}) = op.
6221. METACOROLLARY. Let a be well-defined, let q ( x ) be an a-boundable formula and suppose that o(x, z) well describes a symmetric g-filter x and determines a permutation model for cp(x); (everything w. r. t. TSS, U3); then the following theory is consistent (relative to TSS): TSS’, ( 3 4 ) (u(4)= U r & 4 is disjointed and all elements of 4 have the same power as Ur & q(4) & (Va c F:) (M (a)) & ( 3 ~(38) ) (e is an exact functor & D (Q) = U r & (Vc E Ur) (e”{L}is a cofinal part of cop)). The following statement holds in the model Up as a model in the above theory: 293
6222
AXIOM OF CHOICE
[CHAP. VI SEC. 2
(*) The model class M contains all urelements; there is a partition q of Ur whose elements have the same power as Ur, such that 1) rp%)
1
2) P: c M , 3) there exists a 1-1 mapping of Ur onto a certain set of cofinal subsets of some cardinal as. 6222. Proceeding in the theory TSe extended by (*) we prove that (38) (3x f ppp (a,))d x ) . Lct f be the mapping of Ur onto cofinal subsets of osgiven by 3); clearly f" Ur is a set of individuals (in the sense of 3339), so that x = {f"u, u E q ) is also a set of individuals and x E Ker. The mappingfcan be extended to a n isomorphism between ji: and jj:; since ji: is the same as j: in the sense of Ker and since rp is a-boundable, we have q ( x ) and q ~ ~ "(x).( ~It ~follows ~ ) that ( 3 p ) (3x E PP (a,))q ( x ) holds in the sense of Gt (Ker); thus we have demonstrated the following result: 6223. METATHEOREM. Let a be well-defined, let rp(x) be a-boundable and suppose that o(x, z ) well describes a symmetric g-filter on x and determines a permutation model for rp(x). Then the formula (3x
is consistent with T S
294
+ D3.
E
PPP (on)) rp(x)
CHAP. V I SEC. 31
SUPPORT OF A MODEL-CLASS
6301
SECTION 3
Support of one model-class over another
The notion of a support has interesting applications in the study of modelclasses, especially in the study of a pair of model-classes X , Y such that X E Y. I n this section we discuss these applications. For the sake of brevity we consider the theory TSS, D3 (denoted here by TSS"'); however it would be easy to reformulate the main results of this section for full modelclasses in TSS". First consider two possibilities of composing direct models determined by the F-definitions Gt (M) and Seal. We have the two following diagrams: (TS' denotes TS, D3)
6301
I
295
6302
[CHAP. VI SEC. 3
SUPPORT OF A MODEL-CLASS
6302.
Consider the first diagram. Arrow 1 is air (aTS/d%eal) which is a model of TS"' in TSS,. Arrow 2 is air (dTSS,/dGt(M)) which is a model of TSS, in TSSz.The axioms of TSS, arc as follows: (a) the axioms of TSS"' and the fixing Mcl (M); (b) *-class-variablesfor subclasses of M and other *-notions corresponding to TSS-notions are defined by dTSS/aGt (M); c) 0-class variables are defined for those *-classes which are real in the *-sense;
(d) Xo EO Yo is defined as Xo E* Y o ; (e) all other 0-notions corresponding to TS-notions are defined by their definitions from aTSS/a%eaI*. Obviously, the following are provable in TSS"', Mcl (M), dTSS/dGt (M): Real" ( X " )
=. X * c M&(Vy€M)(X* n y E M )and X* E* Y* = X * E Y* .
Hence the following are provable in TSS,:
6303.
(1) (VX) f(3x")( X (2)
(VX',
=
Y O ) ( x O' E
Xo)
yo
The pair of formulas %Gt (M). 296
.X
5
M & (Vy E M ) (X
A
y EM)] & sec,
= X O E yo)& sec.
(L), (2) is an F-definition which we denote by
CHAP. V I SEC. 31
6305
SUPPORT OF A MODEL-CLASS
Now consider arrow 3 and its terminal theory TSS, and let the notions defined by dTS/a%Gt (M) be denoted by the superscript 0. If we add to TSS, the derivation aTSS/aGt (M) (where new notions are denoted by superscript *) we obtain a conservative extension TSS, of TSS,. The axioms of TSS, which occur sub (a) and (b) above are axioms of TSS,, those sub (c) and (d) are provable from (1) and (2) and those sub (e) coincide with the corresponding axioms from aTSS/a%Gt (M). Hence TSS, is equivalcnt to TSS, and is therefore a conservative extension of TSS,. Thus the diagram commutes and 4 is a faithful model; hence 3 is a model and a TSS-formula p holds in 1 * 2 iff it holds in 3. Now consider the second diagram. As in the preceding discussion we verify that the diagram commutes and that 8 is a faithful identity where the F-definition Gt%(M) is given by: 6304.
(4)
.
= Real ( X )& X c M] & sec, (vx*,Y*) (x*E* Y* = X* E Y*) & sec.
(3) ( V X ) [(3X*) ( X *
=
X)
We shall often have occasion to consider two model-classes X , Y such that X c Y. 6305. DEFINITION (TSS”’).
.
2Mcl ( X , Y ) = Mcl ( X )& Mcl ( Y )& X
_C
Y
Consider the theory TSS‘”, 2Mcl (M, N) and ask if 2Mcl (M, N) is a specification for a reasonable F-definition. 2Mcl (M, N) is certainly a specification for Gt (N) stronger than Mcl (N) and if we denote the notions defined by aTSS/dGt (N) by the superscript 0 then by 3228 we have TSS’”, 2Mcl (M, N), aTSS/aGt (N) F MclO (M) ;
i.e. M is a class of the model and a model-class in the sense of the model. The easiest thing we could do is to construct Gt(M) in G;t(N), i.e. to consider the composition 9 * 10 in the following diagram:
291
6306
SUPPORT OF A MODEL-CLASS
[CHAP. Vf SEC. 3
6306.
But this is of no use because 9 * 10 is equivalent to 9 * 11 and 11 is a faithful identity. Indeed, consider the theory TSS,, in the diagram. We have constants M, N. First, 0-class variables are defined for subclasses of N and all other notions of dTSS/dGt (N) are defined. We have Mcl' (M); we define *-c!ass variables for those 0-classes which are 0-subclasses of M and define X * E* Y* as X * en Y*, etc. But one shows as in the preceding diagrams that 1.1 is a model and that the diagram commutes. Moreover, TSS,, is a conservative extension of TSS, so that 11 is a faithful identity. Hence a TSS-formula 4p holds in 9 * 10 iff jt holds in 9 and therefore the composition 9 * LO is of no interest. A slightly more complicated procedure is required. We recall 1444 telling us that TSS", 2Mcl (M, N), aTSS/aGt (N) I- Realn (M)
(as a consequcnce of Mcln(M)), i.e. if we construct the real model in S t (aTSS/aGt (N)), M will still be a class of this model. Hence it seems to be reasonable to consider the composition 12 * 13 * 14 in the following diagram:
298
CHAP. VI SEC. 31
6308
SUPPORT OF A MODEL-CLASS
Consider the theory denoted by TSS,,. In this theory, 0-class-variables are variables for subclasses of N and 0-notions are defined by dTSS/dGt (N). ' B e d is constructed in the 0-sense; in particular, 0-class variables are defined for 0-classes which are U-real. We have TSS,, t MclO (M) and we continue to define *-notions constructing Gt (M) in the 0-sense. Thus +-classes are defined as 0-subclasses of M, or, equivalently, as those subclasses X c M for which (Vy E N) (X n y E N) . This leads to the following F-definition (called Gt'BGt (M, N)): 6308. ( 5 ) (VX) [(3X*) ( X = X * )
(6) (VX*, Y*) ( X * E* Y*
= .X
E M& (Vy
= X * E Y*)&
E
N)( X
ny
EN)] & sec,
sec.
We see that the diagram commutes (since 16 is an identity). It is easy to show that I6 is a faithful identity; i.e. a formula cp holds in 12 * 13 * 14 iff it holds in 1.5. Moreover, if a formula holds in 12 then it obviously holds in 12 * 13 * 1.4 and hence in 15. In particular, by 5123, the following holds in 15:
(i) (St) ; (ii) (Sl) + ( S 5 ) ; (iii) ( S l & Bd)
+ (S6)
.
299
6309
SUPPORT OF A MODEL-CLASS
[CHAP. VI SEC. 3
We consider one more diagram which will be useful later:
I
= TSS,
Here SpStR (Z, M) is defined as Mcl (M)& Real (Z)& Z & (Vu, u
c M) (Dep (M, u, Z) & Dep (M, u, Z) + .+ Dep (M,
where Dep (M, u, Z)
c M&
_=
- 0, Z))
,
(3r E M) ( u = r"Z) .
GpEt% (Z, M) is the following F-definition: 6310. (7)
(VX) [(3XA) ( X = XLZ)= . X
E
M& ("YE M) (Dep (M, Y n X , Z)]&
& sec ; (8)
(VXC,
YA)
(Xb. €/IY"
3
X"
E
YA)&
sec.
