Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
223 Ulrich Feigner Universit~t Heidelberg, Heidelberg/Deutschland
Models of ZF-Set Theory
Springer-Verlag Berlin. Heidelberg. New York 1971
A M S Subject Classifications (1970): 02 K 05, 02 K 15, 02 K 20, 04-02, 04 A 25
I S B N 3-540-05591-6 S p r i n g e r V e r l a g B e r l i n • H e i d e l b e r g • N e w Y o r k I S B N 0-387-05591-6 S p r i n g e r V e r l a g N e w Y o r k • H e i d e l b e r g • B e r l i n
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PREFACE
This given
set of notes
is part and parcel
in 1970 from February
tics at the State University as an aid to the people
at Utrecht
attending
as the course developed.
those
In fact~
complete
anddetailed
some few places
there
of lectures
and they were lectures.
of Mathema-
intended
proofs
scribblings.
are given
are only short
looking
In most
in these notes but in
indications
This occurred when a result was only slighty
only
They were written
in spite of their official
aspect, they are no more than prelecture oases
of a series
up to June at the Department
to the proof.
touched
in order to
round up the presentation. It was the aim of these
lectures
some of the basic techniques
and results
Set Theory.
This theory
to compress
it into one series
theme the construction the construction ungrounded
is given, Chapter
some basic
of lectures.
concepts
the generalized
and L@vy's
continuum
Mostowski-Specker
and of P.Cohen
Many of these
explanations attention
are included.
and obscurities
of
models
principles
As an
of reflection.
proof
for the axiom
and the axiom of
the methods
in a general
Although
models.
hypothesis
several
sets,
(containing
of Zermelo-Fraenkel
consistency
applications
as our
L of constructible
iII and IV contain
As an aid to the reader
mistakes
model
relative
Chapters
print.
of
of models
We have chosen
and Cohen-generic
constructibility. applications.
of the theory
I), the axiomatization
GSdel's
an exposition
a wide field and it is not possible
of Ggdel's
like x = {x})
(chapter
II c6ntains
of choice,
covers
of Fraenkel-Mostowski-Specker
sets
introduction
to provide
setting
of Fraenkel-
and various
have not yet appeared informal
discussions
we have attempted
in
and
to reduce
to a minimum we should be glad to have our
drawn to any indiscretion
the reader may discover
in the
text. To finish D.van kind
Dalen
to HelSne
Heidelberg,
I wish to express
and the D e p a r t m e n t
invitation
to Rode V r i j e r and
this preface
of ~[athematics
to spend a y e a r at this and K o R a s m u s s e n
Keller
July
f o r typing
5, 1971
my gratitude
to
at U t r e c h t
for the
Institute. Thanks
are due
for correcting the manuscript.
several
misprints
CONTENTS
CHAPTER
!. P r e r e q u i s i t e s
A) R e c u r s i v e
Functions ........................................
2
B) F o r m a l T h e o r i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
C) A r i t h m e t i z a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
D) S y n t a c t i c a l
Models .........................................
E) Z e r m e i o - F r a e n k e l F) The P r i n c i p l e CHAPTER
6
Set T h e o r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
of R e f l e c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
ii. C o n s t r u c t i b l e
Sets
A) The A x i o m of C h o i e e B) The C o n s t r u c t i o n
and the C o n t i n u u m
Hypothesis ...........
23
of the m o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
C)
Z~ F - F o r m u l a e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
D)
Z~ F - d e f i n a b i l i t y
of s y n t a c t i c a l
notions ..................
32
E)
&>F _ d e f i n a b i l i t y
of s e m a n t i c
notions .....................
33
F)
z>F _ d e f i n a b i l i t y
of the
G) P r o p e r t i e s
of t h e c l a s s
constructible
model .............. sets . . . . . . . . . . . .
35
and a L e m m a of G . K r e i s e l . . . . . . . . . . . . . . .
39
I) T h e o r e m of J . R . S h o e n f i e l d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
J) R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
H) T h e o r e m of H . P u t n a m
Additions CHAPTER
III.
A) The
to c h a p t e r s
L of c o n s t r u c t i b l e
34
I and
II . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fraenkel-Mg§towski-Specker
Independence
Models
of the A x i o m of F o u n d a t i o n . . . . . . . . . . . . . . . .
B) The F r a e n k e l - M o s t o w s k i - S p e c k e r
Method ......................
52 57
Independence
of the A x i o m
D) The
Independence
of the g e n e r a l i z e d
Continuum-Hypothesis
f r o m the A l e p h - H y p o t h e s i s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Anti-Chain
of the A x i o m
of C h o i c e
of C a r d i n a l i t y
in ZF ° . . . . . . . . . . . . . . . . . . .
G) A F i n a l W o r d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additions CHAPTER
IV.
to e h a p t e r
III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cohen Extensions
59
(AC) f r o m K u r e p a ' s
Principle .......................................
F) T h e U n d e f i n a b i l i t y
47
of C h o i c e . . . . . . . . . . . . . . . . . . . .
C) The
E) The I n d e p e n d e n c e
45
61 67 73 75
of Z F - M o d e l s
Introduction ..................................................
76
A) The F o r c i n g
78
(Ramified
Relation
languages,
in a g e n e r a l
setting ..................
s t r o n g and w e a k
forcing,
complete
sequences) B) C o h e n - g e n e r i c
sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
VI
(Definitions
of val~ and the extension
the forcing technique:
~
~ZF~
~£,
Hauptsatz of
Digression).
C) The Axiom of Choice in generic extensions .................... ( ~[a]
~ ZF + (AC), Symmetry properties,
96
the Independence
of the countable axiom of choice (AC ~) from ZF). D) The Power of the Continuum in generic extensions ............. (Independence of the Continuum-hypothesis
103
from (AC), indepen-
dence of V = L from (AC) + (GCH)). E) The Independence of the Boolean Prime Ideal theorem (BPI) from the Ordering Principl~ .................................. (Equivalents
of the (BPI), the Order Extension Principle,
Historical retrospect: universal
relational
Mostowski~s model, homogeneous systems~ the independcnee
Orderextension principle
of the
from the Ordering principle).
F) The Kinna-Wagner Choice Principle ............................ (Formulation of that principle (KW-AC) + Ordering principle~ (KW-AC)~
110
independence
G) The Independence
(KW-AC),
122
Proof for ZF
independence of (AC) from
of (KW-AC) from the Ordering principle).
of the Axiom of Choice (AC) from the
Boolean Prime Ideal theorem (BPI) ............................
128
(Outline of a proof that (BPI) holds in Mostowski's model, Construction of the generic model ~ [ a 0 , a l , . . . , A ]
and
proof that (BPI) + 7 (AC) hold in that model, Application: definitions
of continuity).
H) The Axiom of Dependent Choices ...............................
146
(Bernays' and L@vy's formulation of that axiom, proof for ( ~ a ) ( A C ~) ÷ (DC ~) and for ( ~ ) ( D C results:
~) ÷ (AC)~ Independence
Mostowski's model of ZF ° + (DC ~) + ~ (AC~I)~
Jensen's model of ZF + (AC ~) + 7 (DC~)~ Feferman's model ~[ae,a1,...] results).
for ZF + (DC m) + ~ (BP!), List of further
I) A F I N A L W O R D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
167
REFERENCES
168
.....................................................
CHAPTER
I
prerequisites In this short chapter we list those topics, mostly taken from the lower predicate
calculus and Peano Arithmetic,
our readers to be already familiar with.
that we expect
For more details the reader
is referred to [15] S. FEFERMAN:
Arithmetization
setting~ [60] E. MENDELSON:
Fund. Hath.
Introduction
Company, [TS] J.R. SHOENFIELD:
of metamathematics
49 (1960) p. 35-92.
to Mathematical
Princeton-New York-London Mathematical
in a general
Logic~ v. Nostrand 1966
(3rd printing).
Logic~ Addison-~lesley Publ. Comp.
1967. All our considerations
about formal languages,
their syntax
and semantics are carried out in a certain underlying
Metatheory.
Our metalanguage will be english enriched by some mathematical and logical
symbols such as: ~, &,~,~,~,V,~,~
(negation, universal
conjunction,
disjunction,
quantification,
implication,
existential
equivalence,
quantification and equality}~
It is understood that the use of these symbols is known, a semantics (id est: denotes
set theoretical "membership".
set theory as formulated
in the above
The universe of sets as given by our naive set theory
will be called sometimes and relations
is not given. The symbol 6
We adopt naive set theory as our metatheory
(id est: Zermelo-Fraenkel meta-language).
interpretation)
"the real world" and objects~ operations
in it shall get sometimes the adjective
:'actual" in
order to distinguish them from the corresponding objects, operations and relations of some "object-theory".
Hence E is the actual member-
ship relation and ~ ~ m0 is the actual set of (actual) natural numbers.
A) RECURSIVE FUNCTIONS We adopt the usual definitions cursive
(id est: general recursive)
Mendelson [60] p. 120-121).
of primitive recursive and refunctions of ~ into ~ (see e.g.
We recall briefly the definitions:
A function f is said to be primitive recursive
iff it c a n be
obtained from the functions Z(x) = O, N(x) = x+l, U~(xl,...,x n) = xi, the initial functions, by any finite number of substitutions and recursion.
If f can be obtained
application of the ~-Operator,
in this way but with some finite then f is called recursive.
A relation R(xl,...,x n) between natural numbers primitive recursive
(recursive)
is said to be
iff its characteristic
function
0 if ~ R(x,,...,x n) XR(Xl,...,x n) : 1 if is primitive recursive
(recursive,
R(xl,...,x n) respectively).
A subset A of ~ is called a recursively
enumerable
(r.e.) set
iff B is either empty or the range of a recursive function. set B of ~ is recursive
iff both B and its complement
It is well known that A is r.e. form ( ~ y)(R(x,y))
(recursive)
R iff f can be obtained from the initial functions stitutions
in the relation
together with
function XR of R by any finite number of sub-
and recursion
respectively).
in the
R (one can allow R here to
be prilm~tive recursive )° A function f is primitive recursive the characteristic
~-B are r.e.
iff "x 6 A" is expressible
for some recursive
A sub-
(and application of the u-Operator,
B) F O R ~ L
THEORIES
A formal theory T is a triple (L,!,~) where ~ is a formal language, C is a set of consequence operations and V a set of sentences of ~ closed under i, called the valid sentences of ~. Here we shall discuss only those theories ~ where ! is the firstorder predicate calculus and ~ is an elementary language.
Such
languages L can be defined as abstract algebras whose elements are just the well formed formulas,
|70] H. RASIOWA-R.SIKORSKI:
see eog.
The Mathematics
Monografie Matematyczne,
of Metamathematics;
vol. 41, Warszawa 1963.
Let K an alphabet consisting of one sort of variables v0, vl,...,Vn,...(n E ~),countably many constants c n, countably many primitive predicates ~n and logical symbols ~ , ^ , A ( n e g a t i o n conjunction,
universal quantification)
v , ~ ~, ~/(disjunction, quantification)
implication,
are definable
the other logical symbols:
equivalence,
in terms of -],^~).
"countable': means "finite or countably infinite". of K the set At K of atomic formulae,
existential Remark that From these signs
the set T K of terms of K and
the set ~K of all (well formed) formulae are obtained as usual. Certain formulae of ~K are called logical axioms
(in K) and in
order to be definite we take those defined in Mendelson |60], p.57. The set C of rules of inference consists of the "modus ponens" and the "Generalization"
(cf. [60] p. 57).
For any ordered pair (~,A) as just described~
, where ~ is a first-order language
and A is a set of sentences
from ~, one can
define the ~Foof relation PRF(~,A): PRF(~,~)(¢,S)
holds iff S is a finite sequence of formulae
~0'''''~n from ~ whose last element ~n is ~ and for i < n, ~i is either an element of A or a logical axiom or there are j,k < i such that ~k ~ ~]• ~ ~ i (application of modus ponens) or there is a k < i such that ~i ~ A ~k (application of generalization). U
We shall write A ~ ¢ for (~ S)(PRF(~,A)(~,S))
and say that ¢ is
syntactically derivable from A by means of logical axioms and the rules of inference modus ponens and generalization.
Define PR(A) : {~;~ ~¢},then V e PR(A) is the set of valid sentences of the elementary theory (~,A). C) ARITHMETIZATION The method of aritl~netization of the syntax of an elementary theory
(id est: first order theory)
times called "gSdelization".
is due to K. G~del, hence some-
The method consists
every formula ¢ of an elementary language ~K natural number y(~)
in adjoining to
(with alphabet K) a
, sometimes written as U¢q,
such that every
natural number which is in the range of y is uniquely "readable" as a formula of ~K' i.e. has a unique grammatical
structure.
The
definition of y is by induction on the number of symbols in ¢ 6 ~ K (id est: the complexity of ~) and is carried out in detail in Feferman [I~]. It follows that the sets of g6delnumbers
of formulae
# from At K and from ~K are primitive recursive and similarly the set {y(t); t E T K} of g6delnumbers
of terms is primitive recursive.
The proof relation PRF as given above can be put into the form of a number-theoretic
function prf and it can be shown that prf is
primitive recursive Definition:
in ~, see Feferman [15] p. 44.
A first order theory ~ ~ (~,~) there is a r.e.
is axiomatizable
iff
subset A of V such that V e PR(A).
A is called a set of axioms for T. If there is such a finite subset A, then ~ is called finitely axiomatizable. W. Craig has proved that if a first order theory T has a r.e. set of axioms it has a primitive recursive "On axiomatizability
set of axioms
within a system", J.S.L.
(see his paper:
18 (1953) p.30-32).
Let ~, e <~I,[I ) an ~2 ~ <~2,[2 ) be first order theories such that the alphabet K I of ~I is a subset of the alphabet K~ of T~. Hence ~, ~ ~z" If [l ~ ~z, then ~2 is called an extension of ~1 (in symbols: ~i < ~2 ) and ~l is called a subtheory of ~2" Definition:
Suppose that ~I is a subtheory of ~2" If there is a finite set W z of sentences
from ~2 such that
PR([, U W2) e [2, then ~2 is called a finite extension
of ~i" If furthermore K, ~ K 2 (equality of the alphabets of ~I and of ~2), then ~2 is called a finite extension of ~
without new constants.
As usually a theory is consistent
iff not bo±h ¢ and 7 # ,
¢, are derivable from T. Using the numbertheoretic this can be expreNsed by a numbertheoretic wish to define Con for infinite, but r.e.
for some
function prf
function Con too. We sets A of formulae
(in
this moment we identify formulae ¢ with their corresponding g6de!numbers y(~)).
For this reason it is best to m~ke ~re=ise %he p r o o { -
relation prf too. Hence,
let e be the (primitive recursive)
defined on m such that A -~ {~(n); n e ~}. Now let P r f number-theoretic
formula
function
be the
(with free variables x,y) which expresses
the assertion that y is a proof of x on the basis of formulas represented by e
(Prf e is just the predicate corresponding to our
previously defined relation PRF(L,A>).
For a rigQrous formulation
of Prf e see Feferman [~5]~or p. 104 in: [63] R. MONTAGUE:
Fraenke!'s Addition to the axioms of Zermelo; Fraenkel-Festschrift of Mathematics",
"Essays on the Foundations
J e r u s a l e m 1988, 2nd Edition,
p. 9 1 - 1 1 4 . Now Cou~ can be defined to be the following formula: nVV xy Come is a sentence
(Prf (x ^ nx,y))
(of Peano-Arithmetic)
expressing that the set of
formulae represented by ~ is consistent. The following theorem is a generalization of GSdel's second underivability theorem 1957).
(1931) which is due to S. Feferman
Let P be the axiomatic
system of Peano-Arithmetic
(Thesis, (see
[1~]
p. 50). A formula ~ of P defines the set A of natural numbers in P iff ~ has one free variable and for every n e ~: p ~ ¢(n) if n E A, and:
P ~ 7¢(n)
Theorem
if n ~ A. Here n is the n th numeral of P.
(GSdel-Feferman):
Let A be a consistent
set of elementary
formulas, and let e be a primitive recursive
function which
defines A in P. Then P + Con a is not relatively in A
(see Feferman [15] p. 90).
interpretable
D) SYNTACTICAL MODELS A semantical model
(or mostly simply:a model) of a 1St-order
theory T is given by a (meta-mathematical)
set ~ in which some
relations are defined such that the set V of "valid sentences of T" are all "true" in J. In contrast to this a syntactical model of T is given by a transformation
into another ist-order theory T*.
Here is required that the transformed
"valid sentences of T"
are all "valid sentences of T*". In the case that T and T" are axiomatized by A and ~* respectively this means that the transformed axioms A of T have to be derivable from A*. The notion of a syntactical model is very old and due to many authors. A, Tarski has called them "Interpretations", is by the procedure mentioned above "interpreted" [8~] A. TARSKI-A.
MOSTOWSKI-R.M.
ROBINSON:
since
in T*, see:
Undecidable
theories;
(studies in Logic), North Holland Publ. Comp. Amsterdam 1953.
[87] HA0 ~ N G :
Arithmetic
translations
actions of the Amer.
Math.
of axiom systems~ Soc. vol.
Trans-
71 (1951) p. 283-
293.