The reader may easily show that in the last diagram all arrows are ~nodels, that the diagram commutes and that 21 is a faithful identity. We now discuss the notion of a support in the sense of some of the rnodels just discussed. 300
CHAP, VI SEC. 31
SUPPORT OF A MODEL-CLASS
6313
6311. THEOREM (TSS”’, Mcl (A), aTSS/dGt% (A)). If Z is a total support in the sense of Gt% (A) then V = Cstr ( Z x A).
Proof. Let Z be a fixed total support in the sense of Gt%Gt(A) and let R be the fu!l upward extension Fup in the sense of Gt% (A). (The existence of such an R follows from the fact that Z is a total support in the sense of Gt%(A).) We set B = Cstr(Z x A) and observe that R E E A E B and R is a real class in the sense of Gt% (B). We define, in the sense of Ei%(B), a real morphism of C ( R ) onto a complete class C as follows: H”{O} = Cnv(ImbGtm(”)), v >0
+
IZ”{a} = {(x, y ) ; y E C ( R )& E d R( y ) c D (H”u) & x = =
(If”%)” E d , ( y ) } .
Further we let H ” On = F and W ( F ) = C. The c!ass C is complete (it is even a model-class) and we have C E Cstr(Z x A) and D(F) = C(R). We prove C = V. If, on the contrary, V C, then we may choose an element x of least rank in V - C; clearly x c C. We let (Cnv (F))” x = y. Since y is a semiset in the sense of Gt% (A), and since R is saturated-universal in the sense of Gt% (A), there exists some xo such that Ext, (xo) = y . Hence we have (x, xo) E F and x E C, a contradiction. We have proved V c Cstr(Z x A), so that V = Cstr(Z x A).
+
6312. COROLLARY (TSS”’, 2Mcl (A, B), aTSS/aGt%Gt (A, B)). If Z is a total support in the sense of Gt%Gt(A, B) then B = Cstr(Z x A). Proof. We recall that the classes of the model Bir (aTSS/dGl%Gt (A, B)) are the subclasses of A which are real in the sense of Gt (B); in particular, the semisets of the model are the subclasses of A which are elements of B and the sets of the model are elements of A. By the preceding theorem we have [V = Cstr (2 x A)]6t’(B) and hence by 3525 we have B = = Cstr ( Z x A) (see Diagram 6307). 6313. THEOREM(TSS”’, SpStR (Z, A), dTSS/dGpGt% (Z, A), 2TSS/dGt%Gt (A, Cstr (Z x A))). (1) Z is a total support in the sense of Gt%Gt (A, Cstr (Z x A)) ; (2) For any X , X is a class in the sense of GpGt% (Z, A) iff it is a class in the sense of Gt%Gt (A, Cstr (Z x A)). Proof. (1) Let R” be the full upward extension of E in the sense of 3 p 6 t % (Z, A) and let Imb“ be the embedding of the universal class into RA in the sense of this model. Clearly RA is an almost universal, extensional 301
6314
SUPPORT OF A MODEL-CLASS
[CHAP. VI SCC. 3
and regular relation. We define a relation H by the same definition as in the preceding proof and let F = H" On, W ( F ) = B. B is a model-class: it is closed since it is morphic to C (RA) w.r.t. E, RA, it is obviously complete and it is an almost universal class because RA is an almost universal relation. Suppose that x is a semiset in the sense of Gt%Gt (A, B), i.e. x E E P (A) n B; we prove DepG'"(*) (x, Z). For some y E C (RA) we have (x, y ) E F and since the composition of Imb" and F is a n identity on A we have E d R A(y) c W (Imb*), Ext,, ( y ) = Imb"" x and x = (Cnv (Imb'.))" EXtRA (y). Since y E C (R'>) and both R L and ImbA are classes in the sense of QGt% (Z, A), both Kxt,, ( y ) and x are semisets of G p 5 t % (Z, A), i.e. x is dependent on Z in the sense of Gt%(A). Consequently, Z is a total support in the sense of Gt92Et (A, B). Using the above corollary we obtain B = Cstr (Z x A), which completes the proof of (1). (2) In both models the sets are the elements of A; the semisets of the first are the subsets x of A such that x = r"Z for some r E A and the semisets of the second are the subsets of A which are elements of Cstr (Z x A). Clearly every semiset of the fist model is a semiset of the second, since Z is a class of 5t%(Cstr (Z x A)). As for the converse, we have proved in (1) that every semiset of the second model is a semisel of the first. The classes of the f i s t model are the subclasses of A which intersect every semiset of the model in a semiset of the model; the classes of the second model are the subclasses of A which are real in Gt (Cstr (Z x A)), i.e. which intersect every semiset of the second model in a semiset of that model. It follows that both models have the same classes. Since the membership predicate in both models coincides with the actual membership relation we obtain the following 6314. METALEMMA. The following diagram commutes and both 24 and 25 are faithful identitics.
302
CHAP. V l SEC. 31
SUPPORT OF A MODEL-CLASS
6319
6325. COROLLARY. (1) (TSS”’, Mcl (A), i3TSS/dGt% (A)). If Z is a support in the sense of Gt8(A) and x E A then DepG‘”(”)(x,2) iff X E E Cstr (2 x A). (2) (TS”, aTSS/aGt (L)). If Z is a support in the sense of Gt (L) and x E L then DepetcL)(x,2) iff x E Cstr ( Z ) . Indeed, we proved (I) for fixed A and Z in the theory indicated in the preceding theorem (cf. the proof of that theorem); it follows that our statement (1) is provable in the theory indicated sub (1). (2) follows from ( I ) because TSS’” I- Cstr (2) = Cstr ( Z x L). 6316. DEFINITION (TSS”’). Let X , Y be model-classes and let X E Y. We say that Z E X is a supportfor Youer X (Support (2,X , Y ) ) iff
(I)(Vy E Y ) ( Z n y
E Y ) (consequently, Z
(2) (Vx E P ( X ) n Y ) (3r E X ) (x
=
is a real class),
r”Z).
If Z is a set then it is called a set support f o r Y over X ((1) can then be replaced by Z E Y ) . 631 7 . Remark (TSS’”, 2Mcl (A, B), dTSS/dGt%Gt (A, B)). For any Z ,
(L) Z is a support for B over A iff Z is a total support in the sensc of
Gt8Gt (A, B); (2) Z is a set support for B over A iff Z is a scmiser total support in the sense of G t N G t (A, B).
6318. THEOREM(TSS”’). Let A E B c C be model-classes and let B be a model-class with El. If 2 is a support for C over A then Z is a support for C over B. Proof. Clearly 2 c B and (Vy E C ) (2 n y E C). Suppose that x c B and x E C. We must find r E B such that r”Z = x. There exists some a E B such that x E a ; we let f e B be somc one-one mapping of a onto an ordinal a and xo = f ” x . Clearly xo E A and xo E C, so that xo = s”Z for some S E A . If we denote by r the composition of Cnv (f)and s we obtain r”Z = x; hence Z is a support for C over B. We now derive some theorems on set-supports.
6319. THEOREM (TSS’“). Let A c B c C be modeI-classes and let B be a model-class with El. If z1 is a support for B over A and z2 is a support for C over B then there exists a set-support for C over A. Proof. Fix A, B, C. Clearly z1 E A and z1 E B; similarly z2 G B and z2 E C. We may suppose that z2 E A; for if z2 s a E B and iff E B is a oneone mapping of a onto an ordinal then Z2 = f”zZ is a subset of A and, since 303
6320
si
SUPPORT OF A MODEL-CLASS
[CHAP. VI SEC. 3
( z 2 , Z2), Z, is a support for C over B. If we let z = z2 x z1 then z E A and z E C. If .x -C A and x E C then x = r;’z2 for some r2 such that rZ E B, r2 c A. Since r2 E B A P(A) there exists r , E A such that r2 = r;’zl. If we let (it, u, w) E r E ( ( u , u), w) E rl then r E A. If U E x then ( u , v) E r2 for some u E z2 and hence ( ( u , v), w) E r , for some w E zl. Hence we have ( u , v, w) E r, (v, w) E z and u E r”z, so that x c r”z. Conversely, if u E r”z then (u, v, w) E r for some (v, w) E z ; hence (u, u ) E E r2, t i E z 2 and we have u E x. We have proved x = r”z, i.e. z is a support for C over A. 2 tWSt(B,C)
6320. THroREM (TSS’”). Let A C B E C be model-classes and suppose that B is a model-class with Et. If there exists a set-support for C over A then there exists a set-support for B over A. Proof. Again fix A, B, C. By 4134 it is sufficient to show that (Pot) and (Bd) hold in the sense of Gt%Gt (A, B). If a E A then by there exists f~ B which is a one-one mapping f of some ordinal CI onto P ( a ) n B. For 7 < x we let ( y . y ) E r = y ~ f ’ y ; in the sense of Gt%Gt (A, B), r is an exact functor which codes all subsemisets of a. Hence (Pot) holds in the sense of 3 t 9 < 5 t(A, B). Suppose that z is a disjointed support for C over A and that z E a E A. If x c A and x E B then x = r”z for some disjointed relation r E A with D ( r ) E u. For u E a we let f ’ u = r”{u), b = and b hence (Bd) holds in the sense of Gt91Gt (A, B). 6321. COROLLARY. (TSS’”). Let A E B c C be model-classes and suppow that B is a model-class with EL. A set-support for C over A exists iff there exist set-supports for C over B and for B over A . We now state a theorem showing the significance of the class HDf of all hereditarily definable sets in the theory of semisets with the axiom (SConstr). 631717. THEOREM (TSS”). If V = Cstr(x) for some x c On then V has set-support ovcr IIDf, Proof. Let s c 5 E On. To begin with we construct a certain definable (not nccessarily hereditarily definable) “definably complete” Boolean algebra and an ultrafilter on it which is closed under definable intersections. If we now let b,, = P (P (c)) n Df (the set of all definable sets of subsets of t) then b, E Df. For u, L‘ E bo we let (u, u ) E c, = u = P (c) - v ; clearly c,, E E Df. For q E b,. q E Df, 11 E b, we let ( u , q ) ~ f =, u = nq; clearly f, E Df. I f n c define b, = (b,, c,,,fo) and u E -7” E x E u for u E b, then we have U E Z , = P ( 5 ) - ~ I # z , and , q E Z, = r ) q E z , for q =k 0.
;L
304
CHAP. VI SEC. 31
6325
SUPPORT OF A MODEL-CLASS
Let F be the one-one mapping of Df onto On which was defined in 3406. F has the following property: for any x c Df, x E Df = F”x E Df. We let b = F”b, and define the operations on b in such a way that F becomes an isomorphism of b, and b. We have b E Df and since the elements of 6 are ordinal numbers, it follows that b E HDf. If we let z = F”z, then, in the sense of Gt% (HDf), b is a complete Boolean algebra and z is a complete ultrafilter on 6. Let A = Cstr(HDf x z). ByTheorem 6313 z is a total support in the sense of Gt%Gt (HDf, A) hence z is a set-support for A over HDf. To prove that A = V it suffices to show that x E A. We define a function g by setting g’u = U((Cnv (F)” u) for all u E b. Since the sum of a definable set is definable, g is definable and, since g assigns to each ordinal number in its domain a definable set of ordinal numbers, g is hereditarily definable. Hence g”z E A and n(g”z) E A. We also have n(g”z) = x, since a E x iff for all u E zo there exists a E u such that x E a; one implication is obvious while if a 6 x then P (t - {a})E E zo and x is in no a E P (t - (2)). Hence we have a E x = a E U(u) E E
n
x E n(g”z).
UEZO
6323. COROLLARY (TSS”’, x c On, aTSS/dGt (Cstr (x))). Cstr (x) has a set-support over HDfGt(C5fr(X)). 6324. THEOREM (TSS”’). (a) If M = Cstr (x) and if a model-class N has a set-support over M then N = Cstr ( y ) for some y. (b) If x E On and if M = Cstr (x) has a set-support over a model-class N with El then N = Cstr ( y ) for some y. Proof. (a) follows from Theorem 6312; if y is a set-support for N over M then N = Cstr(Cstr(x) x y ) = Cstr(x x y). (b) We may suppose M = V and fix N. By Theorems 6318, 6312 it suffices to find a model-class No c N such that No = Cstr ( 4 ) for some q and such that V has a support over No. By assumption there exists, in the sense of Gt% (N), a complete Boolean algebra b and a complete ultrafilter z such that z is a total support in the sense of Gt% (N). We have x = r”z for some r E N. If we let No = Cstr ( b x r) then No c N and z is a support for V over No. For, z is a complete ultrafilter on b in the sense of Gt%(N,); hence z is a support and, since x = r”z, Cstr(x) is included in a unique model-class for which z is a support over No; consequently this model-class is V. 6325. COROLLARY (TSS”’). If V some y .
= Cstr (x)
then HDf = Cstr ( y ) for
305
6326
[CHAP. VI SEC. 3
SUPPORT OF A MODEL-CLASS
We shall now continue the discussion begun at the end of Chapter 111, Sect. 5; we shall demonstrate the independence of the axiom of constructibility from the axiom of definability. (See also Section 4 of the present Chapter.)
6326. We define an algebra Comb as follows. Let 9 be the smallest cardinal > 0 such that a < implies 2Kb< <,i.e. 9 = Uh’n where h’O = 0,
<
h’(n
I
+-
<
m
1) = 2Kh’n,Comb is the algebra with the base a = n ’ b : : : where
P (9). (See 6101 for the definition of 6:: t .) Consider TSS, D3, (Constr), (S Comb). We put a,,
LE9
=
=
n”’
b i z : , dmi-
1
2 a and +a. Clearly a is isomorphic to a,, larly for >CL, to a < , 0 b::: 0 a>,.
0 a , , and also
6327. LEMMA. (TSS’, E l , Cont). [a) Every exclusive system in a,, has power at most K,; (b) a,, is K,-multiplicative. Proof. (a) As in the proof of Lemma 61t7, a,, may be represented by at (2) where s = ((y, S}; 6 < a & y < o ~ +and ~ } XLS
x ~j
=
.
x E s & D (x)
EI
& (V6) (Extx(6) < m,,,).
We have Norm(j) 5 K,, so that every exclusive system in a,, has power at most K,, by 2524 or by 2526. (b) If L 2 3 then v(b:f,:) 2 K C f l 2 Kafl and, since the ideal 1 (L - x ) is Xu-additive, a,, is K,-multiplicative.
I
e
6328. LEMMA (TSS’, El). If 2 is a Boolean support on a and if u ( Z ) then Z = P(e) n u and for a < 9 we have
=
Proof. see 4329. (a, is ( ( ( u , a ) } ; u E Z~zft).)
6329. LEMMA (TSS, D3, Constr, (S Comb)). (a) All cardinals are absolute. (b) For every CL, i ( 2 , K,) = This can be proved in analogy with the proof of 6118. 306
CHAP. VI SEC. 31
6331
SUPPORT OF A MODEL-CLASS
6330. METALEMMA. Let T(X) be a godelian term such that TSS, D3 t = { a ; TAp (L, Nu+,)}.Then the followi~ig is provable i n TS, D3, Mcl,, (L): for every model-class M ,
t- T (V)
Demonstration. It suffices to show the following in T S , D3, Mcl,, (M), ZTSS/aEt (M):
Since semisets in the sense of Gt (M) arc subsets of M (we arc working in a theory stronger than TS), AP'*(~)(L, ma+ 1) is equivalent to
Bi1t L is absolute, Un is absolutc and, sincc Mcl,(M) absolute and (*) follows.
holds,
.< is also
6331. METATHEOREM. Let T(X) be a godelian term such that TSS, D3 t- T (V) = {a;1Ap (L, X\\,+l)} and denote by (Det,) the following axiom:
(+,
i )(Det, ( r )
Est ( r , F) & r
r (o} 4 Cstr
(0))
.
(Cf. 3532 and 3533.)
Ilp is a model in the following diagram:
Deinonstration. We proceed in the latter theory. By 6329 all cardinals arc absolute so that we have Mcl,*,(Tor); V = L implies Tor = Id* (see 5138 (e)). Therefore Mclc*.(L*) holds. Secondly, the axiom of continuum holds in the *-scnsc (cf. 6118). Consequently if 9* denotes the cardinal of the model defined in the sanic ~vayas 9 but i n die scnsc of the m v d c i ihcn we have 9* = Tmb" 9. 307
6331
[CHAP. V I SEC, 3
SUPPORT OF A MODEL-CLASS
Arrow 1 in the following diagram is the model Up: r
I
TS,D3,€2,Coni, (L)
Mcl,,
1
I
-1
1 1
TSS,D3,Constr, ] L l ( S Comb) + upw TSS/ d Gf *(Toor) dTSSId GlfL) -_ _ TS,D3,€2,ConT,
TSS,D3,Constq
(SComb) + upw
We extend Up by defining 0-notions in the terminal theory of 1 by c?TSS/dGt* (Tor)(arrow 4) and we know that in this way the whole diagram becomes commutative. (Cf. arrows 5, 6, 7, 8 in Diagram 5L12). By 5126 ( S Comb)" holds in 3. Hence let us proceed in T S , D3, E2, Cont, Mcl,, (L), 2TSSlGG-t(L), (S Comb)". Denote by z a complete ultrafilter on Comb" i n the sense of Gt (L). Defining r = Uz we have by 4329 rr{a}= ~ ( ~ n ~ ~ ) , r ~ ( D ( ~ ' ) - ( a } ) = U ( _ ~ n a ~ , ) , 'z n a," = Po ( r T { x } ) n u;, z n a:, = P ( r r D ( r ) - ( E } ) n a;,. We wish to prove that I' is a T-determining well estimated relation. In particular, we want to prove (2) (Vx < 9) (3 E T (Cstr ( r
I'(x}))&
r
x $ T (Cstr (I' (D ( r ) -
{E}))))
.
L is a model-class with absolute cardinals and therefore, by 6330, (2) is eqiiivalcnt t o
CHAP. VI SEC. 31
6332
SUPPORT OF A MODEL-CLASS
; hence the quantifiers in (3) may be replaced by
= Cstr (z n
(Vfe Cstr (z n u:))
(WECstr (z n uYu))
and
respectively. Since Un (f,mu+,, implies f E L we may use 6315 and so these quantifiers may be replaced by (Vf) (DepC (f,z n u:) -+ and (Vf) (Depn (f,z n utu) -+ respectively. Thus we must prove the following: (4) (Va < 9) (~[(vcP) (Dep' (an, z n u p ) -+ (3O) (a" E s" & (VL) (Ed,, (L) < mu+ 1 )))]& & [(Van) (DepO (an,z n ug,) -+ -+
+
(a" E so & ( V t ) (Ext,,
(4
%+,)))]> .