The classical notion of "Interpretation" one has been generalized
of one theory in another • n s " by to "parametrical interpretatlo
Petr H~jek: [29] P. HAJEK:
Syntactic models of Axiomatic Theories~
Polon. [30] P. HAJEK:
Bull. Acad.
Sci., vol 13 (1965) p~ 273-278.
Generalized
Interpretability
Note to a paper of R. Montague;
in terms of models Casopis pro pSstov~nf
matematiki vol. 91 (1966) p. 352--357. H~jek has defined his generalized notion of Interpretation only for finitely axiomatized theories and used the name "syntactic models" for it. W e shall follow H~jek but give the definition for arbitrary r.e. axiomatized theories
(the definition can obviously
be stated even more generally by writing E [2 instead of ~2 ~ etc.). In order to simplify the notation we assume henceforth that the alphabe t K of a first order theory T consists of only one sort of variables
v0, vl,...,
signs ] , ^ , A letter).
a sequence of primitive predicates wi and
(hence K does not contain any constant or any function
This assumption
is no loss of generality.
In the following definition let ~l and ~2 be axiomatized I st order theories such that {Vn; n E ~} is the set of variables of T, and {Wn; n 6 ~} the set of variables of ~2 and let I be an index set such that {~i; i e I} is the set of primitive predicates of ~,. n i will be the number indicating that ~i is ni-ary. Definition: Let ~i
m
< ~ 1 ~ I > and ~2 ~ < ~ , ~ 2 > be I st order theories.
The set of formulas of ~2:
6(wl,...,Wk), ~(wl,''',W k, Wk+l), ~i(wl,...,Wk,...,Wk+n.), i E I, l is called a translation from ~i into ~2 iff the following three conditions are satisfied: (1)
If wj is a bound variable in ~i' then j is odd;
(2) A2 ~Vw/--Vw k e(w~,---,wk); (3)
A2 ~ n w , ' ' ' A W k [ O ( w 1 " ' ' ' W k )
~VWk+l
$(w,'''''Wk+l)]"
Since Ocharacterizes the parameters we shall call 0 the "parametercondition"~ ~ defines (in dependence to the parameters) the objects of the model, hence we shall call ~ the "model-condition". The open formulae ~i will be the images (in ~z) of the primitive predicates ~i
(i 6 I) of T,. Condition (I) is required only in order to avoid
confusion of variables in the next definition (othe~lise we would have to care about a suitable changement of the bound variables in 9.. l It seems to be convenient to introduce a replacement operator rep. Therefore let T ~ (0,¢,{~i ~ i e I}) be a translation of ~, into ~z and let h be the following mapping from the set of free variables v0, v], v2,.., of ~i into the set of free variables we, w,, w2,... of ~z given by: h : vj ~ w2(j+k) (k is the numbe~ of free variables in 0 and is therefore uniquely determined by 7). The definition of repT(F) for F e ~l is by induction on the length of F. Remark that if vj is free in ~i, then h(vj) is a free variable of repT(~ i) (this is guaranteed by the
curious condition
(1) from the definition above).
Definition (of the rep-0perator with respect to a translation T): (i) If F is an atomic formula of the form vj, = vJ2' then repT(F) ~ (W2(k+jl) : W2(k+j2)); (ii) If F is an atomic formula of the form ~i(vj l
,...,v. ), then ]n i
repT(F)
. ); ~i(wl .... ,Wk,W2(k+jl ) .... ,W2~k+]ni)
(iii) If F is of the form F, ^ F2, or 7F,, then repT(F) ~ rep~(F I) ^ repT(F2), repT(F) e -h?epT(F ,) respectively; (iv) If F is of the form A
vj F',
then
repT(F) -- A w 2 ( k + j )[$(Wl,...,Wk,W2(k+j))
~ rePT(F*)].
The action of repT can be described briefly as follows:
in
the formula F of ~i every ni-ary primitive predicate ~i is replaced by ~. and all quantifiers are restricted to $. Remark that the l image of a sentence F under repT is again a sentence if and only if 0 is a sentence, id est: if the translation T is without parameters. Definition. A translation T ~ (0'$'{~i; i e I}) from T ~ (~,,Al) into ~2 e (~2'~2)is called a syntactic model of T, in ~2 iff for every ~ 6 ~I the following holds: ~2
~ Aw,""
AWk[~(w1~''''Wk)
~ repT(¢)]
T is called parametric if k ~ o~ parameterfree otherwise. It is easily seen that the notion of a syntactic model (~ la H~jek) is a generalization of the classical notion of Interpretation, because, if J ~ (¢,{~i; i ~ I}) is an interpretation of ~l in ~2 (in the sense of Tarski [85], and F any sentence, provable in ~2, then (F,¢,{~i;
i E I}) is a (parameterfree)
syntactic model of T, in ~2"
Obviously the following holds: if T is a translation from ~I into ~2 and ~ e L a logically true formula (i.e. ~ ~ ~), then --1
rep~(~) is logically true. This enables to prove by means of the deduction theorem and the finiteness-theorem the following lemma. Lemma. Let T--~(0,$,{~i; i ~ I}) be a syntactic model of TI-~(L,,A,) in T 2 -~ (L 2,A 2) and suppose that A, ~ @. Then A 2 ~
Awl...
Awk[ §(w~,...,wk) ~
rapT(@)] •
The importance of the notion of a syntactic model is contained in the following theorem. Theorem. Let ~l e {~l'~l ) and ~2 e (~2,~#
be first order theories
and suppose that ~2 is consistent. If there exists a syntactic model • of ~l in ~2' then ~, is consistent too. Proof. Assume that T is inconsistent. Then there is a formula F -i of ~i such that A, ~ F ^ 7F (and in particular A, ~ F). Hence by the preceeding lemma:
F A w I .. " A w k |8(w I ,''''Wk ) ~ repT(F ^ ]F)] ' and
(o)
~2
(oo)
~2 5 A w , ' O " Awk[8(w1'''''Wk) ~ rep (F)].
By (2) of the definition of "translation" we get from (oo): (+)
~2 ~
~/w,''" Vwk[O(wl ....
'Wk) ^ rapt(F)|"
From (o) we get using clause (iii) of the definition of "repr""
~2 F
Aw,..,AWk[
(8 v rep~(r)) ^ ( q e ~ q r e p
(r))].
We a~e interested only in the second member of the conjunction and get:
(+) and (++) show that ~2 would be inconsistent too, a contradiction to our hypothesis. Theorem (S. 0rey). Let T, and ~2 be axiomatie 1st order theories, ~2 be reflexive and ~2 contain Peano Arithmetic. If for every finite subsystem D of ~I we have, that D has a parameterfree syntactic model in ~2, then T, has a parameterfree syntactic model in ~2"
10
For a proof see Feferman [15] p. 80. A theory ~ is called reflexive iff the consistency of every finite subtheory ~ of T can be proved within T. Further results on syntactic models of set theory will be contained in the following chapters. E) ZERMELO-FRAENKEL
SET THEORY
We call ZF (set theory of Zermelo-Fraenkel) theory with identity whose alphabet,
a first order
formulae and axioms are
defined as follows: The Alphabet of ZF: One sort of variables x0~ x , , . . . , X n , . . . ( n ~ ) (x,y,z,...
stand for these variables)~
a binary predicate
e,
and logical symbols: ] , v , V
, = and brackets.
The Formulae of ZF: xey and x = y are (atomic) formulae~ and if and ~ are formulae, then Vx(~)
~($),
[#) ~ (~) and
are formulae.
We follow the us~ai conventions which allow us to omit brackets in some cases (see Hilbert-Ackermann
p. 74).
The axioms of ZF: (0) Null-set: V x A y [ 7yex]. (I) Extensionalit~ AxAy[
Az(zex
(II) Axiom
(Axiom der Bestimmtheit): ~ zcy) ~ x = y],
of pairs:
AxAyVzlA
ncu~z ~ u
(III) Axiom of unions
: xv
u = y)l,
(Sums):
Ax VyA~lzcy " Vu~ ~xl, (IV) Axiom o f
Vx (V)
infinity:
Vy~y~x
Power-set
AxVyAzlzey
^ Az(z~x
~
VuCu~x
(Potenzmengen-Axiom) :
~ Au(u~z
~ uex)l ,
^ u ~ z ^ AvCv~,
~ v~u)))
,
11
(VI~) Axioms of substitution Ax~y¢(X~y)
~ A a VbAy[yeb
(vii) Axiom of regularity Ax[
VyysX ~
In the schema
(Ersetzungsaxiome) ~
Vx(Xea
^ #(x~y))].
(Fundierungsaxiom)
Vy(yex
^ Az(-]zsy
~ 7zsx))].
(Vl¢) the formula ~ is supposed not to contain any
free occurence of the variable b. (VI~) implies the axioms of subsets (Aussonderungsaxiome). The system of axioms consisting of axioms
(0) - (VII) is called ZF. The subsystem
(0) - (VI%) is called Z ~ . Hence
ZF ~ ZF ° + "Axiom of regularity". Sets, totalities
and classes have always played an important
r61e in mathematics
though this has not been recognized explicitly
It was Georg Cantor
(1845-1918)
ever used properties mathematics
who has formulated e x p l i c i t l y
the
of sets. According to the methodology of
he investigated the interrelations
ties between these various properties.
and interdependen-
This lead to the socalled
Set Theory. [ 8 J G. CANTOR:
Gesammelte Abhandlungen Springer-Verlag
(Edited by E. Zermelo),
Berlin 1932, Reprinted 1962 by
G. Olms, Hildesheim. Cantor's
Set Theory can be viewed as a consequent
transfer our comtemplations
attempt to
with respect to the domain of finite
sets to domains
including infinite sets.
Exempl& gratia:
the principle,
that the image of a finite set is
again a finite set, yields the axiom of replacement
(Vie). On the
other hand there are principles which cannot be translated,
such
as m < m + 1 for finite cardinals m. Hence it is not too much surprising that between these extremes there is a large class of assumptions
for which it is hard to sayw~ether they are true only
when restricted to finite sets. A very classical example for this situation is the axiom of choice: (AC)
Ax(Ay(yex~
y ~ @) -~ V f ( F n c ( f )
^ Ayexf(y)gy))
There are other problems which do not have an a~alog~e in the domain of finite sets, such as the continuum-problem.
But as in the case of
12 the axiom of choice (AC) our questions are wether the Cantorian principles, which are abstracted from the domain of finite sets, are sufficent to decide them. The original Cantorian set Theory [as based on the "Ideal Kalk~l", whose axioms are extensionality and comprehension ~x(AyycX
~ ~(x))] was contradictory (take Russell's predicate
xex). Since 1908 many axiomatic systems for set theory have been given which seem not to give raise to the Russell-antinomie. Roughly they can be divided into two groups. The first group are type-theoretic foundational systems, such as the Principia Mathematica PM of Russell-Whitehead, the systems NF ("New Foundations") and ML ("Mathematical Logic") of Quine, the systems of N. da Costa (Indagationes Math.
27 (1965)), J. Houdebine (Th~se,
Rennes 1967). The second group are Extensions of Zermelo's set theory (1908):
[88] E. ZERMELO: Untersuchungen ~ber die Grundlagen der Mengenlehre; Math. Annalen 65(1908) p.261-281. [89] E. ZERMELO: Uber den Begriff der Definitheit in der Axiomatik; Fund. Math. 14(1929) p.339-344. such as Johann yon Neumann~ System (Hath. Zeitschrift 27(1928) p.669-752, and J. f~r die reine u. angew. Math.(Crelle)154(1925) p.219-240), the system of Bernays (contained in a sequence of 7 papers in the J.S.L.)vol.2,6,7,8,13,19, a modified version is contained in his book, Amsterdam 1958), the version in G~del's monograph (Princeton 1940), the ZF-version of Bourbaki, Skolem, Thiele, Sonner and others. One of the nicest foundational system has been given by: [~]
W. ACKERMANN:
Zur P~iomatik der Menge~lehre; Math. Ann.131
(1956) p.336-345. Since we will not discuss this system~ we refer the reader to the following articles:
[~]
A. LEVY:
On Ackermann's set theory~ JSL 24(1959) p.154-166.
[56J A. LEVY - R.L. VAUGHT: Principles of partial reflection in the set theories of Zermelo and Ackermann; Pacific J. Math. 11(1961) p.1045-1062. [~]
R. GREWE:
On Ackermann's set theory~ Doctoral Dissertation, University of California, Los Angeles 1966.
13
[~8]
R. GREWE:
Natural JSL
[71]
W.N.
models
34/1969)
REiNHARDT:
Topics
Thesis,
of Ackermann's
p.481-488. in the Metamathematics
University
Reinhardt's Coincides
Set Theory~
of Wisconsin
Abstract:
with ~F"
of Set Theory
19677
"Ackermann's
See also
Set Theory
in the AMS-Notices,
vo!.13(1966)
p.727. Zermelo's plus the
original
system
Z contained
only the axioms
AxVy[Az(ZCy The "Fundierungsaxiom" duced by A. Fraenkel Zermeloschen
~Grenzzahlen
We assume
subject,
and the "Brsetzungsaxiom" (in his article: Math.
the whole
system
that the reader
of ZF-set
theory.
such as J. Rubin:
(0) - (VII)
(Fund.
Comp.
Amsterdam
Ordered
is similarly
is the open formula
are defined
defined
of f and pr2(f) respectively.
for ZF-formulae by Church's
(Holden-
(North-Hol-
~.This
is the cartesian
pairs
than pr,(s)
on the second
yields
of ZF the set of logical
conversion
pr,(f)
written
use abstraction
a definitional
terms
extension axioms
is
principle:
x c {y; Cry)} ~ ~(y/x) where y is the free variable tained
from ¢ by substituting
currences
of ¢ and ¢(y/x)
is the formula ob-
x for y at all places
of y in ¢. It is supposed
s}.
coordinate.
of f, sometimes
freely
is
pr, (s) = ( X ~ V y < X , y } e
thlt f is a function,
the range
We shall
the intersection,
x × y
~ la Kuratowski:
as projection expressing
In such an extension
enriched
x N y
and
If s is a set of ordered
pr2(s)
of ZF.
of x,
of s to the first coordinate:
Fnc(f)
{x; ~(x)}
Set Theory
1968).
pairs (x,y)
= {{x},{x,y}}.
on this
conv~entions:
is the power-set
Rg(f)
on
p.29-47)
for the m a t h e m a t i c i a n
x - y the difference
the union,
is the domain
in his article
16(1930)
with the classical
- A. Mostowski:
For sets x,y,P(x)
Dom(f),
p.230-237).
are many textbooks
x U y
the projection
Math.
is familiar There
Set theory
We shall use the following
product.
86(1922)
der Cantor-
it ZF.
Day 1967) or: K. Kuratowski land Publ.
(VI¢) are intro-
Zu den Grundlagen
Annalen
und Mengenbereiche"
and has called
development
~ zcx ^ ~(z))].
Mengenl~hre,
Zermelo has adopted
(x,y)
(0) - (V)
"Aussonderungsschema":
of free oc-
that x is free for y in ~.
as
14
Let ~ and B denote define
Ordinalnumbers
(~ l a v .
Neumann).
Re(x) = U{~(RB(x)) ; 8 < ~}. For x = ~
k n o w ~ v. Neumann's The M i r i m a n o f f - r a n k
"Stufen"
the following Lemma:
p(x) of a set is defined
V
= Re(~).
to be the first
V e and, the r a n k - f u n c t i o n
p have
properties: _c V6,
1) ~ ~ B ~ V 2) p(~)
= e,
3) p(x)
= U{p(y)
4) If xEyeV e 5) x~V
one gets the well-
and we shall write
such that g e Ve+ 1. 'The "Stufen"
For any set x
+ I; y c x},
or
x _C ycVe,
then xeVe~
o O(x) < e,
e
6) Ve+l
= p(ve),
7) V~ = U{V
; e < X} if % is a limit
8) If A is any set of ordinals, dinal B. 9) x C V --
then
ordinal, U V a = v B for some orseA
p(x)'
10) x ~ Vp(x)+l
.
Proof by induction. Note that V = U{ve; sets,
e an ordinal}
is in ZF ° equivalent
The class ~ J v
can be used to construct
This shows the relative consistency the lemma in section D).
F) THE PRINCIPLE
ReI(C,~)
/~x~
~/x~
ZF in Z ~ .
to Z ~
obtained
and a class-term
from ~ by restricting
in ~ to C ~ id est~ by replacing
change
") insert:
a parameterfree.)
of ZF with respect
~ of the ZF-language
be the formula
or
phabetic
V is the class of all
(see
OF REFLECTION
For a given formula quantifier
, where
to the axiom of regularity.
by A x ( X C C
+ 4) or
of bound
variables
syntactic
model of.
Vx(xec
C let
every
each occurrence
of
^ ~), respectively
may also be needed).
(al-
The rigorous
15 definition
(by induction on the length of ¢) of R e I ( C ~ )
given ana!oguously Definilion:
can be
to our definition of repT(,#) in section D.
Let ~ and C be given. % is convenient for C iff the following is provable in ZF:
(.)
Ax, cc
A x ~c[%(xl
. . .
. . . . .
xn)
~ Rel(C~%)]
n
A partial ordering ~ between ZF-formulae
in pr@nexform is defined
by 01 ~ 0z iff there are finitely many quantifiers V x, ~ . . . ~ V x k such that 02 is obtained from ~l by putting in front of 01 negationsymbols
and these quantifiers ~/x i in a certain order.
known that every formula is equivalent to a formula set of free variables) Definition:
Lemma:
in prenex form.
~ in prenex form is hereditarily (,)
It is
(with the same
convenient
for C iff
holds not only for 0 but also for every 9 ~ 0-
Let 0 be a ZF-formula in prenex form and let {Xi; i e m} be a nested sequence of sets (i.e. X i C Xi+!). rily convenient for X : .U X. 1C~
i
If ~ is heredita-
for every Xn, % is hereditarily convenient too.
Proof (by induction on the length of ~): If 0 does not have quantifiefs, then the lemma is immediate. 9
is obvious.
The case where ¢ has °the form
Now suppose 0 has the form ~/x~(X,X, .... ,Xn). By
assumption 0 is hereditarily
convenient
holds for ~. By induction hypothesis for X. Now (.) follows immediately Theorem:
(R.Montague
- A.L@vy):
for each X n. Hence the same
~ is hereditarily
convenient
for ~/x~ and the lemma is proved.
Every instance of the following
schema of complete reflexion is a theorem of ZF: (CR*)
f~Vs[Lim(8)
^ e < 8 ^ x'eVsi .... XneVs/~ [ % -- ReI(V 8,%)]]
Here e and 8 are Variables ranging over ordinals that 8 is a limit ordinal~
and Lim(8) expresses
i.e. 8 ~ 0 ^ 8 = US. It is understood
that xl ~ .... x n are precisely the free vamlables of #. Proof:
It is sufficient
to prove the theorem only for the case
that 0 is in prenex form.
If ~ is without quantifiers~
is obviously true for 8 = e + ~. The case 0 ~ q ~ trival.
the theorem
is also very
Hence let us suppose that ~(xl ~...,x n) has the form
V x g ( X , X , ~...~Xn).
By the induction hypothesis
a limit-ordinal
8 such that 9 is hereditarily
there is for every convenient for V 8.
Let F(y)xl ~...~x n) be the formula saying that y is the set of sets x
16 of minimal rank such that ~(x,xl ~...,Xn). sequence
of ordinals
Define a countable
80 < 81... < 8 n < ... in the following way:
Let 80 be the first ordinal > s such that 9 is hereditarily convenient
for Vs0.Having
82n+i = Min{y;
defined
8 i for i ~ 2n, put
A XngV82n [ r(y,x,
82n < y ^ A x l e V B 2 n " "
.... 'Xn
) ÷
ycV~l }.
Now let ~2n+2
be the least ordinal > 82n+I
rily convenient
for V82n+ 2. Obviously
such that ~ is heredita-
8 = lim 8 n new
is a limit
ordinal and
v8
: n eUw
v 8n
: n eUw
vs2 n
(see the lemma in section E). In order to show that V 8 is convenient for ~x~(X,X,,...,Xn) , let a,,...,a n g V 8 are elements
of partial universes
Vsi's be contained
be given.
Hence the ai's
Vsi and let these finitely many
in V82 k. If ¢(al ~...,a n) holds,
then Vx~(X,a,,...an )
holds too. Hence Vx[Vy
xey ^ F(y,al ,...,a n ) ^ ~(x,a, ,...,an)]
holds too. It follows
from the definition
an object a in V82n+ 2 such that V82n+2 ~ V8~ a ¢ V~ that R e l ( V s ~ ( a ) ) hereditarily
,it follows
holds
convenient
because
Since
from the induction hypothesis~
(remark that by the construction The direction
are restricted
if Rel(Vs,#)
that there is
for V2n~ hence by the previous
V 8 too). Hence Rel(VB,#). free variables
of 82k+I,
~(a,a, ,...~a n) holds.
ReI(VB,¢)
÷ ~
~(a) is
lemma for when the
to range only over V 8, is obvious~
holds in VS, then there is an object a a V 8 such
that R e i ( V s ~ ( a ) )
holds in V S . By the induction hypothesis n with a e V8 a n ) holds for every sequence (al ,...,a n) ¢ Vs,
~(a,a,, .... Hence V x ~ ( X , a , , .... a n ) holds too for every sequence
(CR*) is called a schema of (complete)
reflexion
it follows that for any closed formula ~, if ~ holds in the
whole universe V, ¢ must hold in some partial universe V 8. Since the converse holds too, the reflection By (CR*) a property
principle
~ of V is "reflected"
is called complete.
to be already a property
in some V s . Let S be the theory based upon extensionality, pairing,
sum set, power set and the following
Aussonderung~
schema of regularity:
17
(V!l~)
~x¢(X)
-~
we want to show, ZF : S + (CR*). t'set-form".
V y [ ¢(y)
+ ] ¢(z))].
that ZF = S + Infinity
+ Replacement
Our axiom of regularity
(VII)
In the V.Neumann-Bernays-GSdel
GSdel's monograph),
^ x N X = 0
13(1948)
p.68)
out
groups
ZF ooes not have class-Variables~
Since
the schema basis
above.
of the remaing
xeu as ¢ in (VIIi). is true.
Obviously
ZF-axioms),
Bemerkungen,
12 (1966) instance
set-form schema Proof:
elosure~
there
closure
Cl(x)
replacement ~(x)).
paper:
= {x} U x u U
axiom for ~(x,y)
yields
v such that
zev ~ -] zet.
t C Cl(u)
and Cl(u)
Now,
\fx(XeCl(u)
^ ~ (x)).
the
Furthermore
a set v such that ~(v) the instance
(VII) t has an
let w be an arbitrary
is transitive,
w of v. Hence
or ~ ¢(w).
hence wsCl(u).
~(v), and
element But wev,
Therefore
since vet. Hence we
~ ~(w)
for every
of the schema of foundation
element
for ~ is
q.e.d.
Let ZF* be the system ZF without K.Hauschild
has shown
in [~6]
that
in ZF*. R . B . J e n s e n - M . E . S e h r S d e r 12(1969)
^
a set t such that xet ~ (xeCl(u) ^
have obtained
proved~
Aw(ZeW Z
^ x = y. Hence
that either q weCl(u)
it must hold that 7 ~(w).
~
is the transitive
we have
¢(x)
of
set x has a transitive
u ...
formula
and the axiom
From the axioms
that every
x U~x
the
in ZF ° .
such that x C y and --
and ¢(u) hold,
be the following
(VII)
are equivalent
it follows~
of v. Since ] wet we have
d. Math.
of ZF. Hence
By the set form of the axiom of regulamity
element
betreffend;
Logik u. Grundl.
is a theorem
(VIIi)
is a set y
of x. Since ueCl(u)
Now let ~(x,y)
(on the
that also the converse
axiom of foundation
'
wey ÷ zey).
in ZF by
p.51-56.
of foundation
and infinity
i.e.
the appropriate
(VII)
das F u n d i e r u n g s a x i o m f. math.
of (VIIi)
of the)
of
for a given set u take
shows,
Let u be any set such that ~(u) holds.
replacement
vet,
since
of the
of the axioms
implies
from the following
Zeitsehrift
Every
(see Bernays,
can be stated
(VII;)
The next theorem
We take the proof
[$6] K. HAUSCHILD:
Theorem:
on the basis
of the axiom of regularit$ (VII$)
see
that the set form and the class-form
are equivalent
A,B~C.
(NBG,
also in a "class-form":
GSdel has pointed
axiom of regularity
"class-form"
and that
is the socal!ed
set theory
one can state regularity
~-+ J.S.L.
^ /~z(ZCY
p.119-133)
the axiom of infinity.
(VII$) have
that the schema
is not provable
shown
(VIIi)
for every
(in: Arehiv math.
is not provable
Logik
in Z + VII).
18
Now we shall show that the schema of complete reflection the frame of S (: general set theory)
equivalent to the conjunction
of the axiom of infinity and replacement. show within S that for given e, V
is within
Since we are not able to
exist (the axiom of replacement
would be needed), we have to take the following different (CR) S c h e m a o f complete reflection: variableSA~ x * ..... Xn: e . .y . z [ y e z ~, Smc~(z) ^
formulation:
Let ¢ be a ZF-formula with free
A .... x,¢z
A z (¢ ~ Rel(z,~))] . n
Here smcS(z)
is the ZF-formula saying that z is a standardcomplete
(i.e. transitive) smcS(z)
AX
Ay(xeyez*
P(x)Ez
model of general set theory S, more precisely,
is the formula: xez) ^
A XA
y (xez ^ yez ~ {x,y} ¢ z) ^ A X (xez
^ Uxcz).
Here P(x) is the power set of x. Since x C yez + xgP(y)ez + xez, z is a model of the Aussonderungsaxiom,
if smcS(z).
In ZF it is
provable that Az[SmcS(z) hence
~ Va(Lim(~)
(CR) a n @ (CR*) are equivalent
^ z = Ve)] , in ZF. In order to show that
ZF = S + (CR), we have to verify first, that in S~ (CR) is equivalent to: (CR m)
m :
A y V z [yez
^ smcS(z)
A A ^ xlez...XneZ
/A% i:l (~i ~ Rel(z,~I ))],
where ~ m e a n s
conjun=tion and ¢i (I < i < m) are ZF-formuiae whose
free variables
are at most x,,...,x n. The implication
is proved by means of the following
(CR) ÷ (CR m)
i ~ ¢.m (i < m) can be implemented within the ZF-language by defining ~ to
be the following formula
idea: the enumeration
(~/stands for disjunction):
m
i:I
= i A 9i).
Since the natural numbers are absolute with respect to standard complete models, we have: /~z[SmcS(z)
~ (Rel(z,~) ~ ik~__l(t : i ^ Rel(z,¢i)))]"
i is the ith-numeral. t in the reflection
m
Since smcS(z) ÷ m C z we may substitute
~ for
formula given by (CR) with respect to ~. Thus
we have proved (CRm). The equivalence in his paper
(p.229):
S ~ (CR) ~ (CR m) is due to A.L~vy,
contained
19
[~6]
A. L~VY: Axiom schemata of strong infinity in axiomatic set theory; Pacific J. Math. 10(1960)p.223-238.
Theorem (Lgvy [~6.]): ZF = S + (CR). Proof: In order to show that S + (CR) C ZF we need to show only that every instance of the axiom schema of Aussonderung is a theorem of ZF. Hence let ¢(x) be a ZF-formula.
If ZF ~ 7 V x ¢ ( X ) ,
then the empty set 0 satisfies the Aussonderungsaxiom for ~. 19 ZF ~ V x @ ( X ) ,
then the replacementaxiom with respect to ~(x,y)
@(x) ^ x = y gives a set b which satisfies the Aussonderungsaxiom for ~. We show that ZF C S + (CR). The axiom of Null-set follows directly from the Aussonderungsaxiom with x ~ x as #(x). Infinity is obvious since every set z with smcS(z) has the properies required by the axiom of infinity. Now we prove the replacement axiom: Let ~(v,w) be given with v, w, xl,...,x n as the only free variables.
%AsAt,(
Vv(V
c x
Let F(x) denote the formula:
(r,s
^ oCr,t
:
.V AwtW
y-
^ ~(v,w))]]
By (CR)~ or equivalently (CR4)~there is a set z such that smcS(z) and for all x~ ,...,Xn,X,V,W e z: [ ¢ ~ Rel(z,~)]
^ [Vw# ^ ! Ax,
~ R e l ( z , V w ~ ) ] ^ IF ~ Rel(z,F)] .. • AXn--AxF - Rel(z, Axl
... A
Xn
AxF)
The formula in the first brackets says that ~ is absolut with respect to z, hence, by the second formula ~ maps members of z on members of z. By the Aussonderungsaxiom ~ maps the members x e z (hence x C z, since z is transitiv) onto a subset y of z. Thus, if xl,...,Xn,X e z, then F. Hence by the third formula: Rel(z,F).
But the univers~l closure of Rel(z,F) is
Rel(z, ~ x , ' " A x % _ _ ~ x r )
thus the fourth formula gives us F which
is the replacement axiom with respect to ¢, q.etd. Corollary:
In the set theory S the axiom schema of replacement in conjunction with the axiom of infinity is equivalent to the schema of complete reflection (CR).
Let~, ~,
and ~ 2 be I st order theories with the same language and
axiomatizable, ~ 2
Arithmetic, t h e n ~ 2 iff:
an extension of ~ ,
containing Peano-
is called essentially reflexive o v e r ~ ,
20
~
~ ¢ ÷ Con(K|
for every
sentence
+ ¢)
¢ of the language
shown that ZF is essentially schema.
Furthermore,
(without
reflexive
('JEin Beitrag
over Zermelo's
of ZF is a finite
Z. A slightly w e a k e r zur Hetatheorie
Logik u. Grundl.
der Mengenlehre",
d. Math.9(1963)p.291-314).
ZF which has wellordered NQ£ice,%hat following
Lemma.
models
result which theory
states
Proof by induction
Zermelo's Proof.
Suppose
id est:
ZF-set
set theory
sumption
theory
is (as Bernays extension
~ Rel(w,Rel(u,#)).
theory
with
and a finite
extension
of Z,
for #l .... '¢n (n < w) sentences of
Z. Since
there
are partial
By the preceeding
is convenient
has one free variable
of
¢i ,...,¢ n. By the as-
ZF = Z + ¢, # holds
Let e be the least
lemma V
of
Z.
Let ¢ be the conjunction V. By (CR')
¢.
is not a finite extension
such that ¢ is true in V6.
Rel(Vy~¢)
universes
in the
VB,
8 > ~,
such ordinal
(> ~).
for Rel(Vy,%)
(The formula
Y). Now let ~ be the following
ZF-formula: Vy|Lim(y) Then ReI(V
^ ~ < Y ^ Rel(Vy,#)]
,~) is the following
VyIY
e Ve ^
Lim(Y)
Since y e V e implies Rel(Ve,~)
formula:
^ ~ < Y ^ Rel(Vy,¢)].
y < e and ~ was minimal
is not true and hence
other hand V
give the whole
of ¢. Since
ZF-theory~
we infer that
~ is not true in V . On the
is a model of Z (since
of e, also a model
of
axiomatizable".
that ZF is not a finite
+ Cn
¢ is consistent
whole universe
f. math.
has shown:
- Here we shall prove only the
that ZF is consistent
ZF = Z + ¢i+...
ZF-language.
by K.Hauschild
new constants)
on the length of the ZF-formula
(R.Montague):
of Zermelo
Z.
If u C w, then Rel(u~¢)
Theorem
(without
has
extension
Zeitschr.
is not finitely
axiomatizable.
~]
Aussonderungs-
Hauschild
in contrast to ZF the NBG-set
finitely
of Zermelo-set
extension
result was obtained
"The system ZF and every extension
has shown)
R.Montague
he has proved that no consistent
new constants)
set theory
of~l.
Lim(e))
Z F ~ ~(via
q.e.d.
and, by the choice
(CR'))~
Z + ¢ cannot
21 !~.Corollary:
ZF is not finitely axiomatizab!e.
2. Corollary:
ZF is reflexive
(i.e. the consistency of every
finite subtheory of ZF can be proved within ZF). The first corollary was obtained
(by different methods)
also by
K.Hauschild [B6]. A.L@vy has generalized these corollaries
and
in particular the results of R.Montague in his papers [~6] and [~7] A.L~VY: On a spectrum of set theories~
Illinois J.Math.4
(1960)p.413-424. In [~6]L~vy has extended Montague's
results to a transfinite
Hierarchy of set theories:
Z, ZF, ZMI, ZM2,.., ZM~,... where ZMa+ 1 results from ZM a by adding suitable principles.In [@}]
Lgvy constructs
any two consecutive
(strong) reflection
further Hierarchies
between
theories ZM , ZM + I. The construction
is
carried out between ZF and ZM: ZFI,~ZF2,..., for every ZF-definite
ZF 6+1, ZF6+2
...
ordinal 8. Here ZF 8+I is roughly the set
of all sentences ~ which are true in natural models (Ve,6) , where e is an inaccessible number such that V e contains at least 6 many inaccessibles,
i.e. 6 + I
many ZF-universa.
Lgvy shows that ZF I
is essentially reflexive over ZF and that ZF 8+2 is essentially reflexive over ZF 6+1. In another paper of A.L~vy: (Proc. A.M.S.
9(1959)p.942-945,
"Comparision of Subtheories ~' Errata vol.10) it is shown that
the numbertheory of ZF is an essentially
in£inite extension of the
numbertheory of Z and that the same holds for ZF + "There are inaccessible
cardinals" and ZF. More precisely let T be the usual
interpretation
(parameterfree
syntactic model) of Peano Arithmetic
in set theory. Then let Z / ~ Z ~ repT(¢)}
and Z F / ~
~ {¢ formula of Peano-Arithmetic;
similarily defined.
tially infinite extension of Z / ~ is an essentially
Then: Z F / ~
is an essen-
and (ZF + ~/ee i n a c c e s s i b l e ) / ~
infinite extension of Z F / ~ .