Instead of proving this it suffices to prove the following i n TSS,D3, Constr, (S Comb) + upw: if 2 is a total Boolean support on Comb then
(5) (Va < 9) (l[(Va) (Dep (a,2 n uu) -+ (3s) (a E s & & +
W) (Ed, (4
[(Vo) D e P (0.
ma+ 1)))l
( 3 s ) (g c s & (VL) (Ed,
(L)
< mu+ I)))]) .
zn
+
In TSS, D3, Cors:r. (S Comb) + upw, fix Z and a < 9, and add the definiOur problem tions aTSS/i?Gupp ( Z n ua) and aTSSldGupp (Z n reduces to proving 1Appr (ma+ 1, ma+ 1, ma+1) in the sense of Gupp (Z n a,) and Appr (ma+lr ma+,, ma+*) in the sense of Gupp (Z n In the sense of Gupp (Z n ua) we have a total Boolean support on Cant::: and therefore by 6106 I Appr (aa+ ma+1, ma+ 1) holds in the sense of Gupp (Z n ua). To prove Appr (ma+1, ma+1, ma+1) in the sense of Gupp (Z n a,,), suppose that Appr (Ka+l, Xa+lr Xa+l)fails in the seme of Gupp (Z n n (u..~ 0ua<));then there exists some a E a ,, which is not approximable. Observing that u,, is N\",+,-rnultiplicative,we see that c is dependcnt on Z n uca. But since p(a,,) 5 Ka+ it follows that Appr (Xm+ Sa+1, Na+ holds in the sense of Gupp (Z n uca), a contradiction. Hence in TS, D3, E2, Cont, aTSS/dGt (L), (S Comb)= we have shown that r is a T-determining relation. It remains to show that r is estimated. Define F = (bz::)" x { a ) ) ;evidently r c F; since (Cont) holds f is a good
u
U
estimation for r. Evidently, r f (0) $ L. This completes the demonstration. By 6331 and 3534, we have the following 6332. METACOROLLARY. The axiom (E3)& 1 (Constr) is consistent relative to TSS, D3. 309
6400
I3OOLCAN VALULS A N D FORCING
[CIlAP. VI SEC. 4
SECTION 4
Boolean values and forcing
mi i :i li n al sect ion is co iicer t i ed w1t h a met am at he in at i cal no ti o n which corresponds to the intuitive notion “generalized truth value of a formula”. We give an algorithm which enables 11s to define, for each closed TSSformula cp of a certain type, a godelian term F,(b) in TSS’ such that the following is provable in TSS’: I ) If b is a complete Boolean algebra then F, ( b ) E b.
2) If b is a complete. Boolean algebra and Z is a total Boolean support on b then cp = F,(b) E Z. The element F, (b) will be called the cp-value in 6. With thc aid of this notion we can, for example, give a uniform reduction of consistency problems w.r.t. the theory TS, D3, E2, a T S S / G ‘ t (L), ( 13d)‘*(L)to consistency problems w.r.t. TSS”’, (Constr). We shall show how to find for any SF cp another SF @ such that cp is consistent with TS, E2, ?TSS/d,Gt (L), (Bd)“’(L) if and only if @ is consistent with TS, (Constr). I n case @ is provable in TSS, (Constr) it follows that q is consistent with TS, E2, dTSS/dGt (L), (Bd)Gt(L); the results of Section 1and other similar results may be obtained in this way. The formula $C always has the form “there exists a complete Boolean algebra with such ar?d such property”; in other words. we give here a “canonical form” for conditions to be imposed on a complete Boolean algebra in order that the existence of a total Boolean support should imply that a particular formula cp is valid i n the sense of the model Up. The “canonical form” in question is the form (36) (F, ( 6 ) = 16). We shall also investigate properties of operations on cp-values and we shall then consider the matter more generally in rhe case when the argiiments of these operations are not complete Boolean algebras but separatively ordered sets. Thus we finally arrive back, in a modified form, at the origifial notion of fprcing due to Cohen. 310
CHAP. VI SEC. 41
6403
BOOLEAN VALUES AND 1‘ORCING
In 431 1 a disjointed relation which is a set and whose dorrain is included in a complete Boolean algebra b was called a b-set. If Z j s a total Boolean support on 6 then for every semiset 0 there exists a b-set I’ such that 0 = r”Z. In particular, if x is a set and if we let x(’) = { ( y , l,}, y E X } then i\ a b-set and we have x = d b ) ” Z . Hence we make ihc following 6401. DEFINITION (TSS‘). Let 6 be a complete Boolean algebra. A 6-set r is called a b-constant if r = x x (I,} for some x. 6402. METADEFINITION. Let x and y be set-variables and let set-variable.
0
be a semi-
(a) x E y and x E 0 are pure SNF ; (b) if y and I) are pure SNF then all formulas of the forms 1 q ,(‘p & $), (q v $ ) a r e p u r e S N F ; (c) if q~ is a pure SNF then all formulas of the forms (VX) y , (Vo) q ,( 3 ~ q) and (30) y are pure SNF ; (d) every pure SNF can be obtained from pure SNF sub (a) by a finite number of operations sub (b) and (c). We now give a n algorithm which associates, with each pure SNF y with iz free set-variables and m free semiset variables, a normal constant [q] in TSS’ such that it is provable in TSS’ that [q] is a mapping which assigns to every complete Boolean algebra 6, to cveiy n-tuple of b-constants and to every m-tuple of b-sets an element of b. 6403. METADEFINITION. We define in TSS‘:
[x E a]’ (z‘p), r , 6) = rzrif r is a b-set
.
31 1
6404
[CHAP. VI SBC. 4
BOOLEAN VALUES AND FORCING
(More exactly, if cp has free variables u l , 0 , i l l , has free variables u l , 0 , w l , 0 , then lcp&
$1’ ( r l ,
0 , sl, 0 ,
t,,
0,
b) = A
0
for sets or semisets and
19%’( r l ,
wn’( r l ,
0 , sl, 0,
I,,
0, 6
)
A
0, b )
;
I)
similarly for disjunction.) Suppose we have already defined [cp] and that (Vx) rp, resp. (Va) cp are formulas; we define
6404. Rernurk. If k, 0 are b-constants and r, 0 are b-sets we shall often write $cp(li, 0 , r , o)]b instead of [rp(x, .,a, o)]’ ( k , 0 , r, 0 , b ) , i.e. we “substitute k , for xl, rl for ul’’etc. Notice however that [cp(x, 0 , 6,.)]I is the name of a mapping which has ( k , 0 , r , 0 , b ) in its domain. All constants [cp] arc defined in TSS’ by normal formulas; hence it is provable in TSS‘ that [cp] is a real class. If cp is closed thsn we may define an operation F,(b) by setting F,+,(b) = [cp]’ b. In TSS’ we call [cp]’ ( k , 0 , r , 0 , b ) “the cp-value of k , 0 . r, 0 in b”. 6405. Extrrnple (TSS’). If b is a complete Boolean algebra and r is a b-set thCil
I’hI‘Ub
( D ( r ) ) = I(3.X)
(X = ff)]’
(r, b) .
P i m f . To be precise, the formula (3x) (x = 0) is not a pure SNF since . By [1(3x) (x = a)] we mean [(ZIX) (vJ’) ( ( y E x & y E a) v WC have
it contains 7 v I# s & J’ # (
312
~
3
.I)
CHAP. VI Let u,
=
6407
BOOLEAN VALUES AND FORCING
SEC. 41
A r,, A A - r,,. We prove the following: ,,EX
YCX
(1) for every x , u, is layered by D ( r ) , (2) for every u layered by D ( r ) there is an x such that u 5
ti,.
(By (1) and (2), I(3x) ( x = o)] i s the sum of all elements layered by D ( r ) . ) (1) is evident; for, if y E x then ry 2 u, otherwise ry 5 - - u - ~ . (2) Put x = { y ; 2) 5 r,,], then u S u,.
6406. METATHEOREM. If q ( x , 0 , o, 0 ) is a pure SNF then the following statement js provable in TSS’: Let b be a complete Boolean algebra and let Z be a total Boolean support on 6; if k = x ( ~ )0, and if o = r”Z, then
Demonstration. By induction on the length of q. We give here the proof for (Ve) q(e, x ) under the assumption that the statement holds for rp(e, x). We proceed in TSS‘. If (Ve) q(e, x ) for some x then rp(r”Z, x ) for any &set r, so that (Vr) ( r is a 6-set + [ q ( r , x‘”)] E Z) and hence A[q(r,x‘”)] E Z . The converse is proved in the same way.
r
6407. METATHEOKEM. The following statement is provable in TSS’ for any pure SNF q : If b is a complete Boolean algebra and if u E b then
Demonstration. By induction on the length of rp. We give the iriduction step for (Va) q. We have A
[(Va) q]’ (k, r , b ) = =
A{u
A
11 A
A{[q]’ (s, k, I’, 6); s is a 6-set)
=
f(p]’ (s, k , r, 6); s is a 6-set} =
1 U,k I u, r I u, b I u ; s is a 6-set} = = A{[q]’( 1 , k I u , r I u , 6 I u ) , t is a ( b I u)-set} =
=
A{[q]’
(S
313
6408
BOOLEAN VALUES A N D FORCING
[CIIAP. VI S t C . 4
6408. METATIIEORIIM. Let cp be a closed pure SNF. The formula q ~ ' * ' ~ ) is coxistent with T S , 113, E2, i?TSS/dGt (L), (Bd)"'(L) iff the formula (36) (,[cp)ij' b 0,) is consistent with TSS, D3, Constr.