These results
depend essentially on the existence of certain truth definitions with some particular properties. There are interesting finitely axiomatizable ZF. If we admit in the schema of Aussonderung A It x - ( t llpl " ' ' A p n A y V x
subtheories of ' . . . . Pn
))|
22
only those
formulas
¢ in which
occur
the form
solely
i.e.quantifiers
having
Vx(x
), then we get a weaker
A(x
limited
£ y ÷...
quantifiers,
) as well as
A
e y ....
of Eo-Aussonderungsaxioms cribed
is finitely
to Tarski
(in:
(in L~vy's
axiomatizable.
'TThe present
terminology)
Mostowski
state
A generalization
of axioms
are E
k is any fixed natural to E.J.Thiele
the Bernays-GSdel [84]
E.J.THIELE:
axiomatization
operations).
this result
but a proof has
of this result:"every
or H -formulas n n number ~ is finitely
(the finite
as just des-
credits
..." p.20),
never been published. of ZF which
system than Z. The set
set
for some n < k, where axiomatizable"
is essentially
is due
given by
See the paper:
Uber endlich
axiomatisierbareTeilsysteme
Mengenlehre;
Zeitsch.
lagen d. Math.
f. math.
14(1968)p.39-58.
der ZF-
Logik u. Grund-
CHAPTER
Construetible
Sets
In this chapter ralized
continuum
tibility
cy of these three
cerning
we discuss
hypothesis
V = L and present
conclude
the axiom of choice
(GCH) and G~del's
GSdel'S
statements
the chapter
with the axioms
M e of e o n s t r u c t i b l e
OF CHOICE AND THE CONTINUUM
Mathematicians
used the hypothesis
sets we are able to choose one element.
With respect
a philosophical
(in an article
first perceived
Annalen
the doubtfulness
avec laquelle
as a m a t h e m a t i c a l l y
the question
by the axioms
there
the ZF-axioms
describe
concerning
isw~ether
tical
theorems.
quences~ Ann.
59(1904)
of c o n c e n t r a t i o n
(AC) has
65(1908))
when thinking?)
of sets given contemplations
learned
"strange"
mathemaconse-
(Zermelo,
and we are unable
Math.
to "imagine"
(is this only a question
Hence there
are two important
questions: (I) Is the axiom of choice
(AC) provable
Or (2) is (AC) at all events
consistent
that
(AC) is intuiti-
in proving
set can be well-ordered" Ann.
une indi-
are, but we have
of the set of real numbers
arbi-
or not. We believe
it seems to be indispensable
and Math.
a well-ordering
He wrote:
of some formal
way our intuitive
On the other hand the
who
has been formalized
in the universe
in a correct
It
differen-
de lois une loi
of set. The axiom of choice
such as "every
live;
on a fait correspondre
choice-sets
are
see p.210)
of such a hypothesis.
vely not so clear as the other ZF-axioms to use it because
short
of ordinary
p.182-229,
e.g. based on axioms
are always
the notion
in their
As soon as set theory
theory
of the
to 19 th centur~ mathematics
une infinit4
~ une classe
classe".
collection
from every member
37(1890)
"Mais on ne peut pas appliquer
language,
We con-
sets.
on integrability
Math,
traire
of H.Putnam
question whether human beings
tial equations,
vidu de cette
of ZF-set theory.
that for any infinite
able to arrange infinite choice sequences was G.Peano
consisten-
of the 19 th century have very often and without
of nonempty
it is merely
axiom of construc-
HYPOTHESIS
any skruple collection
(AC), the gene-
proof of the relative
with a proof of some results
the h i e r a r c h y
A) THE AXIOM
II
of refutable with
ZF?
in ZF?
24
The same questions may be asked in the case of the Continuum Hypothesis:
(CH): 7~/x(~O" "
x, hence (assuming (AC)):
< ~ < 2~°), (CH) ~ 2 ~
~
= M,
is the cardinality . The hypothesis
of
(CH) was
the simplest conjecture concerning the cardinality of the continuum of G.Cantor who spent many years in order to prove it without obtaining a proof. The generalization
of (CH) is:
(GCH) General Continuum Hypothesis:~ (AH)
Aleph - Hypothesis:
These assumptions
A~(~
~
A x ~
Vy(~
< ~ < 2x);
= ~+i).
are often used (mostly in Topology and Algebra)
in order to decide questions about the cardinality of certain spaces or structures. pected consequences.
As in the case of (AC), the (GCH) has unexWe give an example
(due to W.Sierpi~ski
(Ren-
diconti del Circolo Mat. di Palermo, Serie II, vol.l(1952)p.6-10)): "The euklidean space E of 3 dimensions
is a union of sets
E, ,E2 ~ E3 such that whenever A is a line parallel to the i th axis of coordination
(i = 1,2,3), then A N E i is finite"
Classically
it is known, that (GCH) implies
(AC) and (AH),
see the imDortant survey (in particular p.314): [531A.LINDENBAUH-A.TARSKI: Communica%~on sur les recherches Th4orie des ensembles~
de la
Comptes Rend. de $~eances de la Soc.
Sci. et lettres, Varsovie
19(1926)~ Classe III, p.299-330.
This paper does not contain any proofs. The first proof for (GCH) (AC) (in ZF °) was published by W.Sierpinski p.137-168). Math.
5(1954) p.332-33
33(1945) (Archiv der
. In full ZF (id est ZF ° + regularity),
(GCH) and (AH) are equivalent, every
(Fund. Math.
Another proof has been given by E.P.Specker
set of type P(M )
since (AH) obviously implies that
can be well-ordered
[x ~ y ~ Min((x U y) -
(x ~ y)) 6 y is a total ordering of 2 Ms ] , which in turn implies the axiom of choice as H.Rubin has shown in 1960 (see the Book "Equivalents page 77).
of the (AC)" by H. and J.E.Rubin, Amsterdam 1963,
In 1936/37 G~del in his lectures given at the University of Vienna (Austria) has proved the consistency of (AC) with ZF. G~del has extended this result by showing that the (GCH) is consistent with ZF too. The first announcement ~
of this result appeared in 1938:
K.GODEL: The Consistency of the Axiom of Choice and the Generalized Continuum Hypothesis; Proc. Nat. Acad. Sc.24(1938)p. 556-557.
25
A more detailed version appeared in 1939: [Z~] K.GODEL: Consistency-Proof for the Generalized ContinuumHypothesis; Proc. Nat. Acad. Sci. USA, 25(1939)p.220-225. The proofin[Z~]consists
in defining by means of a transfinite
Hiearachy Ma a semantical model of ZF + (AC) + (GCH). In lectures presented at Princeton 1938/39 GSdel has proved the Consistency by means of a syntactic model of NBG + (AC) + (GCH) in NBG. The syntactic model $ is defined by means of a hierarchy F e of construetible sets. GSdel's Princeton lecture notes appeared first in 1940 in print: ItS] K.GODEL: The consistency of the axiom of choice and of the generalized Continuum-Hypothesis; Princeton University Press 1966 (7 th printing), Annals of Mathematics Studies Nr. 3 . In our presentation of the consistency result we shall follow [Z~] but with some modifications due to R.B.Jensen (unpublished Seminar notes, Bonn 1967) and to C.Karp: |@~] C.KARP: A proof of the relative consistency of the Continuum Hypothesis~ in~ "Sets, Models and Recursion Theory", Proceedings Leicester 1965, North-Holland Publ. Comp. Amsterdam 1967. B) THE CONSTRUCTION OF THE MODEL Definition. A model
~
~ (M,E)
of set theory is called "standard"
iff M is transitiv (in the sense of the meta-theory) and the binary relation E coincides with the actual membership relation 6 when restricted to M. The model L will be constructed by means of a hierarchy of sets M e . The construction may be motivated as follows: Let A be a set of nonempty, pairwise disjoint sets. A choice set C for A is a subset of ~JA, hence an element of P ( U A ) , power set of U A .
the
Hence one might think, that the axiom of choice
is provable from the axioms of union and Powerset. But the problem is, in order to exhibit such a choice set we need to have a ZFformula @(x) which, by the Aussonderungsaxiom, describes one choice set. We do not know in general, how to obtain such a description. In order to get that (AC) holds in our model, we shall construct it by taking only those sets which are definable in a certain ramified language.
26 Definition:
Let
~
be a first order language
of relations,
functions~
(allowing all manner
predicates
and individual
constants to occur in the alphabet of 0~
(A,...)
£ ) and let
be a relational system appropriate
for
, with domain A. A subset S of A is said to be definable
in C~ iff there is a formula @ of
£
& (.~[,~~[a]}.
one free variable x, such that S ~ {a; a 6 A Let Def(0t) be the set of subsets definable Theorem
(K.GSdel):
Every standard model g ~
9~ with the same ordinals
with
in ~ .
of ZF contains a submodel
in which ZF + (AC) + (GC H)
hold. Proof:
Define i n ~
a hierarchy M s of sets for ordinals ~ o f ~
in
the following way: M0
:
M l = U {Ms; e < i} if I is a limit ordinal, Ms+l= Def(M e) where Def(M e) denotes the set of all parametrically and subsets of the structure (Ms, 6, ~) all elements of M
and subsets of M
structure (M~, 6, X) x6M~ : m s The language
2
of~
e
definable
or equivalently
which are definable
elements
the set of in the
.
has besides the ordinary ZF-symbols
dual constants ~ for each x E Me and x is interpreted i n ~
indivia by x.
Define finally: 6 On9~ }
L : U{Me
where On 9~ is the class of all ordinals structible in~
A set is called con-
iff it is a member of some M s . Since the Ma's are defined
they all are subsets o f ~
~-class
in~.
of constructible
and hence L C ~ .
sets o f ~ .
L is called the
We claim that in L the axioms
of ZF and (AC) and (GCH) hold. This will be proved in several steps. First n0%ic~ , that a ~ 8 implies M~ ~ M~. Furthermore: then M
if e < 8
Q M8. This holds since obviously M e e Me+ 1 _C M 8. For a
eonstructible
set let ord(x) be the least ordinal e such that x 6 M s .
Obviously ord(x) can never be a limit ordinal. Lemma:
If x is constructible ord(y) < ord(x).
and y 6 x, then y is constructible
and
The sets M s are thus transitive.
Lemma: Every ordinal e o f ~
is constructible
and ord(e)
= e + I.
27 Proof:
First we show that or4(~) ~ ~. Suppose %here are constructible
ordinals s such that ord(s) are constr, ordinal.
= 8 ~ s. Then s E MB C Ms, hence there
ordinals s such that e E Me . Let ~0 be the least such
If e0 = ~ + 1, then so 6 Ms ° = Def(My), hence either s0 6 My
or so C My. But since y6 e0 and My is transitiv, cases): y e My If ~o
- a contradiction,
so would not be minimal.
= ~..)So, then So e M 0 for some P < S o ,
since Mp is transitiv;
id est pE e o .
again a contradiction.
(if such ordinals exist)
For every ordinal 8 6 ~o therefore Hence,
if e0 is a limit ordinal:
Then P 6 Mp
Hence we always must
have ord(s) > s + 1. We shall show that ord(e) the first ordinal
we obtain (in both
= s + I. Let s0 be
such that so ~ Ms0+1.
8 6 MS+I, which is 8 6 Def(Ms).
so C ~ { M ~ ;
8 e so} = Ms0.
In
order to show that so 6 Mso+l holds, we have to show that so is a definable subset of M
se
. Consider the formula ¢(x):
u, Au%V
U^
U E X ~- V E X] (x is totally ordered by 6 and transitiv). subset A of M
O.o
. Since M
so
is transitive
#(x) defines a certain
and e is interpreted by
the actual membership relation E, A must be precisely the set of those ordinals, which are in Mso, But these are by definition at least the elements tPansitivity
~ of so. If y 6 M
fop y > eo, then by the so of Moo, so 6 Meo which cannot happen as it was shown
above. Hence A consists precisely of the elements of So.Id est e0~-A and ~0 is definable
in Ms0 and hence an element Of Ms,+1.
If s0 is a successor ordinal, Since S 6 MS+ I = Mso
say so = 8 + I, then s0 ~ 8 U {8}.
(by minimality of se), hence definable, we
have obviously s0 6 Ms0+i too, q.e.d. Since M0 = 0, hence @ 6 M,, the axiom of Null set is true in L. Since extensionality holds in ~ elements
are constructible
and elements of constructible
too, the axiom of extensionality holds in
L. The axiom of pairs is also obviously true in L. Lemma: The axiom of sum-set holds in L. Proof:
Let a be a constructible
set and s an ordinal o f ~
a 6 M s. Since M s is transitive b = U subset) Me
in M s. The formula
Vv(Y
such that
{x; x 6 a} is contained
(as a
e a ^ x ~ y) with constant a from
defines b as a subset of Me. Hence b E Def(M ) = Ms+ 1 C L.
Lemma: The axioms of regularity and infinity hold in L.
28
Proof.
We have shown,
that every
Since the axiom of Infinity But ~ satisfies regularity
ordinal
holds
in~and
is constructible.
in~1~6, m is i n ~ ,
the axiom of Infinity
holds
of~
hence
in L.
in L. Since the axiom of
L C~and
the epsilon-relations
of L
and~
coincide,
the axiom of regularity
Lem~a:
The power
set axiom holds
Proof.
Let a be a set of L and b be the set of c o n s t r u c t i b l e
in L.
sets of a. Since a is a set o f ~ of a in the sense o f ~ . hierarchy mapping
Me, MI,...
x ÷ ord(x)
of replacement
, b is a subset
Therefore
Ms,...
f r o m elements
and one constant
/~u(U
Define
of b to ordinals.
{ord(x);
bound =. Hence b C M e_. Let 8 = Max{e, a E M e . The formula
ing~).
sub-
of the pQwerset
b is a set o f ~ ( s i n c e
was defined
is true i n ~ ,
must hold in L as well.
the in~
the
Since the axiom
x-E b} has a least upper ord(a)},
then b _C M 8 and
e x ÷ u e a) with one free variable
a from M 8 defines
b as a subset
b 6 MS+ I and b is constructible./~he
x
of M 8. Thus
proof of the following
lemma
I
uses techniques 1961 paper,
which
are essentially
rediscovered
The axiom of replacement
Proof.
Consider
y and parameters x in L there
holds
in L.
¢(x,y,a, ,...,ak ) with
free variables
a~,...,a k which are in L. Suppose
is precisely
one y in L such that #(x,y,!~,..oak).
sets y such that there
are sets x with
in~
Max{e,
ord(ak)}.
ord(a),
most simplest refection (GRP):Let
ord(al),...,
We shall
show that b 6 M~o+I.
way by applying
(in~)
{ord(y);
e. Define
y E b} has 80 =
Then a, al,...,a k 6 MB0
and
This can be done in the the following
generalized
principle: {We; e an ordinal}
that for I : [ J ~ W = ~]{We;
be a nested
it holds
e an ordinal}.
ZF, then for every ~
a supremum
and x 6 a.
element y of b is construc-
tible and since b is a set in the sense of~Y~6, by the axiom of replacement
of construc-
#(x,y,...)
We have to show that b is c o n s t r u c t i b l e . E v e r y
b C MSo.
x and
that for every
Now let a be any set in L and let b be the collection tible
in his
in 1963 by P.J.Cohen.
Lemma:
a formula
due to E.J.Thiele
ordinal
sequence
that W l ~ ~ ] { W e ;
e < k}. Let
If #(x, .... ,x n) is a formula of ~ there
is a 8 > ~ such that
A
/\;./\nLX,,,Xn
of sets W e such
C el W, , "
29 It can be shown exactly in the same way as we have proved the principle of complete reflection
(CR*), that every instance of
(GPR) is a theorem of ZF (from the hierarchy V e we needed only the facts~ that for I a limit ordinal: ~
~
÷
C_ VS)
V
holds i n ~ 6 .
.
Sinee~
Vl = U { V e ;
e < l} and
is by assumption a model of ZF, (GPR)
We shall apply (GPR) for the hierarchy M e and the
formula #(x,y,xl ~...,x k) with free variables x,,...~x k replaced for the constants a, ~...~a k (alphabetic change of bound variables in ~ may be needed.
Hence for the ordinal 8o previously defined
and # there is and ordinal Yo > 8o such that A
A
A ~..~ [x,y,x, ..... Xk e Myo ÷ ( R e I ( L ~ ) " Rei(Myo,~))]. y x, k Since ord(al) ~ 80 < Yo,...~ ord(a k) ~ 80 < Y0 we have that al ~...~a k are elements of My0 ; hence we obtain:
For ¢(x~y,a, ~...~[ k) let us simply write 9(x,y) The interpretation of #(x~y)
in the sequel.
in My0 consists of those pairs
(x~y) 6 My° x My ° such that Rel(M~o,~(x,y)).
By the last formula it
is equivalent to say tha%the interpmetation of ~(x,y) in M consists of those pairs (x,y) 6 M 2 such that ReI(L@ (x~y))~°Hence Yo the formula Vx(X
e ~ ^ ~(x,y))
with one free variable y and constants a~a ''''~!k defines the set b in My°. Thus b E My0+1 and b is constructible, Remark.