+
+
Iknionstrcitioii. (1) Suppose that TSS, D3, Constr, (3) ([q]'6 0,) is ([rp]'b = 1,) i s provable. (Take 5 = b co:isistcn!. I n this theory (3) (T y j ' b ) where EyT'b 0,; then [ q ] ' S = Is.) Denote by Val,(b) the forinLila [ q j ' b = l,,. By the Support Principle 5215, the theory TSS, D3, (loijstr, ( S Valq ( b ) )+ upw is consistent and by 6406 y is provable in this theory. IJsing the model U p as a model i n the last theory we see that T S , 113, E2, i?TSS/dGt(I,), (Bd)"*'L', v " * ( ~is) consistent.
I
+
I
(2) Suppose conversely that the theory TS, D3, E2, dTSS/dGt(I,), ( Bcj)-t".' , ybttL) ,is consistent; we investigate the model Gt (I,) in this theory. I n the sense of this model there exists a total support (by E2); hence, by ( lki)Gt(L), there exists a total Boolean support Z on a n algebra b (in the sense of 5 t (I,)). Obviously, D3 and (Constr) hold in the sense of the model. By 6405 we have [ y ] ' b E Z i n the sense of the model and hence [q]' b =l= 0, i n [he sriise of the model. Consequently the theory TSS, D3, Constr, (3b) b $: 0,) is consistent. Th.- demonstration of the following slightly more general metatheorem is left t o the reader:
([(fa'
6409. MI.:TATIIEOREM. Let rp be a closed pure SNF. The formula v ' * ( ~ ) is consistent with TS", FMcl (M), dTSS/aGt (M), (E2)Gt'M), (S6)"*('' iff the formula (36) ([rp]' b =k 0,) i s consistent with TSS, E2. 'i'hus we obtain the desired reduction of consistency problems for formulas of the form y"(M).We shall show that in TS", FMcl (M), dTSS/dGt (M), ( s ~ ) ' Qevery ~ ) closed set formula is equivalent to some formula of this form.
6410. M t r a i ~ r E o RM.~ If q I$ a closed set formula then there is a pure SNI. I )\uch that rp 3 I / / ' * ( ~ ) is provable in TS", FMcl (M), aTSS/ZGt (M), ( s6)ZNM). I ) e i J i o i i \ f f . t r t i o r t . We investigate the theory TS", FMcl (M), (;TSS/ZGt (M) ( S6)''(M). This theory will be denoted by T S ! 111 the course of thedemonstrat i o n . Recall the godsliail term Fopw (K,S ) defined in 5109. By the proof of 41 19, theie is a goidelian term TU (u) such that TSS' t TSupp (0)--* - + Iotunbr ('IU(0)). Hence fix a constant GO in T S ! by TSupp" (&), definc a constnnt R" by Fupw" (E' I, TIJ' (&)) and add the definitions ?TS/C91" (R- ) (cf. arrow\ 4, 10 in 51 12). We denote the notions defined
CHAP. VI SEC. 41
641 1
BOOLEAN VALUES AND FORCING
by these definitions by the superscript *. Let cp be an arbitrary SF (not necessarily closed). We show by induction that there is a pure SNF Ic/ which is normal and such that
(x*
=
E d R D (xu)& 0 ) + ['p*(x*,
0 ) EE
I/I"(x",
0,
.")I
is provable in TS!, TSupp" (GO), dTS/a%" (FupwO (ED,TUn(aD)).For q atomic we have x* E* y* E <xD,yo) E FupwD (EL', TU" ( 0 " ) ) . Since both Fupw and TUare godelian terms, the last formula is equivalent to a formula Ic/O (xu, y", G O ) of the desired form. For the induction step cf. 1146. In particular, if cp is closed, then cp = q* is provable in our theory (by 5128) and therefore cp = I~/"(G") is provable in TS!, T SuppO (G"). By 1245 we obtain TS!
t (Van) (TSuppO (a")
-+
[q
3
$D (rsO )]) .
Consequently, cp is equivalent in T S ! to either of the following formulas:
Since TSuppO (u") is equivalent to (Ven) Dep" (en, on) which is obviously equivalent to a pure SNF, the demonstration is complete. Moreover, note that in fact we have found two formulas of a relatively simple form equivalent to q: the former has the form [(Vo) (3g) $(a, Q)]" and the latter the form [(3o) (Ve) F(a, e)]" where both $ and 5 are normal pure SNF's. The following metatheorem is analogous to Metatheorem 6409 but refers to provability instead of consistency.
6411. METATHEOREM. Let cp be a closed pure SNF and suppose that n(b) is an SF such that TSS' t (n(b) + b is a complete Boolean algebra). Then TSS', (E2) t (n(b)&2 is a total Boolean support on b) -+ q iff TSS', (E2) k X(b) --* [rp]' b = l b . Dernonstration. The first expression follows from the second by Metatheorem 6406. Suppose the tkst expression holds; we shall prove n(b) + + [cp]' b = 16 in TSS', (E2) (or in TS, E2, since the formula in question is an SF). Suppose that b is a complete Boolean algebra and that d b ) & & [ q ] ' b 1,. Let z be an ultrafilter on b which contains the element - ([q]'b) and let R be the ultraproduct relation given by b and z. In the senbe of % (R),5 is a complete Boolean algebra and we have )@'(n (nota-
+
315
6412
B 0 0 1 . E A N VALUES A N D FORCING
[CHAP. VI SEC. 4
tion as in 5203); there exists a complete ultrafilter Z on 6 such that [cp]’ 6 4 2. Let Z be a fixed ultrafilter with these properties. By (i) the formula cp holds in the sense of Gupp (25) * %(R); the formula l c p also holds, since in the sc‘nse of this model, the cp-value in the algebra b does not belong to the total Boolean support. This contradiction completes the proof. 6422. COROLLARY. If a formula cp is provable in TSS’, (E2) then TSS’, (E2) t (Vb) ([(pi’b = 1,).
Recall that if p is an isomorphism between two complete Boolean algebras 6 , and b2 then we may define a 1-1 mapping fi between b,-sets and b2-sets as follows (see 4324): r2
= (Vs) (Vu E b,) ((s, u) E r1 = (x, p’u) E r2) .
J’r,
=
6413. METATllEOREY. If (p is a pure SNF then the following is provable i n TSS’: If p is an isomorphism between complete Boolean algebras b , and b2 and if r , , 0 are b,-sets then
P’(Jcpl!’
(F-1,
0, h,))
=
[Cpi’
0962)
(fi’F-13
*
Demonstration. By induction (exercise). 6414. META1)EPINITION. A constant F defined in TSS’ by a godelian term is called a Boolean-valued function if the following is provable in TSS‘ for suitable natural numbers nz and n:
+ n + 1.)-tuples whose last element is some complete Boolean algebra b, whose first m elements are b-constants and whose remaining n elements are b-sets;
1 ) (a) D (F) is the class of all (tn
(b) F’(k,
0,
r,
0,
6 ) is always in 6 .
2 ) If h is a complete Boolean algebra, and if k, are h-sets and u E h, then
(a)
11 A
F’(k,
0,
r , 0 , b ) = F’(k
Ib u.,
0
are &constants, r ,
0,
r
Ib
0
s,
0
I
u, b u),
I’ s & -+ F’ (li, 0 , r, 0 , b ) = F’( k , 0 , S , 0 , b ) . (b) 3) If p is an isomorphism between complete Boolean algebras b, and b2 and if k , 0 are b-constants and r , 0 are 6-sets then
31 6
CHAP. VI SEC. 41
6415
BOOLEAN VALUES AND FORCING
We have shown that for every pure SNF q the function [q] is a Booleanvalued function. We now prove the converse. 6415. METATHEOREM. If G is a Boolean-valued function then there exists a pure SNF cp such that TSS’ t G = uqj .
Demonstration. We restrict ourselves to the case m = n = 1, since the other cases are analogous. The domain of G consists of triples < k , r, 6) where k is a b-constant and r is a &set. We define a formula q(x, cr) as follows:
(*) (Vb, e, r) (3(b, p) & r is a b-set & r”e
=D
+
G’(x(~’,r , b ) E e)
where 3(b, 2) is the formula “b is a complete Boolean algebra and 2 is a total Boolean support on b”. First we prove in TSS’ that (*) is implied by
(**) (3, g, r ) (3(b,e) & r is a b-set & r”e
= cr&
G’(x(~),r, 6)
E
e) .