The proof given above that (L,e)
carried out informally.
q.e.d.
is a model of ZF is
A detailed proof defines first i n ~
language £$Ttwhich contains stant x for each set x o f g ~ .
a
besides the usual ZF-symbols a conIn order that £ ~ i s
a class of
one has to arrange that all the symbols of the alphabet of £9T5 are sets o f T ~ . K . B o J e n s e n
in his lecture notes (Springer~
"Modelle der Mengeniehre"
gives a detailed solution.
1967)
The satisfac-
tion relation ~ between structures (s,e ~s ) (which are sets off*g) and formulae of £~% can be defined in gT6(see Mostowski's "Constructible sets"). Then the class L can be defined i n ~ transfinite recursion.
by means of
30 In order to show, that < L, 6 ) is a model of (AC) and (GCH) we shall show that a much stronger axiom, the soealled axiom of eonstructibility" implies
V : L", hol~s in (L,£ > and that "V = L"
(AC) as well as (GCH).
So far we have only shown, that
"L" can be defined within any given standard model~TL. shall show that the statement set is eonstructible:"V (pure) ZF-language,
Now we
informally expressed as "every
= L" can be expressed by a formula of lhe
so that V = L is a certain ZF-sentenee.
We
shall prove a stronger faet~ namely that V = L is in ZF equivalent to a E,-formula
Defintion.
(this result is due t o
C.K&~p~]).
(A.L~vy): A formula ¢ of the ZF-language
is a E0-for-
mula iff ¢ contains no unbounded quantifiers; iff ¢ has the form
V
¢ is ~,,
.~ with ~ a 7.o-formula; ¢ is ~I
iff ¢ has the form
A X ~ for a To-formula ~. If a ZFx formula F is in ZF (provably) equivalent to a Z0 (ZI ,HI)
formula,
then r is called a ~ZFc~ZF,nZF
respectively)
formula,
r is A zF iff r is both ~Zr and n zF
Lemma I: If ¢ and ~ are zZF, then so are ¢ ^ ~, ¢ v ~,
A(x~ Proof.
~¢)
By hypothesis:
and
V(×~ z
ZF ~ ~ ~
Vx¢ ,
^ ~).
V y ~ with ~ o .
We have to show that
ZF ~ V x V y $ ~ V z ( V x e z V y c z $ ) . The part "÷" is obvious; for the 'part "÷" take z : {x~y},z a Variable not in $. Hence V x ¢ is ~>F. Now it follows that also ¢ ^ $ and ¢ v ~ are 7.ZF. Next we show that :
The part "÷" is obvious.
Ad ~'~": assume x,y are free in ~. Define
a function F: F(x) = {y; $(x,y~...) ^ p(y) : Min{~; V v ( P ( V ) : ~ ^ ~(x,v,...))}} By the axioms of sumset and replacement (.) follows with : U{F(x); l ZF
x e z}. It remains to show that
Vx(x
s z ^ ¢) is
f o r ¢ -~ Vy~ with SZo. But x e z ^ ~ is Zo, hence V ¥ ( x 6 z ^ ~)
is E, and therefore
Vx( VyX
e z ^ ~) is l ZF as was shown above.
3! C o r a l l a~ r y
2: If ¢ and ~ are A ZF , so are ¢ ^ ~ ¢ v ~, ¢-* ~, A x ( X £ Z-~ ¢) and V x ( x e z ^ ¢).
Lermna 3: If ZF ~ Vz V n A x ( ¢ ( x ) - ~ (a)
Ax(¢
(b)
V __x(~_ ~
where Proof.
÷ ~ ) -- A ~) ~
"~" obvious.
the lemma follows
"+":
yt ez
Vy
it is supposed
x e Unz),
gz
yneYn_ 1
AXeyn(¢(x)-+
~)
... V y neYn_l V x e y n(¢-* ~)'
that y, ''''Yn are not free in ¢,~.
since x e L j n z o
Vy, e z . . . V y n e Y n _ l ( X e Y n )
immediately.
Lemma %: The f o l l o w i n g
ZF are E0 -formulae:
formulae
(2): y : {x,z}, y : ~]x,
y e Dom(f),
(6): y : x-z,
(8): y = Rg(f),
y : f ~ z, (I0): y : f"z, Fnc(f),
(I): x = y,
(3): y = (x, , .... x n ) , (4): y : x n z,
(5): y : x ~ y, y : LJx,
y = Dom(f), (13):
..A
then
(ii):
(7):
y e Rg(f),
(9):
x : y x z, (12): y : f-l,
(14): y = x U {x} and
(15):
Ord(x)
(x is an
ordinal ). Here (x, ,x2 ) m e a n s Dom(f)
the ordered
the d o m a i n of. f, Rg(f)
of f to z, f"z = {f(y); x cartesian (1) and
product
anaioguous
and Fnc(f)
(9),
expresses
(3) follows
(10)~
differenee~
the range of f, f ~ z the r e s t r i c t i o n
(8) are obvious,
to (7);
- set t h e o r e t i c a l
y s z} the image of z under
(2) are immediate~
lenhma 3% (5) and
pair,
f for z C Dora(f)=
that f is a function.
from
(7) uses
(2),
(4) follows
from
(3) and lemma 3~ (8) is
(12) and (13) are proved by means of
lenm~a 3. In all cases use a p p r o p r i a t e
defining
formulas
(see C.Karp
[4z] e g ) Lemma
5: If ¢(y~x, ..... x n) is l ZF and Z F ~ A then
¢ is
Proof o Since ZF ~ n is ~ZF. Lemma
AZF.
~ ~
• ..
x,
A
~(y,x,
xn
~..,X
n).
y
V z ( Z ~ y ^ ¢(z, .. )), n ¢ is EZF ~ hence
6 : S.u b.s t i.t u.t i o n s p rinciple" . Let G~ ~ . . . ,G n be m - p l a c e
functions
such that Yi = Gi(xl '''" 'Xm) is E ZF. Then if (ul ~... ~u n > eR is A~ ~ (G, (x, ,... ~x m) .... ,Gn(X, ~... ~x m) } e R is A ZF too. 1
52
Proof.
By lemma 1 the proof of the claim follows from
E R . V y . . . V y n (
''''Yn ) ~ R^ 1 ~ n
and the formula obtained by replacing
Yi: Gi(~))
(in both places of occurence)
eby~. Lemma 7: Let x : G(yo,..~,y n) and z : H(y,,°..~y n) be n + I resp. n-place
z~F-functions
and (Yo~..
,y
) e R be a
n + i - place A,ZF-relation ~ then F isna n-place E~F-func tion~ where
F(y,,...,y n) = {G(yo~...~Yn);
Y0e H(Yl ~-..~Yn)^
(yo,...,y n ) ~ R}. Lemma 8: Ordinal recursion: z~F-function
Let x : G(yo,...,y n) be a n + 1-p!ace
and F the function defined by ordinal recur-
sion on y, as F(yl ~...~y n) = o if 7 0rd(y,), G ( F ~ yl,y,,...,yn)
if 0rd(y,),
F(yl,...,yn):
then F is Z~ F.
For a proof of Lemmata 7~ 8 the reader is referred to Jensen or Karp [%Z]p.25-26.
F ~ yl
= the function whose domain is Yl and whose
values are.F(~,y2,...,y n) for ~ e Yl. No%iCe %hat the formulae By ordinal recursion
is A> F~ and analogously exponentiation).
Ord(x),
Lim(x), x ¢ my x = m are z S F
it follows that z = x + y (ordinal addition) z = x.y and z = x y (ordinal multiplication~
Further y = C(x), where C(x) = x U
{C(z); z e x}
is the transitive closure of x, is A~ F (by lemma 4~ lemma 6 and lemma 8).
D) z~F-definability
of syntactiqal notions
A sequence of symbols is defined to be a finite sequence such that Z = {y; Fnc(y) of symbols,
^ Dom(y)em}
(function)
is the class of all sequences
z e Z is A~ F (by lemma 4,6,8).
Lemma 3 and since x e
is z$F gives moreover that z e Z is Z$ F. If Z u = {y e Z; Rg(y) ~ u}, then y e Z u and y = Z u
are both A~ F. Let uv denote concatenation
of the sequences u,v; then z = uv is A~ F.
33
Arithmetization In o r d e r
to e n a b l e
us to c o n s i d e r
language
as finite
sequences
object
language
with
x i = {<
, 0 >}
= {<<0,2>
, 0 >}
= {<<1,2>
the formulas
we must
special
finite
of the Z F - o b j e c t
identify
the A l p h a b e t
of the
sequences:
(x i is the
i th variable),
, 0 >}
=
{<<5,2>
, 0 >}
A" = {<<2,2>
, 0 >}
( =
{<<6,2>
, 0 >}
E"
= {<<3,2>
, 0 >}
) =
{<<7,2>
, .0 >}
= {<<4,2>
, 0 >}
x =
{<<x,0>
, 0 >}
6
C
(E For every
set x e V we have hence
The symbols domain
is a c o m p r e h e n s i o n
of the A l p h a b e t
{0} and values
a constant
i,g e m. The
(y is a v a r i a b l e ) ,
y = i, y e Konst
Fml of all f o r m u l a e
usual)
class
and we get that
of a v a r i a b l e
x e Fml
z by a c o n s t a n t
is A ~ F - d e f i n a b l e .
If Fr(~')
(above)
formulae
des...)
functions
is d e f i n e d (using
t in a f o r m u l a
~',
set of free
of form i) are
by r e c u r s i o n
lemma
with
y = xi, y e Vbl
(y is a c o n s t a n t
is A~F
is the
ensemble
x in our o b j e c t - l a n g u a g e
are by d e f i n i t i o n
,
all z~F.
The
operator:
(as
8). The r e p l a c e m e n t
in symbols
variables
~'(z/t),
in ~',
then
•
y = Fr(~
) is A F, hence
variables)
is A~ F too.
subformulas A similar are
are elements result
y ~ S (y is a sentence,
Finally
holds
let Fml u be the
has
no free whose
of u. Then y e Fml u and y = Fml u are A~F.
for
sets
S u of s e n t e n c e s
whose
subformulae
in u.
E) A Z F - d e f i n a b i l i t y
of s e m a n t i c
For any set u and any s e n t e n c e u ~ ~"
is d e f i n e d
U ~ %~"
as usual
" ~ U ~ #', U ~ A " ~ ' ~
u ~ £~E'z¢"
the relation:
founded, Define
~
eS u the
A
satisfaction
relation
u ~ xez ~ xez,
U ~ ~'(X/x)
XCU
and f i n a l l y
--
~ u ~ ¢'(Z/x).
AS a c o n s e q u e n c e that
notions
by recursion:
X
KTu
i.e.
set of f o r m u l a e
we g e t - t h a t ~
and A~ F and then
KT u to be the
u ~ ¢"
is A~ F (this
is a u - c o m p o n e n t applying
lemma
Z
(zeVbl ^ y
= E'z¢"
of ~
7).
set of all c l a s s - t e r m s
= {Y; y . ¢ ' e F m l u ^ V
is shown by p r o v i n g
(: subformula)
in u:
AFr(¢')=
{z})}.
is well--
that
34 Then yeKT u and y = KT u are both A~ F. Now let u[t] be the set {x; xgu ^ u ~ ¢'(Z/x)} for the comprehension term t = E'z¢'EKT u. Then y = u[t] is A~ F (the u-valuation of t) Lemma 9: y : Def(u) is A~ F. Proof. Def(u) is defined to be the set {u[t] ; teKTu}. Since a : KT u is A ZF the lemma follows from the fact that y = u| t] is A ZF using lemma 7. F) The z~F-definability of the constructib!e model The class L (i.e. the constructible, inner model) is defined by the following recursion: if
x
:
~,
Def(L(z)) if Ord(x),Ord(z)^x : z+l, L(x) :{
U{L(z); zEx} if Lim(x) otherwise
and L = U{L(x); Ord(x)}. Lemma 10. y : L(x) is A~ F. Proof. Define the following 2-place function G(x,u) by: G(x,u) : if x = ~ , G(x,u) = Def(u(z)) if Ord(z)^x = z+l^Fnc(u)^zcDom(u), G(x,u) = O{u(z); zcx} if Lim(x)^Fnc(u).x C Dom(u), G(x,u) = ~ otherwise. Obviously: L(x) = G(x,L~x) and Lemma 4 yields v = u(z) is A~ F, hence by lemma 6: y = Def(u(z)) is A> F since y Def(v) is A~ F by lemma 9 Thus lemma 7 gives us that y = G(x,u) is A~ F and lemma 8 yields that y = G(x,L~x) is A~ F too. Corollary 11: zeL(x) is A~ F. Proof: Since zey is A> F, the corollary follows from lemma 6 and I0. Corollary 12: zeL is Z~ F. Proof: Since zgL ~ V
(Ord(x)^zeL(x)) the corollary follows from lemma x 1, !emma 4 and corollary 11.
35
G) Prgperties
of the class L of constructible
Theorem (G6del [~S]):
sets
There is a ZF-formula ¢(x,y) which wellorders
the class L. Outline of proof.
Remark that by induction one can show that
each L(x), for Ord(x), has a definable wellordering W(x). ly ~ : W(0)
Obvious-
is such a wellordering of L(0). Suppose we have defined
W(x) for all ordinals x
< e. If Lim(~), then
W(e) = {(x,y)i VBVy[(B < Y ^ xBL(B) V(8 = Y ^ (x,y) e W(B))} L(~) = Def(L(8));
^ yeL(y)
wel!orders
the elements L(a)[t]
sets (or elements) ZF-language
v
L(e). If e = 8+1, then
of L ( e ) a r e
definable
sub-
of L(8). The language FmlL(8) which describes
these subsets has besides the wellordered constants x for xeL(B).
wellordering
^ ~ yBL(8))
sequence of the usual
But W(B) induces obviously a
between these constants and hence KTL(8)
is wellordered.
Thus the whole language FmlL(8) which is used to obtain "definable" subsets of L(8) is wellorderable
and the wellordering
between the
formulas can be used to obtain a wellordering of L(e) = Def(L(8)). The relation W(e) can be defined thus in terms of W(8) and the valuation
(interpretation)-function
for terms t of TKL(8)
(for a
rigurous proof see C. Karp [@~] p. 18-17). Corollary:
The statement V = L implies the axiom of choice
(AC).
A
Remark:
V = L is an abbreviation
for /\x xeL: every set is con-
$tructible. Let ~ be the relation of cardinal equivalence
(equipollence).
Using (AC) one shows that for infinite ordinals ~+e~<,oe~
n (all n£m). Therefore one is able to show by induc-
tion that e > m ~ L(e) ~ e holds. Lemma (Mostowski):
(in ZF): If R is a binary relation on a set A
such that (A~R) is well-founded and extensional,
then there
is a function f and a transitive
set D such that f is an
isomorphism of (A,R) on (D,e~D).
Moreover,
C(x) C A, then f(x) = x. Further, ZF Z 0 -formula, then
if R is e~A and
if ~(x1,°..,x n) is a
36
Ax~ ~A.•.A~n J A relation
9(f(xl ) ..... f(x n) ) ~ Rel (A,~(x I ..... x n) )] .
R on A is well-founded
and extensional The function
iff
iff A B [ B
/~y~A[ /\zsA(< z,x)ER -- < z,y)sR) -- x = y].
AeA
f can be defined
(along the well-founded
able (since A is a set) relation f(x) = {f(y)
by Mostowski Mostowski's
and localis-
R) by recursion:
~ (y,x)eR}
Let D be the range of f then f is the required the rest see Karp [~Z]
~V~,gB( y,x)R]
~ ~ ~ .~Vw£B ~
isomorphism.
For
p. 14. The first part of !emma was obtained
in Fund. Math.
36 (1949) p. 143-184,
book on "Constructible
tions are due to R. Montague
sets"
see also
(1969, p. 20). The addi-
(Eull. AMS 61 (1955), p.442)
and
L6vy [~a]. The following
theorem
[~8] A. LEVY: A hierarchy AMS, nr. Theorem
(A. L6vy):
is contained on p. 52 (theorem of formulas
57 (Providence
in set theory,
36) in:
Memoirs
of the
1965).
Let ~ be a Z~F-formula
having no free variables
except x,y. Then ZF + (AC) Here HC(x)
~ V x % ~ ~/x[HC(x) < Max{N o,HC(y)} ^
is the hereditary
%(x,y)].
cardinal of x, id est: the cardinal
number of the transitive
closure of x: HC(x) = C--~). In the proof one can assume that # is Z0ZF , since if # is equivalent to Vz~(X,y,z)
with ~ z ZF 0
~
then let ~*(u,y) be the following
E~F-formula:
Vx(XSU ^ Vz(zSu Then take u = {x,z} to establish
^ ~(x,y~z)))
ZF ~ Vx~ ~ Vu~*.
Then the theorem
for ~*(u,y)
(with u instead of x) yields the theorem for ~(x,y). ZF Hence we may assume ~ to be ~0 • Let a = C(x) u C(y) and a is Z~
transitivs. Since for transitive Axea
Ayea[}
~
Rel(a~)]
sets a and E0 -formula
Nence assuming
~(x,y) we get Rel(a,%)
Thus
is true in the structure (a,E[a,y)
Vx~(X,y)
~(x,y):
(proof by induction on the length of %). and therefore
Rel(a,Vx}(X,y)).
= ~.
Obviously
37
the axiom of extensionaiity
(I) is true i n ~ .
By the L6wenheim-
Skolem theorem here is a subset b of a such that C(y) ~ b,
=b ~ Max{~0,d(y)} --in ( b , e ~ b , y )
= Max{M0, HC(y)}
(here (AC) is used,
such that V x ~(x,y) ^ (I) is true since the languages appropriate to
these structures have besides the usual ZF-symbols a constant [ for each seC(y)) By the axiom of foundation,
e~b is well-founded and hence by the
preceeding !emma of Hostowski set D such t h ~ f
is an isomorphism between (b, e~b) and (D,g[D) and
f(y) = y, id est (b,E[b,y> (b,g~b,y>~
there is a function f and a transitive
~{D,e~D,y).
Since
Vx~(X,y)
holds in
it holds also in ( D , g [ D , y > since ~ is z~F~ hence:
ReI(D, V ~ ( x , y ) )
and there is an red such that Rel(D,%(v,y))° Since E0ZF we conclude, as before, that #(v,y) holds.
D is transitive and %
It remains to show that HC(v) ~ Max{M0,HC(y)}. HC(v)
= ~
But
~ ~ = ~ < Max{M0,HC(y)}
since veD, hence C(v) C D since D is transitive,
further ~ = ~ since
f is a bijection from b onto D, q.e.d. Theorem (GSdel):
In ZF + (AC) it holds that for every initial ordinal
> m it holds that P(e) ~ L C L(e+)~ wheme P(~) is the (whole) powerset of e and ~+ the next cardinal after e. Proof.
First remark that every ordinal is constructible
(same proof
as in section B). If yeP(~), y C ~, then ~ ~ e and HC(y) ~ ~. ZF Consider the formula Ord(x) ^ ysL(x). Since Ord(x) is E 0 and ZF ygL(x) is At (see lermma 4 and corollary 11 in sections C and F), ZF the formula under consideration is E z (by lemma 1 in section C). If yeP(a) N L, then
Vx(0rd(x)
x[HC(x) ~< Max{M0,HC(y)}
^ yeL(x)) and hence by T" ~evy's theorem:
^ 0rd(x)
^ yEL(x)] . Since HC(y) < e and
~ e, we get HC(x) ~ ~. But x is an ordinal, hence ~ ~< ~. Thus yeL(x)
for x < e+. The sequence L(o),L(1),...,L(B),...
hence L(x) C L(e +) and therefore ysL(~+), Co__rrpllary i. ZF Coroilary Proof.
~
2. ZF ~
is nested,
q.e.d.
V = L ~ for every initial ~, P(~) ~ L(e+). V = L ~ (GCH).
We know already that
(AC) follows from V = L in ZF. Hence
38
given any set x~ x is equipollent to an initial ordinal ~ and P(e) g L(e +) implies ~(x) : P ( ~ P(~)¢L and therefore P(~)EL(~)
<~--~W)
: ~+. But V = L, hence
for some 8 ~ e+. But then P(~) C L(y)
for some y < 8 ~ e+. But P(~-~ < L(y) = ~ and therefore y < e+ cannot be the case (we would get ~ = ~ < P - - ~ ~ ~ : ~). Hence y = ~+ and P(x) ~ ~+ as desired, q.e°d. So far we have shown, that (in ZF) V : L implies the (AC) and the (GCH). Now we shall return to our model constructed section B. In order to establish that
in
(AC) and (GCH) hold in it
we need only to show, that V : L holds in it° Lemma
Let ~(xl,...,x n) be a E?F-formula ~ and C a transitive class,
then
zr
A,
c...Axn
6
*
Proof immediate from the fact that ReI(C,~) ~ ~ for z~F-formula ~. Lemma:
ZF If ¢(x1,...,x n) is a NI -formula and C a transitive class, then
ZF [- f x ¢C..O A x aC[e~ -+ Rel(C,q~)] !
n
(dual to the previous lemma)o Corollary:
ZF If ~(x1~...,x n) is A, and C a transitive
ZF ~ A x , ¢ C - . - A Absoluteness-lemma
class, then
x gC[@ ~-~ Rel(C,9)]. n
for L: ZF ~ Ax[XCL ~ R e i ( L , x E L ) ] ,
Proof.The formula xeL is defined to b e Y , (Ord(y) ^ xsL(y)). Since Ord(y) is Z ZF 0 and xeL(y) is A~ZF both formulae are L-absolute according to the corollary,
resp.
a remark made above.
Hence the relativi-
zation of xeL to L reduces to V. (Ord(y) ^ xeL(y) ^ yeL) under the y assumption xeL. But since ZF ~ ~ y ( O r d ( y ) ~ yeL) we see that ReI(L,xeL)
~ x¢5~ q.e.d.
Corollary:
The axiom V =L holds in the submodel < U M ,E).
This follows~
since the predicate
"x is construct ibie " " is absolute
39
with respect to the class L = U M . This finishes the proof for G6del's consistency theorem. GSdel's original proof for P(m ) ~ L ~ L ( ~ + l)
(see |Z~] p. 221)
was by direct computation that the order of any constmuctible set m of w e is strictly less than m~+l arguments). clarified)
The difficult
sub-
(using some cardinality
proof of GSdel has been simplified
(and
later by using the LSwenheim-Skolem theorem (for such
a proof see either P.J. COHEN's book: Hypothesis"~
"Set theory and the Continuum
Benjamin.New Y o r k - ~ s t e r d a m
book "Th6orie axiomatique des Ensembles" Univ. de France,
Paris 1969, p. 115 ff.).
1987, p. 95 ff, or the by J.L. KRIVINE
(Presses
I think that the most
elegant proof is by using L6vy's theorem and the Z~F-definability of L, which was discovered by R.B. JENSEN and C. KARP [~Z].
H) A Theorem of H. Putnam and a Lemma of G. Kreisel Is there always a constructible
set of order ~+I which is not of 9 order e, when e is less than ~ . H. Putnam has answered this question in the negative~
in his paper:
[6~] H. PUTNAM: A note on constructible
sets of integers~
Notre Dame J. of Formal Logic~ vol. 4 (1963) p. 270-273. It is clear that there is always a constructible
set of order e+l
which is not of order ~, for M e itself is a member of M~+ 1 but not of M e. But there is not always a set of integers
in M~+ 1 which is not
already in M e. In fact~ Putnam shows that it can happen~ that "for a long time"
(a &*-ordinal)
we get no new sets of integers
in the
2
hierarchy of constructible appears.
sets, and then "pop!" a new set of integers
In order to state the results of Putnam we have to explain
the notation ~ .
We know, that L contains all ordinals,
know whether m, is absolute
(m0 is certainly absolute,
Z ZF) and hence the first uncountable 0
by
L
ordinal
but we do not since x=~ 0
ml, may be smaller than the first uncountable ordinal
of the whole universe V.
is
in the sense of L, denoted in the sense
40 I
Theorem
(H. Putnam)
(in ZF): These are ordinals e < m~, such that
there is no set of integers
in L(a+l)-L(e).
Proof: Consider the structure (L(m +2)~e)
and let T be the set of
....
I
sentences true in this structure.
By GSdel's theorem P(~0 ) N L ~ L(m,),
hence every constructible
set of integers
is constructed at some level
B for 8 < ml' Thus L(ml+l)-L(~ l) does not contain any set of integers. Therefore the following statement
(abbreviated by 4);
V [Ord(a) ~ ~x(X ~ ~^xeL(a+l)-L(a))] holds in (L(~,+2),e) , which is a substructure of
Now apply the
LSwenheim - Skolem theorem in L and get a substructure (D,e) (L(~:+2),£) , which is countable sentences of T are valid. By Mostowski's (D*,e)
are isomorphic.
Write $ ~ ~ ( ~ ) , a
is isomorphic to a transitive
and ReI(D,¢) ~ ReI(D*,¢),
set D*.
since (D,e)
and
Thus we know that ReI(D*,¢) holds.
then ReI(D*,#)
a e D*. N o t l ¢ ~
that # is a
Thus ReI(D',¢)
implies ¢ by
for transitive ~lasses.
in the sense of L and in which all
Hence in particular ~ holds in (D,e).
lemma (D,e)
But (D,e) ~ ¢ *ReI(D,~)
of
implies ReI(D*,~(~))
~ZF
I -formula
and that ~(a)
for some ordinal is A~ F, hence ~ZF !
one of the preceding absoluteness-lemmata
Hence
ReI(D*,¢) -- V ReI(D*,~(a)) ~ V ~(e) esD* aeD* But D is countable
in the sense of L and D" (in L) isomorphic to D.
We get that ~eD" ~
"a is countable in the sense of L" and we infer L that these are ordinals a < m, such that there is no set of integers in L(a+l)-L(e), q.e.d. H. Putnam has even obtained a stronger result. Kleene's analytical, hierarchy.
Let A 2i be Z*z N ~ 2 in
Noiice that every A~-ordinal
is
countable and ~ m~ (Shoenfield). Theorem
(H. Putnam [69~
p. 271): Let ~ be a A~-ordinal.
There is an
ordinal ~ < L ! such that there are no sets of integers L(~+y)-L(y)
in
for any y ~ 8.
The proof is like the one given above but using e+8 instead of ~+I throughout.
No£1¢e that A l-ordinals 2 are absolute for transitive models
A set ~ is called a At-ordinal 2
iff the ordertype of the ordinal e is a
A~-wellordering of a set of natural numbers.
41
So far we have discussed
questions concerning the constructible
hierarchy and the strength of the axiom of constructibility have shown, that "V=L" implies the
whether V=L enables to ~rove in ZF more numbertheoretic one can prove in ZF. A numbertheoretic which all quantifiers unrestricted
are restricted
addition,
theorems than
formula # is a ZF-formula
(see Feferman ~5]
formula ¢ contains besides
logical symbols and brackets only symbols for ordinal ordinal multiplication
and the successor relation.
has r e m a r k e d that ZF and ZF+V=L have the same numbertheoretic (G. KREISEL:
Some uses of Metamathematics;
Philosophy of Sci. 7 (1956/57) p. 161-173 G. KREISEL: 371-391
La pr~dicativit~,
(see p.
Bull.
theorems
British Journal f. (see p. 165), and
Soc. Math.
Every numbertheoretic
vable in ZF • ¥ ~ L is provable
France 88 (1960) p.
formula # which is pro-
in ZF.
Proof. Suppose there is a numbertheoretic ZF + V = L ~ ¢
G. Kreisel
386)).
Lemma:(G.Kreisel):
model~
in
to e such that the corresponding
formula is a formula of Peano-arithmetic
p. 50, e.g.). Hence a numbertheoretic variables,
"V=L" and
(GCH). Now we pose the question
~ut Z F ~ ¢ .
formula ¢ such that
Then ZF + ] ¢ is consistent and has a
m (M,R) . Since~
need not to be standard we cannot
apply G~del's theorem prima facie in order to obtain a submodel of ZF + V = L. But a careful inspection of the proof shows that in fact a l s o ~
has a submodel ~
that N is a R-transitive
~ (N,R)
subclass of M. Hence ¢ holds in ~ .
But # is Z~ F since ordinal-addition,
multiplication
and ~ has only restricted quantifiers. of N: R e l ( ~ , ~ ) are restricted
÷ Rel(~,@) to~
~ ~ implies R e l ( ~ , ¢ ) .
etc are A~ F
Hence by the R-transitivity
(this is a formula in which quantifiers
, respect to ~
~ ~, a contradiction
of ZF + V = L such
and e is replaced by R). But
Thus R e l ( ~ , @ )
which in turn implies
to our a s s u m p t i o n ~
] ~, q.e.d.
I) A Theorem of Shoenfield Our basic question was whether (AC) and (GCH) are provable or refutable in ZF. GSdel's result shows that neither be refutable in ZF. We do not yet know wether are provable or independent
(AC) nor (GCH) can (AC), (GCH) or V : L
from ZF. Since this question is difficult
to answer we shall ask whether all sets of integers are constructible
42 or even more restrictive w~ether all H* or Z* sets of integers in -n ~n Kleene's analytical hierarchy for n ~ k are constructible. How large is the upper bound k? We shall show that every Z~ and evemy K~ set of integers p o s s i b l ~ result.
is constructible
The r e s u l t s of Rowbottom,
that the existence ones)
(J.R.Shoenfield)~
of some large
cardinals
This is the best
Gaifman an@ S i l v e r s h o w (e.g. m e a s u r a b l e
imply the e x i s t e n c e o f n o n c o n s t r u c t i b l e
S o l o v a y p r o v e ~ that in the set theory w i t h large cardinals 0~
o£ ~ G ~ d e l - n u m b e r s o f sentences
here by results ~umber~
of J . S i l v e r ) i s
see: R. SolovaY-
true in
a n o n c o n s t r u c t i b l e ~, ~ ' 3 - s e t
i s contained
[76] J.R. SHOENFIELD: Foundations
R.Mo
the ~ set
L (which can b e defined of natural
A n o n c o n s t r u c t i b l e A~-set ~ i n t e ~ e r s ;
Amer. Math. Soc., vol. 127 (1967) p o 50-75. are g o i n g to prove,
or Ramsey
s e t s of integers.
Tr~a~s.
- S h o e n f i e l d ' result, w h i c h we ~
in:
The p r o b l e m of predicativity;
of M a t h ' s , d e d i c a ~ e d
In: Essays o~ the
to AoFraenkel, J ~ r u s a l e m
1962,
p.132 - 139. Definition:
a set of integers or a set of sets of inte-
gers is Z I (~l respectively) iff X is definable in the standard ~n ~n model of the 2-nd order arithmetic by a formula in prenex form having n alternating tential
(universal,
2-nd order quantifiers
beginning with an exis-
resp.) one. In the defining formula are parame-
ters allowed.(s¢~: [ ~ ] F ' I ~ 3 ~£) Theorem (Shoenfield): function
If #(e,8) is ~ - p r e d i e a t e ,
K n L ~ ~, if the parameters Outline of proof:
8o a constructible
(from ~ to ~) and K = {~; Q(e,So)} ~ ~, then lie in L.
By a result of Kleene
(Amer.J.Matho77(Ig55),p.405-
428, see § 28) there is a number e such that (')
~(e,8) ~ (R ~'~ is a well-ordering)
where R ~'8 is the predicate which is recursive number e. Since arithmetical predicates a
w e l l - o r d e r i n g ~' is a g~F-formula,
in e,8 with GSdel-
are absolute and since "x is
we conclude that Q(s,8)
is ab-
solut with respect to the transitive class L. Shoenfield shows that V #(e,8)
is also absolut.
V~%(~,80)
Then we have for constructible
~ ReI(L,V#(~,80))
-- ~ L R e I ( L ; * )
"
~L*(~,80
80: ).
This proves the theorem. Corollary:
If ~(e,8,y) parameters
is an arithmetical
predicate such that all
in # lie in L ,and Yo is constructible~
then
43
A VaeLAseL~(~,B,y0). structible :
Again using the theorem, we have for ao c n-
V~ -1 ~C~o ,B,~o ~ ~
VB~ L 7
~C~o ,~,~o ~.
Contraposing and adding a quantifier V a i L ,
we get
This proves (I), and (2) follows by duality. Theorem (Shoenfield):
Every Z~- mm~ every ~
constructible Proof:
-sub~e% of
is
is arithmetical,
then
a
set.
It follows from (1) that if ¢(a,B,x) {x; V a ]~8¢(a,B,x)}
: {x; V a e L
A 8eL@( a,B,x)}e L(~)
for some ordinal E, computed as in the proof for the replacementa~iom in L. Analoguously the theorem follows for hl-sets from (2) of the corollary,
q.e.d.
Remark. Using the Kondo-Addison theorem Shoenfield has furthermore shown that the A~-sets of integers are just the constructible sets of integers whose orders are A~-ordinals.
Another characterisation
of A~-sets is contained
Transactions AMS 117(1965)
in B.Scarpellini,
p.441-450. J) REMARKS P.J.Cohen has shown that, if there are transitive standard-models of ZF, then there is a minimal such model (see h~s paper: A minimal model for set theory; Bull.Amer.Math. Soc.69(1963)p.537-540,
see
Mostowski's
review in the Math. Reviews 27(1964)~41,p.9).
consequence
is that if ~ is a ZF-formula whose consistency with ZF
The main
is provable by means of the minimal model or GSdeis constructible model, then there is no ZF-formula F(x) defining a transitive standard~odel
such that ZF ~ Rel(G, ~ #) if G = {x; F(x)}. Thus the
method of transitive
inner models will not give a proof for the in-
dependence of the (AC) or the (GCH) or "V = L". For a nice presentation of this result of P.J.Cohen see also C.KarpI~]p.18-21, the paper:
and
44 [65] A.MOSTOWSKI:
On models of ZF-Set Theory satisfying the axiom
of Constructibility;
Acta Philosophica Fennica, Fasc.18
(!965)p.135-144. A proof for ZF + V = L ~ "there is a A~-set which has not the properry of Baire"(theorem announced by GSdel in | ~ ] )
has been published
by A4dison: [ ~ ] J.W.ADDISON:
Some consequences
of the axiom of constructibility;
Math.46(1959)p.337-357. Another interesting paper in this area is: V.A.LJUBECKII: laries of the hypothesis of noncountability tible real numbers
Some corol-
of the set of construc-
(Dokl.Akad. Nauk. SSSR - translation in: Soviet
Math. Dokl.vol.g(1968),p.1195-1196).