Suppose (**) holds and let 3, Q and i: be such that 9(6,4) and f”p = 0. We prove that G ( X ( ~F,, 6) E a. Since both p and Q are total supports, there exist u E e, ii E 3 and an isomorphism p between b u and 6 ii such that p”(e lb u) = Q u and ~ ’ ( r u) G i: 16 u (cf. 4326). We have
Is
1
Ib
I
1) First suppose that there exist b, x and r such that [q]’ ( x ( ~ ) ,r, b )
- G’(x(’)), r, b )
+ ob .
Then there exist 6, x and Y such that
317
6416
[CHAP. VI
BOOLEAN VALUES AND FORClNG
SCC. 4
We shall show that (***) leads to a contradiction. By Metatheorern 5218 there is no loss of generality in supposing that b has a total Boolean support e. Under this assumption we have [q]’(x(’), r , 6 ) E e and hence q ( x , r”e); if we set r”e = c then q ( x , c) and consequently we have 3(b, Q), r”Q = and G’(s(’),I’, b ) = Ob, a contradiction. 2) Suppose conversely that there exist 6, x and r such that
Then there exist b, x and r such that
G ’ ( d b ) :r , b )
=
1, ,
We shall show that this leads to a contradiction; as before there is no loss of generality in supposing that b has a total Boolean support e. Under this assumption we have l < ~ ( xr ,” ~ and ) setting r”e = c we sec that (*) does not hold. Hence (**) does not hold and so there exist no 5, F and @ such that 9(T, @),F”, = c and G ( x ( ~ )i, , 5) E @. On the other hand, we have 3(b. Q), r”Q = 0 and G’(x(,), r , 6 ) E Q, a contradiction. This completes the demonstration. By Mctatheorem 64L3 we have 5416. METATIKOKIM. If cp i s a closed pure SNF then the following is provable in TSS’: If p is an automorphism of a complete Boolean algebra then p ’ ( [ q ] ’ 6 ) = = [q]’6 . 6417. D EINITION ~ (TSS’). An element u of a complete Boolean algebra 6 is called rigid in b if p‘u = u for every automorphism p of b. Clearly the rigid elements of b form a wbalgebra of b which we denote by brig. We have shown that [q], E brigis provable for every closed pure SNF cp. We shall now investigate the algebra brigand the brig-sets. First we prove two useful lemmas.
6428. LEMMA (TSS’). Let r and s be b-sets and let u holds iff r, A u = s, A ii for every x. 318
E
h: u
5 lr
= s]b
642 I
BOOLEAN VALUES AND FORCING
CHAP. VI SEC. 41
Proof. We have [r
= S] =
A((rXA )s,
v (-r,
A
-sx))
=
A((rx v -sX)
X
A
( - r , v sx)).
X
The rest of the proof is left as an exercise.
6419. LEMMA (TSS’). Let b be a complete Boolean algebra and let q be a partition of b. If h’u is a b-set for each u E q then there exisfs a 6-set r such that u 5 [ r = h’u]b for every u E q. ( [ r = h’u], is an abbreviation for [(Vx) ((x E a& x E e) v (x q! o& x $ e)]’ ( r , h’u, b ) . ) Proof. Set rx = V ( u A (h’u),) for each preceding lemma.
US4
XE
IJW(k‘tr) and use the u=l
6420. METATHEOREM. If q is a pure SNF with one free semiset-tariable then the following is provable in TSS, (E2): For every b and every b-set r there exists a b-set i such that
Demonstration. We prove the above statements in the theory TSS‘, (E2). Let r be a b-set. If u = [ q ( r ) ] = Ob then we are finished, since Obebrig. Otherwise let R = V( p ’ u ;p is a n automorphism of b ) so that ii E brig. If {pe;ci < A> is an enumeration of all autornorphisms of b we define u0 = u and 17, = pLu - V up. By Lemma 6419 there exists a b-set P such that
5 [F
B
for every a!. Since 11, 5 [q(pAr)] we have ii 5 [ q ( i ) ] . In addition, we may suppose -ii 5 [ i = O(”J. Since cp(0) is a closed pure SNF (or is equivalent to such a formula), we have [cp(O(b))] E b,*, by Metatheorem 64l.6. Further we have [F = O(b’IJ A [cp(t)J = [i: = O(b’I] A A [q(Ocb))]; hence U,
= &r]
since both ii and [q(O(”)] are in brig.(Cf. 6406 and 5218.)
6421. METATHEOREM. If q(o) is a pure SNF then the following is provable in TSS’, (E2): 319
6422
BOOLEAN VALUES AND FORCING
[CHAP. VI SEC. 4
Let b be a complete Boolean algebra and let Z be a total Boolean support on b. If (3!rs) ~ ( 0then ) there exists a b-set r such that D(r) c brig and rs = r ” Z . Demonstration. We prove the above statement in the theory TSS’, (E2). Let r be a b-set such that cp(r”Z) and [cp(r)jb E brig. If we set [(p(r)jb = u then we can assume that D ( r ) c b u . For each automorphism p of b we have [(p(fi’r)],, = u E 2 so that cp((fi’r)” Z ) , i.e. (8”)” Z = r”Z. If we let rl = U { P ’ r ; p is a n automorphism of b)} we have r y 2 = r”Z if in addition we let Fx = V{u;(x, v) E r l } then F is a 6-set, t ” Z = r”Z = 0.
1
As a corollary we have the following 6422. THEOREM (TS, dTSS/aGt (L)). If z is a set support for V over I, and if z is a complete ultrafilter on a n algebra b in the sense of Gt (L), then 2 n (brig)’t(L) is a set support for HDf over L. Proof. We denote by 6,. 1) We have ( z n b,) E HDf; for, by 4327 we have z n b, = b, n n(q; 4 is a Boolean support for V over L on b} = b, n T (V, b,) = b, n T (pa, b,) for some a, where T (V, 6,) is the godelian term corresponding to n{q;q is a Boolean support for V over L OD 6). Hence z n b, is a Boolean support in the sense of St%Gt (L, HDf).
2) It remains to prove that z n b, is a total support in the sense of Gt%Gt (L, HDf). Let q ( q ) be the formula which says that “q is the first total support for HDf over L in the definable well-ordering of HDf” (cf. 3406 and 6320). There is a unique 4 such that ( ~ ( 4 and ) so by Metatheorem 6421 there exists ? E L such that D(t) c b,and i ” z = F”(z n b,) = 4 . Hence D,.‘WL) ( q , z n b,) and z n b, is a total support in the sense of Gt%Gt (L, HDf). Remark. Using this theorem we can easily establish the consistency of the equality or inequality of the classes L, HDf and V. We have TS, D3 II- L E HDf G V. In Chapter TI1 we showed that L = HDf = V is consistent and in 6332 we showed that L HDf = V is consistent. In TSS”’, E t we can prove that the only rigid elements of Cant: are the zero and the unit. Hence in the model Up as a model of TS, D3, E2 in TSS, D3, (Constr), (S Cant:) we have L = HDf (since HDf has a support over L which is a singleton). Thus 1, = HDf V is consistent. Finally, starting with the theory
+
+
TSS”’, L 320
+ HDI = V ,(S Cant:)
CHAP. VI SEC. 41
6424
BOOLEAN VALUES A N D FORCING
which is relatively consistent w.r.t. TSS, and using the model U p , we obtain the consistency of the theory TS, D3, E2, Mcl (M), (HDf has a singleton support over M), aTSS/aGt (M), L 9 HMG*(M' =M V. In this theory, H f l ' ( M )= HDf is provable, so that we have L 9 HDf V.
+
+
In the following we shall consider separatively ordered sets instead of complete Boolean algebras. Disjoint relations are replaced by antimonotone relations (cf. 4253). 6423. DEFINITION(TSS'). Let a be a separatively ordered set. We say that r is an a-set if it is a n antimonotone relation and if D ( r ) E a ; we say that an a-set is an a-constant if x, y E a implies Ext, (x) = Ext, ( y ) . If z is a set then z(") is the unique a-constant r such that W ( r ) = z, i.e. z@) = =zxa.
6424. METADEFINITION. We define in TSS': Let a be a separatively ordered set and let u E a. If z1 and z2 are sets such that z 1 E z 2 then we write
If z is a set and r is an a-set such that ( z , u ) E r then we write ( u , a, z,r ) E ForcXE,.
If cp and 11/ are pure SNF's for which Forc, and FoxJ, are already defined then we let
= 1 (3u 5 u ) ( ( u , a, r, 0 ) E Forc,) , (u,a, r, 0 ) E Forc,&* = . ( u , a, r, 0 ) E Forc, & ( u , a , r, 0) E Fort* , (u, a, r, 0) E Fort,,@ = . (u, a, r , 0) E Forc, v ( u , a , r, 0 ) E ForcJ,. ( u , a, r ,
0)
E Forc,,
(Here we should make a remark similar to that made in connection with the definition of cp-values of conjunctions and disjunctions in the Boolean case)
(u,a , r, 0) ( u , a, r,
E
0) E
Forqyz), Forqy,,),
= (Vz) = (Vs)
( ( u , a , z ( ~ ) r, ,
0) E
Forc,)
(s is an a-set -+ (u,a, s, r ,
. 0) E
Forc,) ,
= (32) ((u , a, z("), r, 0 ) E Forc,) , (u, a, r , 0) E Forc(30)(p = (3s) (s is an a-set & ( u , a, s, r, 0) E Forc,) . ( u , a, r,
0)
E Forq,,),
321
6425
r,
BOOLEAN VALUES AND FORCING
Krrnark. The formula ( u , a, r , 0 in a”.