45
Additions to chapter I 1) The result, that the schema of foundation (VIIi) does not follow from the axiom of foundation (VII) in Zermelo's set theory Z has been obtained independently by R.B.Jensen and Schr~der (see p.17) and by M.Boffa: Axiome et schema de fondement dans le syst~me de Zermelo; Bull.Acad. Polon. Sci~ vol.17(1969)p.113-115. 2) McNaughton (Proc.Amer.Math. Soc.5(1954)p.505-509) has proved that neither Zermelo's set theory Z nor Zermelo-Fraenkel's set theory ZF are finitely axiomatizable. Montague has extended McNaughton's result by proving that neither ZF nor any consistent extension T of ZF without new constants is a finite extension of Z (see p.20). R.Montague has fi?st announced his result in abstract: ZermeloFraenkel set theory is not a finite extension of Zermelo-set theory; Bull.Amer.MathoSoe.62(1956)p.260.
We have given reference to a
paper of K.Hauschild in which McNaughton's result has been obtained by different methods. There is a further paper by K.~auschild in which the non-finite axiomatizability of ZF is proved: K.Hauschild: Das Zermelo-Fraenkel'sche Axiomensystem der Mengenlehre; Wiss.Zeitschr.der Humboldt Univ. Berlin,Math.Nat.Reihe~ vol.15(1966)p.1-3. In this paper Hauschild shows that every finite subsystem of ZF possesses countable models the domain of which can be denumerated by a formula belonging to the formalism of ZF. Additions to chapter ii A modification of GSdel's consistency p r o o f | ~ ] h a s
been published
by L.Rieger: L.Rieger: On the Consistency of the generalized continuum hypothesis. Rozprawy Matematyczne~ voi.31 Warszawa 1963 (45 pages)~ Polska Akademia Nauk.
CHAPTER I!I
FRAENKEL - MOSTOWSKI - SPECKER MODELS This chapter is devoted to the study of the socalled "Bermutation Models". They provide a method for obtaining independence results. It was for the purpose of obtaining the relative consistency of 7 (AC) and negations of some weakened forms of (AC) that A.A.Fraenkel first introduced the notion of a permutation model: |202 A.FRAENKEL: Der Begriff "definit" und die Unabhingigkeit des Auswahlaxioms;
Sitzungsberichte d. Preussischen Akad.
Wiss.,Phys.Math. Klasse,vol.21(1922)p.253-257. |Z~] A.FRAENKEL: Ueber eine abgeschw~chte Fassung des Auswahlaxioms; J.S.L. ~(1937)p.i-27. |Abstracts: C.R.Acad. Sci.Paris vol. 192(1931)p. I072, and: Jahresberichte der DMV 41(1931) part 2, p.88]. At that time (1922) the distinction between Mathematics and Metamathematics was not yet clear enough and Fraenkel's proof fails in so far as questions concerning the absoluteness of certain notions are not treated. This failing has been observed and correcied by A.Mostowski, partly in collaboration with A.Lindenbaum: |5%]
A.LINDENBAUM
-
A.MOSTOWSKI: Ueber die Unabh~ngigkeit des Aus-
wahiaxioms und einiger seiner Folgerungen; C.R.Soc.Sci. Lettr.Varsovie, ClasseIII, vol. 31(1938)p.27-32. 16~] A.MOSTOWSKI: Ueber die Unabh~ngigkeit des Wohlordnungssatzes vom Ordnungsprinzip; Fund.Math.32(1939)p.201-252. Fraerlkel has acknowledged the corrections and refinements obtained by Mostowski (see Fraenke!'s review of |54]
in the J.S.L.4(1939)p.
30-31). The major deficiency of the Fraenkel-Mostowski method is that it does not apply to "ordinary" set theories, id est, to set theories in which all the elements are sets and in which the axiom of foundations holds. The Fraenkel-Mostowski method is applied to set theories which admit the existence of "Urelements", id est objects which ar~e not sets (have no elements and are distinct from the empty set ~). These set theories are obtained from "ordinary" set theories (e.g. ZF) what essentially amounts to a weakening of the extensionality and dropping the axiom of foundation. Such a set theory, if obtained from ZF, will be called ZFU (the "U" indi-
47
cating:
w i t h urelements).
E.P.Specker.
The FM-method
His method,
towski-Specker
method),
exis%ence
of unfounded
sequences
of the type
@he soca!led applies
sets,
has been ameliorated
FMS-method
or other similar phenomena [SZ]
E.SFECKER:
which permit
id est, they permit
the
the existence
of
(n E ~)
such as x = {x}~
Zur A x i o m a t i k
Auswahlaxiom);
(Fraenkel-Mos-
to set theories
..... 6 x n e .... E x3 e x~ £ x,
by
der Mengenlehre
(Fundierungs-
und
Zeitschr.f.math. Logik und Grundlagen
d.
Math.vol.3(Ig57)p.173-210. The set theories from
to which the FMS-method
ZF by w e a k e n i n g
of these methods Theory" [~9]
the axiom of foundation
are obtained
only.
e.g.
For a discussion
we refer our reader to the book "Foundations
by A.Fraenkel
A.LEVY:
applies
- Y.Bar-Hillel
(Amsterdam
The Fraenkel-Mostowski
in Set Theory;
Method
in: Symposium
of Set
1958)p.49-54,
for Independence
on the Theory
and to
Proofs
of Models
(Am-
1965)p.221-228.
sterdam
In the sequel we present that the existence
Specker's
of "reflexive"
method.
We start with a proof
sets x = {x} is relative
consis-
tent.
A) THE INDEPENDENCE
OF THE AXIOM OF FOUNDATION
The notion
of a w e l l - f o u n d e d
considered
by D.Mirimanoff
and vol.2_~l(1920)p.29-52). has added to the axioms der Beschr~nktheit) the formulation (Frankel
set (ensemble
A.Fraenkei of Zermelo
asserting
(Math.Ann.8_66(1922)p.230-237)
an axiom of restriction
that every
it in a way similar
kels paper
in the Journal
by the axiom:
x ~ @ ÷
list of ZF-axioms).
f.Math.141(I911)
1899,
Vollst~ndigsee also Fraen-
p.76).
(1925) who replaced
Vy(y
Fraenkel's
e x ^ x N y = @), (axiom[vii)
J.v.Neumann's
but
the Z F - f o r m a l i s m
to Hilbert's
of the ~'Grundlagen der Geometrie~ yon Neumann
(Axiom
set is well-founded
not within
keitsaxiom
It was Johann
was first
(L'Ens.Math.19(1917)p.37-52,p.209-217~
was u n f o r t u n a t e l y
formulated
ordinaire)
axiom excludes
"axiom" in our
the existence
of
"extraordinaire ~' sets as it was the aim of Fraenkel~s
axiom,
is formulated
independently
introduced
within
the Z F - f o r m a l i s m
this axiom and introduced
for it (Zermelo,
Fund.Math.166(i930)
. Later Zermelo the name
see p.31).
and
"Axiom der Fundierung" The consistency
of
48 the axiom of fundierung
(VII) with ZF ° was proved by J.v.Neumann.
The consistency of 7 (VII) with ZF ° was first proved (independently, using different methods)
by P.Bernays [J.S.L. 19(1954)
see p.8a-84 -
this result was announced by Bernays already in 1941(J.S.L. p.10]
and by ~.P.Specker
in his Habilitationsschrift,
6, see
Z~rich 1951,
see [8~]. The method for obtaining the indepence of the Fundierungsaxiom (VII) has been modified and simplified by many authors [ e . g . L . R i e g e r : 357; P.H~jek: M.Boffa:
Czechoslovak Math.J.[(1957)p.323-
Zeitschr.math. Logik u.Gr.Math.ll(198S)p.103-115;
Zeitschr.ma%h. Logik u.Gr.Math.1__~(1968)p.329-334].
of nice independence results [ %]
in the last 14 years
M.BOFFA:
Les ensembles
is contained
A number
in:
extraordinaires;
Bull. Soc.Math. Belgique,
vol.20(1968)p.3-15. We s h a l l
present
Bernays'
independence
proof
but
with
the
modifica-
tions due to L.Rieger and M.Boffa. Permutations
of the universe.
no free variables
Let ¢(x,y) be a ZF-formula having
except x and y. Suppose ~ satisfies the following
conditions: (P.1)
ZF ~ ~ x A y ~ z [ ~ ( x , y ) ^
zr
Ax A
At
cx,y) ^
, a ,
xVy, X,y
¢(x,z)-y
cz,y
: z]
+
x = zl
^ AyVx, X,y .
Then ~ is said to define a permutation of the universe V. In this case define the class-term F = {(x,y ) ; ¢(x,y)} and ( x , y )
E F ~ y =
F(x). Define a binary predicate 6 F by x E F y Def.~ F ( x ) For a ZF-formula
~ let
~ y ~
Rep(F,~)
Vzl~(x,z) be the
^ z c yl
formula
obtained
from ~ by
replacing the symbol c by 6 F at all places of occurrence. T h e o r e m 1. Suppose F = {(x,y ) ; ¢(x,y)} defines a permutation of the universe,
then Rep(F,-)
ZF, id est ZF Proof: Ad(O):
~Rep(F,~)
is a syntactic model of ZF ° in for every axiom ~ of ZF °,
The real empty set B = F(a) satisfies the requirements
of the relativized
axiom of Null-set.
Ad(I):
suppose that z E F x
z E F y for all z. We prove x C y: Take a e x. By (P.3) a = F(b) for some b. Hence F(b) e x, i.e. b E F x. The hypothesis
yields
b E F y, hence a = F(b) e y. The part y C x is proved analogously. By the axiom of extensionality
of ZF we get x = y as desired.
49
Ad(II):
Let x and y be given.
F(u) e z. Hence either F(u) holds by (P.2)
6F y ~
: F(x) or F(u)
Vu(Z6 F
Vu(F(z)
But the latter
Kd(III):
e u ^ F(u)
u ^ u6 F x) as required.
Then u 6 F z ~
= F(y).
iff either u = x or u : y.
given and put y = {F(z); z
Put z : {F(x),F(y)}.
Let x be
e x)}; then
The set y exists by
virtue of the axioms of union and r e p l a c e m e n t
of ZF. Ad(IV):
Define
f(O)
(in ZF) a function
{F(f(O))},
f(n+l)
f with domain m by:
= f(n) U {F(f(n))}.
Finally,
i E ~}. We have that z e F x ~ z : f(i) f(i) UF{f(i)} F i.e.
, w h e r e C F expresses
"Inclusion"
set y = {F(t); fore:
of the r e l a r i v i z e d
For a set x the power-set t _C x}.
In fact:
u e F z e F y ÷ F(u)
u E F x. Ad(VI): variables
e b* -
axiom of infinity.
in the sense of the model
is the
z E F y ~ z _C x (by (P.2)),
there-
x and y and let ~(x,y)
any set. Define a* = {F-l(x);
Ay[y
Hence x satisfies
be a Z F - f o r m u l a
for every x there is p r e c i s e l y
(of ZF) c o r r e s p o n d i n g
for some i and f(i) C F f ( i + l ) =
e z ^ z _C x. Hence F(u)
Let F(x,y)
e x, i.e.
with p r e c i s e l y
be Rep(F,F(x,y)).
x e a}°
is a set b* such that
e a* ^ ~(u,y))] then b satisfies
cement
to ~ and a. Thus the t h e o r e m
axiom c o r r e s p o n d i n g
2: Suppose F : {(x,y ) ~ ~(x,y)} the universe
V. If ( V , e )
then (V~6 F > satisfies ZF + (AC) Proof:
Consider
~ Rep(F,
the c o r r e s p o n d i n g
satisfies
of
the a x i o m of choice,
the a x i o m of choice too,
id est:
of s are not F-empty and pairwise that
~'notion in the sense of the model ~' is meant,
id
Define
a* is a set of non-empty~
e y}; F(y) pairwise
g a} disjoint
is a set b~ such that:
the condition:
sets
(with respect
b* N y is a singleton
each y s a*. Now b : {F(x); x e b*} i s a satisfies
a permutation
"F" in front of a qotion N indicates
a* = {{x; F(x)
By (AC) there
defines
repla-
is proved.
a set s such that s is ~'not empty" with respect to
(the suffix
est: Rep(F,N)).
the r e l a t i v i z e d
(AC)).
6 F and such that the F-elements F-disjoint
Let a be
By the a x i o m of r e p l a c e m e n t
Define b : {F(y); y e b*},
Theorem
two free
Suppose that
one y such that ~(x,y).
to ~ and a* there
Vu(u
=
proper F - i n c l u s i o n
with respec% to 6F).
the r e q u i r e m e n t s Ad(V):
= ~, f(1)
define x = {F(f(i));
F-choice
to e).
for
set for a, i.e.
50
Ay[Y
e F a ÷~z
(zeF b ^ z EF y ) ] .
~ h i s p r o v e s theorem 2. Definition.
A set x is c a l l e d r e f l e x i v the c o n d i t i o n of r e f l e x i v e
Ay(y sets
iff x = {x},
id est
e x ~ y = x) holds. contradicts
T h e o r e m _3. If ZF ° is c o n s i s t e n t ,
iff
The e x i s t e n c e
the a x i o m of F u n d i e r u n g .
t h e n ZF°
+ . ~ / ~. z
= {x})is a l s o
consistent. Proof.
Let
#(x,y)
(x ~ 0 ^ x ~ 1 ^ universe
be the f o r m u l a x = y).
~ defines
s u c h t h a t o n l y the o r d i n a l s
F be the o n e - t o - o n e F(0),
Hence
(x = 0 ^ y = 1) v
hence
1 ~
of ZF ° + V x a theorem
function
1 and 1 :
x = {x}
a permutation
0 and 1 are ; #(x,y)}.
{1} F. H e n c e R e p ( F , - )
in ZF. The r e l a t i v e
in c h a p t . ! ,
Corollarv:
{(x,y)
s e c t i o n D. p a g e
If ZF is c o n s i s t e n t ,
(x = 1 ^ y = 0) v
T h e n F(1)
Hence
from theorems
consistency
follows
t h e n ZF ° + " n e g a t i o n
the a x i o m of f o u n d a t i o n Our n e x t q u e s t i o n
violated flexive
sets or w h e r e
nals
Both questions
E.P.Specker
Theorem
and
Consider
in all o t h e r
lar we have: Hence
F(0)
c a n be
set of re-
c l a s s of r e f l e x i v e
sets
w i t h the c l a s s of all o r d i in the p o s i t i v e w a y by
t h e n ZF ° + " t h e r e
exists
s u c h t h a t R is e q u i p o t e n t
a set R
w i t h m" is
too.
F(x)
h e r e ~*
{x} is n e v e r
F(1)
{3} eF { 3 } ,
permutation
F of the u n i v e r s e :
= y iff x = {y} for y e m" and F(x) is d i f i n e d
to be ~ - {1}
= 0, F(2)
= 0 e 1),
etc.
= {2},
id est:
and o b v i o u s l y
F({2})
In p a r t i c u -
= 2, etc.
{0} E F {0}o
F(x)
= x
= {0,2~3,4~..}.
in ~* and F is w e l l - d e f i n e d .
= 1, F(1)
1 E F 1 (since
{2} e F {2},
sets
the f o l l o w i n g
cases;
f r o m ZF ° + (AC).
[~]).
= {x} iff x ~ m,,
F o r x e m*,
result.
Zeitschr.math. Logik u.Gr.d.Math.
4: If ZF is c o n s i s t e n t ~
consistent
consistency
a countable
a proper
have been answered
of r e f l e x i v e
Proof:
exists
exists
correspondence
(see a l s o M . B o f f a ,
14(1968)p.329-334
+ "negation
too.
is w h e t h e r the a x i o m of f o u n d a t i o n
there
is in o n e - t o - o n e
3 and G ~ d e l ' s
there
from
of the a x i o m
and ZF ° + (AS)
is i n d e p e n d e n t
in s u c h a f o r m w h e r e
which
F(x)
2 and
= 0 e 1 =
is a s y n t a c t i c m o d e l
of the a x i o m of f o u n d a t i o n ~' is c o n s i s t e n t follows
Let
9, q.e.d.
of f o u n d a t i o n ~' is c o n s i s t e n t
This
of t h e
interchanged.
Further
e ~* ÷ x = { F ( x ) } ={x~ F .
51
We shall show that there F-set of reflexive The unordered F(y)}.
The ordered
(x,y)F f(n+l)
F({F(x),
{n+2})F;
(see the proof
n ~ ~}F
between
n ~ ~}, the F-set of F-natural {n+2}
: {F({n+2})}
Corollary:
On*
sets such that there
between
the following
By t h e o r e m
holds.
Hence
(GCH).