0)
E
[CHAP. V I SEC. 4
Fox, is read “ u forces cp at
6425. METALEMNA. For each pure SNF cp the following is provable i n TSS’: If u forces
ri, and v
5 u then u forces cp. (Exercise.)
6426. METATIIEOKEM. For each pure SNF cp the following is provable in TSS‘: Let a be a separatively ordered set and let 2 be a complete ultrafilter on a which is a total support. If q, 0 are sets and I’, 0 are a-sets then q ( q , 0 . r”2, 0 ) holds iff some u E Z forces q at q@),0 , r, 0 . Dentortstr’ati~~n. We proceed by induction the assertion being clear for atomic formulas. We consider the case of negation. Suppose ~ c holds. p Then n o u E Z forces rp and by 3302 there exists u E Z such that no u 5 u forces rp; hence u forces l c p . Conversely, if there exists u E Z such that DO 1: 5 ii forces cp, then l r p holds; for if cp holds then some u E Z forces cp and there exists w E Z such that w 5 u and w j u and w forces both cp and ~ q a contradiction. , We now consider quantification. Consider a fixed u € 2 . We have u forces (Vx) cp iff 11 forces p at z(‘) for any set z but the latter holds iff (Vx) cp. The remaining cases are treated similarly.
6427. DEFINLTION. Let a be a separatively ordered set and let If r is an a-set then we define
U E U .
6428. METAIHEOREM. For each pure SNF cp the following is provable in TSS‘:
Let a be a 5eparatively ordered set and let u =< U E U . If r , 0 are a-sets then u forces cp at I’, 0 in a iff L; forces cp at r u, 0 in (Seg,, (u), 2 >. The demonstration is by induction and is left as an exercise.
,I
6429. METATHEOREM. For each pure SNF cp the following is provable in TSS’, E2: Let b be a complete Boolean algebra and set a = i. If r , 0 are b-sets and s. 0 are a-Fets such that (x, u ) E s = v g r x then the cp-value of r, 0 in b is the join of all u E a which force cp at s, 0 in a. 322
CHAP. VI SEC. I?
BOOLEAN VALUES A N D FORCING
6429
Demonstration. Working in the theory TSS', E2, we let u=
and c =
Iqn*( r , 0 , b )
V{w; ( w ,a, s, 0 ) E Forc,) .
The assumption u < u leads to a contradiction. Indeed, by considering the algebra b ] (2: - u) we may assume u = Ob and z' = 1,; by Metatheorem 5418 we may assume that there is a total Boolean support on b; hence by Metatheorem 6406 we have cp while on the other hand by Metatheorem 6426 we have 19.Similarly the assumption u > c leads to a contradiction. Thus u = v. as was to be shown.
323
Bibliography
Abbreviations: BAPS = Bulletin de 1'AcadCmie Polonaise des Sciences, CMUC = Commentationes Mathematicae Universitatis Carolinae (Prague), NHPC = North Holland Publishing Company, Proc. Symp. = Proceeding of Symposia in Pure Mathematics, Volume XI11 - Axiomatic Set Theory (Part 1 - 1971; Part 2 - to appear).
=
J. L. BELL& A. B. SLOMSON, Models and ultraproducts. NHPC 1969. P. BERNAYS & A. A. FRAENKEL, Axiomatic set theory. NHPC 1958. L. BUKOVSKP, The continuum problem and powers of alephs. CMUC 6 (1965) 181- 197. P. J. COHEN,Set theory and the continuum hypothesis. W. A Benjamin 1966 W. B. EASTON,Powers of regular cardinals. Annals of mathematical logic 1 (1970) 139-178. The K. G ~ D E L , consistency of the axiom of choice and of the generalized continuum hypothesis. Princeton University Press 1940. P. ZIAJEIE, Logische Kategorien. Archiv fiir Math. Logik und Grundlagenforschung 1 3 (1970) 168-- 193. A. HAJNAL, Consistency theorem connected with the generalized continuum problem. Acta Math. Sci. Hung. XI1 (1961) 321-376. T. JECH& A. SOCIIOR, On @-model of the set theory & Applications of the @-model. BAPS 14 (1966) 297- 303 & 351-355. H. J. KEISLER, Limit ultrapowers. Trans. Amer. Math. Soc. 107 (1963) 382-408. A. L ~ V& Y R. VAUGHT,Principles of partial reflection in the set theories of Zermelo and Ackermann. Pacif. Journ. Math. 11 (1961) 1045-1062. A. R. D. MAITHIAS,Surrealistic landscape with figures. In: Proc. Symp. Part. 2. K. MCALOON,Consistency results about ordinal definability, Annals of mathematical logic 2 (1970) 449- 467. A. MOSTOWSKI, An undecidable arithmetical statement. Fundamenta Math. 36 (1949) 143- 164. J. MYHILL & D. SCOTT, Ordinal dcfinability. In: Proc. Symp. Part. 1, 271-278. J. B. ROSSER, Simplified independence proofs - Boolean valued models of set theory. Academic Press 1969. D. SCOTT & R. SOLOVAY, Boolean valued models of set theory. In: Proc. Symp. Part. 2.
324
J. R. SHOENFIELD, Mathematical logic. Addison-Wesley 1967. R. SIKORSKI, Boolean algebras. Springer Verlag 1964 (Second edition.)
E. SPECKER, Zur Axiomatik der Mengenlehre. Zeitschrift fiir Math. Logik und Grundlagen der Mathematik 3 (1957) 173-220. C A. MOSTOWSKI & R. M. ROBINSON,Undecidable theories. NHPC 1953. A. TARSKI P. VOPENKA,General theory of V-models. CMUC 8 (1967) 145-170. P. VOPENKA& B. BALCAR,On complete models of the set theory. BAPS 15 (1967) 839- 841.
P. VOPENKA& P. H . ~ E KPermutation , submodels of the model 611-614.
r.BAPS
19 (1965)
325
index
A
absolute (a formula ISa,) 1267 absolute cardinal 5133 absolutely cofinal 5133 absolutely equivalent 2201 absolutely subvalent 2201 additive (a cut is nta.) 2514 almost regular relation 5105 almost universal class 1417 almost universal relation 1415 antimonotone relation 4252 approximable 5301 atom 2420 atomic 2420 atomless 2420 automorphism (of a class) 3307 axiom FI, F 2 1102; A1 1103; Bl--B7 1107; C1, C2 1410; A2- A7 1422; C3 1446; D1 1450; El 2302; Cont 2325; D2, E2 3101; Ul--U3 3143; E3 3420; Constr, SConstr 3510; CConstr. 3528; S1 4109; S2 4124; Pot 4127; S3 4131; %d 4133; S4 4221; St 4224; S5 4236; S6 4240; Sb 5215 axiomatic theory 1211
13 base (of a complete B. algebra) 2433 Boolean embryo 4201 (complete Boolean algebra see complete; Boolean support see support) 326
bound (a variable) 1205 boundable (-boundable formula) 3338
C calibre 2511 canonical ordering 2409 canonical congruence 4203 canonical quasiordering 4205 Cantor algebras 6101 cardinal 2206 cardinality (of a set) 2305 class 1101 closed class I441 closed formula 1207 closure under godelian operations 3401 cofinal 2232,4226 collapsing algebras 6120 commutator 3304 compatible (derivation) 1255 complement (of a class) 1108, 1110; (in a complete B. algebra) 2403 complete Boolean algebra 2402 complete class 1323 complete filter 2425 complete subrelation 3217 complete ultrafilter on a B. embryo 4215 composition (of models) 1226 comprehensive 1419 connective 1202 conservative extension (of a theory) 1242 consistent (theory) 1220
constant 1202 constructible sets 3504,3522 constructing function 3504 contradictory (theory) 1220 converse 1108 countable 2216 cut 2514 D decoded relation 1308 definable sets 3403,3407 definition (of a predicate, operation, sort) 1244; (of an F-like language) 1262 dense 4245 dependent 1460, 4103 derivation (of a theory w.r.t. a language) 1254 describe (a complete B. algebra) 5213 (see also well describe) determine a model 1256,1263,1264 determining (T-determining relation) 3533 direct translation 1238,1254 direct model 1240, 1264 disjointed relation 4101 distributive (an algebra is x - y -- z-distributive) 5322 domain 1108 draft 2513 E
economical functor 5162 empty class 1110 equivalence (a relation) 2203 equivalent (theories) 1217; (models) 1229; real classes 2201 (see absolutely equivalent) estimate of an independent relation 3535 exact functor 1408; &exact functor 5339 exclusive system 2501 extension (of a theory) 1113, 1217; (in a relation) 1128 (see also induce) extensional relation 1130; (strongly e. r.) 1132
F faithful 1232 field 1110
filter (on a complete B. algebra) 2415 (g-filter see group) finite 2216 k i n g (of constants) 1244 formula 1101, 1205; (PUP-formula) 1437; (T-formula) 1113 (see also restricted, normal, seminormal) free (a sequence of variables) 1115; (a variable) I205 fruitful 3351 full (upward extension) 5111; (modelclass) 5119 function 1127;(b-function) 5302 fundamental 1250 (cf. 1102)
G generate 2432 godelian operations 1109 godelian term 1121 good definition 3336 group 3304 group filter 3306,3332
H have a permutation model 3344 hold in a model 1224 homomorphism (of a complete B. algebra) 2427
I ideal 2514; (i. of subalgebras) 6201; (i. of nuclear subalgebras) 6202 identical model 1225 image I127 independent relation 3532 individuals (set of) 3339 induce(a theory) 1248; (a model) 1248 infimum 2407 intersection 1110 invariant (ofa set) 3315 inverse (of a model) 1231 isolated ordinal number 2126 isomorphic theories 1231 isomorphism of relations 1310; of complete B. algebras 2427 iterable 2144; Sm-iterable 4225 327
J
join
2409
L language 1203 like (E-like relation) 1132; (F-like language) 1250; (V-like class) 1314 limit ordinal number 2126 linear ordering 2103 local 2142; (Sm-local) 4225 logical axioms 1209
M mapping 1127;(b-mapping) 5302 maximo-lexicographical ordering 2241 meet 2403 membership 1101 model (syntactic) 1150, 1219; (m. constructed in another model) 1226 model-class 1442; (m.-c. with absolute cardinals) 3226; (m.-c. with E l ) 3230; (with S1, with D2) 5116 (see also full) morphism 1310 multiplicative (K,-niultiplicalive) 2501
N natural numbers 2128 norm of a cut 2514 normal formula 1114,1123 normal F-definition 1305 nowhere constant relation f406 0
one-to-one mapping 1127 operation 1202 ordered 2104,2405; (ordered pair) 1I04 ordering 2103 (see also reflexive ordering) ordinal 2105 ordinal number 2107
P pair I103 (see also ordered pair) pairing relation I132 partial algebra 2413 328
partition 2422 partition filter 2424 partitionable 2501 permutation 3301 power-class 1127 predicate 1202 principal algebra (in an ideal) 6209 principles (provability, consistency, unprovability) 1222 product 1110 proof 1212 proper axiom 1251 provable formula 1212
Q quantifier
1202
H range 1110 rank 3106 real class 1402 real F-definition 1455 reflecting (system) 3120; (G-reflecting class) 3123 reflexive ordering 2405 regular (cardinal) 2238; (kernel) 3131; (relation) 1405 relation 1106; (b-relation) 5302 relativizing quantifiers 3337 representable I115 representation of membership 1108, 1110 restricted formula 1324 restriction 1108 rigid 6117 rules of inference 1210 S
saturated (part) 1128; (p-saturated element of an ordered set) 2506; (saturated-universal relation) 5107 saturation 1128 segment of an ordered set 2501 selector 2301 semiset 1401 separative ordering 2439 set 1102; (b-set) 4311
set-formula 1325 set images property 2144 set-universal relation 3220 similar languages 1203 similar semisets 4114; (strongly similar semisets) 4317; (b-strongly similar) 4318 singleton property 2518 singular cardinal 2238 sort of a variable 1202 specification I264 strictly (equivalent, subvalent) 2201 (see also absolutely) stronger specification 1265 stronger theory 1217 subalgebra 2427 subclass I l l 0 subgroup 3304 sublanguage 1203 subordinate sort 1214 subvalent 2201 sum of a class 1127 support 1462; (Boolean s.) 4220; (locally semiset s.) 4123;(locallyserniset Boolean s.) 4235; (semiset s.) 4130; (semiset Boolean s.) 4239; (total s.) 4108; (total Boolean s.) 4220 supremum 2121,2410 symmetricg-filter 3322 symmetric sets 3319
T term 1113, 1204 theory: €-theory 1250; T E 1102 TC 1109; TSS 1423; TS 1446; TSSe
5112; TSe 5122 (see also axiomatic, fundamental and the index of symbols) total support see support totally universal relation 4118 tuple I112
U ultrafilter 2415 ultraproduct (u. class, u. relation) 5201 union of a class 1127 union of two classes 1110 unit element 2403 universal class 1108 universal selector 3116 universal sort 1214 universum 2132, 2133 urelement 3141 V
valid (formula in a model) variables IIOI,1202
1224
W
weaker (model) 1229 well describe (a filter) 3342 well founded relation 3212 well ordered 2137 well ordering 2136; (regular w. 0.) 2137 (Sm-w. 0.) 4230 well symmetric filter 3326 L,
zcro element 2403
3 29
Index of Symbols
3107 3116 Ap 6110 Appr 5304 AReg 5105 at 2520 AUn 1415 AUncl 1417 Aut 3307 Aut 3313 Bd 4133 Heinb 4201 Beth 5341 C 1110 Cant 2536, 6101 Card 2206 c a r d 5133 Cba 2402 cf 2234 cf 5143 Cg 3401 Cg, 3501 Clos 1441 CIS 1110 Cn 2206 Cnv 1108 Cnv, I108 Coll 6120 A
Ac
330
Coinp 1323 Conf 2232 eonf 5133 Cstr 3504 Cstr, 3522 1) 1108 Dec 1308 6 , 5322 Dep 1460,6309 Dep, 4103 net,. 3532,6331 Df 3403 air 1238,1254,1263 Dr 4101 E 1108,1110 e 3301 Elk 1132 Econ 5102 Eq 2203 Ex 2501 Exct 1408 exp 2309 Ext 1128 Extl 1130 F,, ..., F, 1109 FF 1326 FMcl 5119 Forc 6424
Fup 5112 Fupw 5109 Fupw 5111 g 3301 Gen 2432 HDf 3407 HRefl 3123 HS 3319 3 b 1230 Imb 3223,5109,5111 Tnd 3339 inf 2407,4207 Inv 3315 k, 5203 Ker 3131,3134,3203 Ker’ 3 140 Ki 6202 L 3508 LOrdg 2103 M 1102 Max 2121 Maxlex 2241 Mcl 1442, Mcl,. 3226, Mcl,, 3230, Mcl,, 5116, MCl,, 5116, 2Mcl 6305 ,LL 2501, p 2511 p a t 2506 Min 2113 % 1305 Ncon 1406 v 2501, v 2513 N F 1114 Norm 2514 NSF 1326 0 2128, coo 2128, 0, 2213 On 2107, On, 2126, On,, 2126 Ord 2105 Ordg 2103, Ordg, 2405 P 1127, Pfi, 2225, PKa 2315
B 3131, Px 3134. Yo, 3401 p a 3134, p: 3134, p: 3140 Part 2422 PCant 6115 PInv 3315 Prg 1132 Pt 2425 rx, rXy 5301 Real 1402 Seal 1455 Refl 3123 Reg 1405 Re1 1106, Rel, 1106, Relb 5302 RF 1326 Rg 3113 RSF 1326 %Gt 6303 RUpw 3223 RWOrdg 2137 S 3319 Sat 1128 SAT 1128 Satunvr 5107 Sd 2226 Seg 2501 sEctl 1132 SetUnvr 3220 SF 1325 Sel 2301 Sim 4114 Sm 1401 SNF 4120, (pure SNF) 6402 SpStR 6309, GpEiYZ 6309 sSim 4317, sSim,, sSim, 4318 Gt 1316 Gt%Gt 6307 s u p 2121 sup 2410,4210 Supp (or Supp,) 1462, (cf. text 331
after 4107), Gupp 1463, Supp, 4123, Supp, 4130, SUPP, 4220, SUPP, 4235, SUPP, 4239, SUPP, 4107 Support 63 16 Sym 3322 TC 1109 T E 1102 ThrU,, Thru, 43 14, ThKU, 5322 Tor 5111 Totunvr 4118 T S 1446, TS’ 1450, TSL, 1450, TS‘ 5122, TS” 3105 T S S 1423, TSS-, 1424, TSS’ 1450, TSS‘, 1450, T S S b i s 2135, TSS” 3105, TSS‘ 5112, TSS”’ before 6301 TSupp 4108 Uic 5201 Ulr 5201 Un 1127, Un, 1127, Unb 5302 Unv 2132,2133 Up 5122 Ur 3141 Urel 3141 V 1108 Vlk 1314 w 1110 W F 3212 WOrdg 2136 WSym 3326 f 1101 &, V , + , 3 , l , V , 3 , = 1101,1202 3 ! , ( } 1103 < > 1104,1112 -, 1108 E, c , + , O , - - , n, u, x 1110 (x: cp(x>} 1125
p
332
U, X ” x X ’ Y 1127 I- 1212
1219 914 * !Ill2 1226 9Xl < 9X7 1229 aTpL 1254 AL 1237 dT/aA 1263 z, <, &, K, Ka 2213 X 2305 7 2310 X 2319 [ X , Y] 2401 I,, OB, A ~ , - 2405 I B J x 2419 4, A 2422 b, 2515 yl”
<,
4
2201
AB 2403
n1 XES
0 2521
2527 2530 b 4530 s s 3217 p . g, p - ’ 3302 [PI 3304 3313 (LIT 3337 1 3349 -, 4203 $ 4205 CT jb u 4303, 4304 ? 5139 5 5141 1 5143 (+ upw) 5216 r [ ~ ]5337 [ ] 6403 r l a i c 6427 llt
b
5
--
G
431 1