This
permutation
is a one-to-one
of all ordinals
of the universe:
to add
like those
consult
the papers
be strengthened follows
proved
: x
an~
of theorem
is indepentent
3 the
(GCH)
from ZF ° + (ACk +
by adding V : L, since obviously
from V = L. For further
in theorems
of M.Boffa
(AC) Zo ZF ° in th.4 and
in the model
the axiom of foundatibn
results
and R"
as in the proof of th.4.
Further,
the axiom of foundation
a proper
: y iff x : {y} for y e On" and F(x)
2 it is possible
above.
cannot
sets
too.
: On - {1}. Now proceed
papers
the class
exists
Here On is the class of all ordinal-numbers
its corollary
: {F(f(n));
and some F-reflexive
class R of reflexive
: {x} iff x e On i, F(x)
Remark.
n e m}F
then ZF ° + "there
Consider
otherwise~
numbers,
n e ~}
If ZF is consistent,
is consistent
F(x)
The function
{n+2})F;
~F : {f(n);
: ~,
for n e ~.
correspondence
Proof.
are the sets f(0)
of th.1).
: {F((f(n),
in the F-sense
is hence
F(y)})}
in the sense of the model
: f(n) U {F(f(n))}
is a function
the
of x and y is {F(x),
pair in the sense of the model
numbers
g : {(f(n).
between
sets n = {n} F for n ~ I and ~. pair in the model-sense
: {F({F(x)}),
The natural
is a F - c o r r e s p o n d e n c e
(those
consistenc U
3 and 4 and its corollary
already
cited and Boffa's
in the C.R.Acad. Sc. Paris v o l . 2 6 4 ( 1 9 6 7 ) p . 2 2 1 - 2 2 2 , v o l . 2 6 5 ( 1 9 6 7 )
p.205-206~ Boffa's Theorem
vol.266(1968)p.545-546,
second C.R.-paper (M.Boffa): consistent,
contains
Let ( s , < )
the following
be any partially
then so is ZF ° + (AC)
set t such that ( s , < ) In particular
(s,~)
(e.g.
totally
a dense
vol.268(1969)p.205).
and ( t , e )
inaccessible
cardinal
ordered
ordered
+ "there
numbers
set.
If ZF is
is a transitive
are isomorphic". ordered
set
set).
it is known that
totally
fine result:
can be taken to be any linearly ordered
In particular
in ZF + (AC)
"the class
by C (see G.Sabbagh,
+ "There are strongly
of G r o t h e n d i e k - u n i v e r s a
Archly
d. Math.
is
2_~0(1969)p.449 ~
52
456).
U.Felgner
AxVy(~ there
< ~ ^ In(~))
it is consistent
that given
any partially
is a set t of Grothendiek-universa:
(t,C)
~@
has shown that
are isomorphic
M°Boffa-
(see U.Fg.Archiv
A weak axiom of foundation,
and
METHOD
with the existence
of re-
sets.
=U{P(Rs(A));
sets.
Define
that e ~ B ÷ Re(A)
C Rs(A)
transitiv.
It is not provable
that
in the preceeding
R0(A)
= A,
8 < e} for e > 0, and W(A)
One proves
section~)
as in the consistency every
set (s, ~ )
d.Math.20(1969)p.561-566:[i~],
compatible
Let A be a set of reflexive Re(A)
ordered
such that ( s , ~ )
G.Sabbagh [5] •
B) THE FRAENKEL-MOSTOWSKI-SPECKER
flexive
with ZF ° + (AC) +
for V = U
relative
(WF) A x i o m of weak foundation:
e e On}
and that all sets Re(A)
Vx(V
= W(x))
but it is consistent
proof
set is wellfounded
= U{Re(A);
Ve).
are
(see the corollary with ZF ° (same proof
In W(A)
it holds
that
to A:
__VA( A x X
~ ~ ÷
V,(y
e x ^ (y N x : ~ v
2
v Agy All models
considered f r ~ m n o w
this axiom. Ay(y
chapter)
the axiom
Vx(V __
{y}))).
will satisfy = W(x))
^
g x ÷ y : {y})).
Automorphisms mapping
of the universe.
of ~efiexive
be any permutation
~(x)
: {~(y);
The uniqueness follows can be defined: automorphism If w permutes
Filters
and let
from A onto A), ~* of
~ so that
and define
y E x}
: Min{e;
: (~,)-I
x C Re(A)}.
This
determined
and
of permutations
(~,~)*
of rank,
shows that every
by a permutation
of A.
then = ~;.
of A and the automorphism
are isomorphic
p(x),
group
and we need not to dis-
them.
of subsroups.
H a subgroup
Let x g Re+I(A)
from the fact that a notion
as Aut(V,e))
between
mapping
we have extended
A and ~* is its extension,
the group
e T(y).
for V, id est V = W(A),
one-to-one
suppose
T of V is uniquely
(-1),
of V (written
O(x)
of V is a one-to-one
in a unique way to an_ a u t o m o r p h i s m
is done by induction:
acts on all sets of Re(A).
tinguish
sets
of A (i.e.
then ~ can be extended V. This
An automorphism
T from V onto V such that x g y ~ T(x)
Let A be a basis
Hence
on (in this
From now on we assume
=
If G is any
(multipiicatively
of G, and g e G, then g-!Hg
is called
written)
group,
a conjugate
53
subgroup
of G, c o n j u g a t e
Definition.
A non-empty a filter
with respect
set F of s u b g r o u p s
(of s u b g r o u p s
conditions
H • F ^ g ¢ G ~
(ii)
H,
g-lHg
e F ^ H2
¢ F ~
that H1is
H, N H2
filter
determines
a model
Definition
F of s u b g r o u p s
of the m o d e l . ~ [ G , F ] .
{T(y);
y ¢ X}.
denote
the t r a n s i t i v e
shall
Obviously
of ZF °.
(Specker
÷ H[y]
=
let C(x)
: {x} O x U U x
U...
e F)}.
elements
of M and the m e m b e r -
Ad(0):
list
universe
some p r o p e r t i e s
V. We
(In ZF ° + V : W ( A ) ) : ~ [ G , F I
If x and y are
Since
F is a filter:
hence
HI {x~y}]
= G e F and
sets o f ~ ,
H[x]
N H[y]
0 C M,
Ux
~ ~(y)
T(x)
: x and T(y)
:
e ~(x)
for T e HIx].
{z;
Hence
Ux
the
@ e M and ~ s a t i s f i e s
T is
then H|x|
But
identical
= l(Ux)
H[x]
~::z
= ~{z;
e F and H|y] N H|y]
e F.
~ H[{x,y}] ~
s h o w that
For
z e y e x ÷
~ s H[xl: But
Y ¢ H[x]
e y E x) -- V y ( T ( z ) z e y e x)}
~-l(Ux)
C Ux
mapping):
= TT-I(Ux)
(e).
e F. We shall
an automorphism.
similar
from
a set o f ~ .
¢ y ~ x)}.
~ x. Thus:
T(Ux)
follows
e F. But H[x]
hence
Vy(Z
since = y'
¢ y e x)} c U x,
(if 1 d e n o t e s
in~
e F. By (8) is {x,y}
Let x be a set o f ~ ;
T(z)
-~ ( M , ¢ )
in~.
The a x i o m of e x t e n s i o n a l i t y
~HIUxl.
e F.
of ZF ° .
S i n c e H[ ~]
HIM
of {M,e ) .
class. of M, t h e n x e M iff H[x]
the a x i o m of N u l l - s e t
Ad(III):
T"x
of G. A g a i n
C(x)
is the one of the w h o l e
|SZ]p.196):
is a m o d e l
Vy(Z
is a s u b g r o u p
are thus
and F a f i l t e r
~ <M,E >
(e) M is a t r a n s i t i v e
Ad(II):
show
G of A u t ( V , £ )
T"x : x} w h e r e
of x, i.e.
But first we
(8) If x is a s u b s e t
A d(I):
. We shall
subgroup
that
is a m o d e l
Proof.
of H~
of any
Let G < A u t ( V , c )
e C(x)
of~[G,F]
~WLlc,r]
Theorem
H[x]
"model"~IG,F]
prove
¢ F.
¢ F.
= {T ~ G;
closure
M : {x; ~ y ( y
of the
three
of ZF ° .
on G. For any set x let H[x]
ship-relation
H2
a subgroup
that e v e r y
Sets
iff the f o l l o w i n g
e F.
S F ^ HI ~ H2 ^ H~ ~ G ~
H e r e HI ~ Ha m e a n s
Now d e f i n e
of G)
of a g r o u p G is c a l l e d
hold:
(i)
(iii) HI
to H.
C ~(Ux)
C Ux.
÷ s y ~ x)
= {~(z); follows.
Hence
54
Thus
T e H[Ux]
• Now by
(ii)
of the
filter-definition
H[Ux]
e F.
By (IB) U x ~ M. A d(IV): (II)
By i n d u c t i o n
and
(III)
requirements A_d(V):
one shows
already
proved).
and T e H[x] , then T(y) y C x and y e M,
The same holds
We shall
for T - 1 . H e n c e
in it.
of p r o v i n g we
First e
one
G:
If y C x
Moreover,
1 below)
if
and t h e r e f o r e
N M)
P(x)
= P(x)
that
first
N M (as above)
N M e M. The
set P(x)
axiom relativized
the r e p l a c e m e n t
schema
with
to~.
is true
that the A u s s o n d e r u n ~ s s c h e m a
to show that
and
N M
in
holds
no free v a r i a b l e s
if x~ ,...,XrfY ~ M
is a set z e M such that x e z ~ x e y ^ R e l ( M , ~ ( x , x l ,...,Xn))]
shows
by i n d u c t i o n
on the
length
of # that
for e v e r y
x,xl ,...,x n e M ÷ (Rel(M,~(x,x, ,..,x n)) ~ R e l ( M , ~ ( T ( x ) , . , T ( X n ) ) ) ) .
In our p r e s e n t
case,
since
H~xl] ~..,H[x n] ,H[y] we can a s s u m e under
C P(x).
Let ~(x,x, ,...,x n) be a Z F - f o r m u l a
Ax[
T
prove
P(x) N M e M.
lemma
of the p o w e r - s e t
directly
shall
the
T(P(x)
t h a n x,xl ,... ,x n. We have
then there
the
N M.
6] M] . Thus
s a t i s f i e s the r e q u i r e m e n t s
T(P~)
e M (see
N M) C P(x)
H[ x] C H[ P(x)
other
(using
~ e M and m s a t i s f i e s
show that
C_ x; h e n c e
then T(y)
T(P(x)
Instead
Hence
all o r d i n a l s
of the a x i o m of infinity.
Let x be any set.
the m o d e l
that M c o n t a i n s
that
e
x, ,..,x n,y e M, thus
F, h e n c e
H[x,]N..NH[Xn]
all of xl ,..,Xn,Y
automorphisms
~ E H0).
N H[y]
are H 0 - s y m m e t r i e
Consider
the
= H0
e F,
(i.e.
invariant
set
z = {x; x e y ^ R e l ( M , # ( x , x , ,...,Xn))}. In o r d e r
to show tha%z
H0-symmetrie. by
(8) that Hence
H0 ~ H[y]
by d e f i n i t i o n
T £ H0 ~ H[x i]
as in (III))~
and
hence
T(x i)
T e H0 ÷ ~(z)
shown,
since
that
e F which
z is
implies
that
to prove
: xi,
_C z. H e n c e
~-~[G,F] that ~
z g G, then
since
) , . . . , T ( x n)).
But
and t h e r e f o r e T(z)
= z (proved
q.e.d. is a m o d e l satisfies
a lemma
i. If x e M and
e T(y)=y
x g z we have
Rel(M,~(T(x),T(xl
(1 ~< i < n),
Z. In o r d e r
a x i o m we need
of H0. Also,
z is H 0 - s y m m e t r i e ,
So far we have
Lemma
to prove
(ii) H[ z]
T E H0 and x e z. Then x e y and T(x)
"[(x) e z. A l t o g e t h e r :
set t h e o r y
(8) e n o u g h by
z is in M. take
R e l ( M , ~ ( x , x l ~...,x n)), since
e M it is by
T h e n H0 ~< H[ z] , h e n c e
z(x)
e M.
of Z e r m e l o -
the r e p l a c e m e n t -
55
Proof
by i n d u c t i o n
x e M + H[x]
e F. We c l a i m that
H[T(x)I Hence Thus
on the e - r e l a t i o n .
> T H[x]T -I.
take o e H[x] . T h e n
(TOT-1)T(X)
T~T "I e H IT(X)] . It f o l l o w s
definition
that H[~(x)]
x g Ro(A))+, then in Ro(A)~
hence
y £ C(T(x))
T(x)
(i) and
x C M, h e n c e
T(x)
C M. Thus,
(id est of T also
+ y = ~(x).
g F and we get x e M.
~ M for all y E RB(A)
= T(X).
(ii) of the f i l t e r
of the a c t i o n
But y e C ( T ( x ) )
= H[T(x)]
= tO(X)
if x is r e f l e x i v e
is by d e f i n i t i o n
reflexive.
÷ H[y]
and y ~ M ÷ T(y) implies
e F. Now~
by
= TO(T-IT)(X)
Hence
If x e Re(A)
for 8 < e, then x g M
by
(8), H[T(x)]
e F implies
• (x) g M, q.e.d° Now we r e t u r n presence
to the p r o o f
of the a x i o m
schema
of r e p l a c e m e n t
A
AAwl+(u,v) ^
schema
of S p e c k e r ' s
of s u b s e t s
is e q u i v a l e n t
+ v :
theorem.
In the
(Aussonderung)
the a x i o m
to the s c h e m a
+ AyV
AuAv(U
y^ v
where with
#(u,v)
is a Z F - f o r m u l a .
no free v a r i b l e s
xl ~...,x n
e
other
M and all u , v , w
Rel(M,~(u,w))
implies
t : {v g M;
in ZF,
by the r e p l a c e m e n t a x i o m
Further
t C M~ h e n c e
contains
the
id est T(Z) products.
the p r o o f
identical
Hence
H[z]
mapping:
sets
was
z : U{T(t);
1+ thus
T g G}.
z C M. Since G z is G - s y m m e t r i c ,
(8): ~ e M. Thus schema
constructed
holds.
that n e i t h e r
relative
is a s u b c l a s s
under
the r e l a t i This
finishes
to the set A of
of W(A)
A nor the e l e m e n t s
if the f i l t e r
e A ~ H[x]
e F]
= ~eRe(A)-
of A are always
. But in all a p p l i c a t i o n s
m e t h o d we just w a n t
condition: .Ax[x
Put
theorem.
in the m o d e l ~ [ G , F ]
(iv)
y e M. D e f i n e
t C z. F u r t h e r
such that~[G,F]
This w i l l be the case
. A s s u m e that for n Rel(M,~(u,v)) ^
since as a g r o u p G is c l o s e d
f o r m of the r e p l a c e m e n t
Mostowski-Specker
be such a f o r m u l a
z is a set since G is a set.
: G e F, and by
of S p e c k e r ' s
It is r e m a r k a b l e
tional
of ZF,
T(t) C M+ by !emma
The m o d e l ~ [ G , F ] reflexive
t is a set.
: Z fo~ all T e G~
vized weakened
sets
Let y be a set,
z).
g y ^ Rel(M,¢(u,v)))}
By the r e p l a c e m e n t a x i o m Again
M we h a v e that
e
v = w.
Vu(u
Now let ~(u,v)
than u,v,x, ,...,x
E
of the F r a e n k e l -
to h a v e A as a set i n ~ .
F satisfies
the
following
addi-
56
It is easily (M,E)
seen
(see the proof
and F satisfying
(since HI A] Further
(i),
of lemma
1) that
(ii)(iii)(iv),
for~[G,F]
-~
A _C M, hence A c M
= G e F) by (8), holds.
, if F satisfies
axiom of foundation
(i) .... ,(iv) then i n ~ [ G , F ]
(WF) holds
the w e a k
(if in the surrounding
set-theory
(WF) holds). By definition identical
mapping
Klxl
where
= {r
e G;
"t I ' x
for any set x:
Ix}
=
I x is the identical
of T to x. Remark K[x]
T e HI x] ÷ x"x = x, but x need not to be the on x. Define
on x and "r 1" x is the r e s t r i c t i o n
mapping
that always
K[ x] ~ H[x]
~ G. If H[x]
need not to be in F, but if K[ x] e F then there
ring of x in ~ [ G , F ]
, if the axiom of choice holds
e F then
is a wellordein the surroun-
ding set theory. Lemma
2: Every
T e G acts as the identity
M (] ~J~V~ Proof by induction Lemma
on the M i r m a n o f f - r a n k
3: (In ZF ° + (WF) of subgroups
well-founded
Proof.
contains
(AC) holds
in the surrounding
(i),
(it),
the axiom of choice
set theory
2: H[x]
= K[x|
(by lemma
is contained
in~[G,F]
are just the well-founded
set theory')
one-to-one
f from x onto e is also a w e l l - f o u n d e d
mapping
also i n ~ [ G , F ]
Lemma
4: (In ZF ° + (WF) of subgroups
and hence ~ is in ~ [ G , F ]
sets
and the set,
. + (AC)):
If G ~ Aut(V,c)
of G satisfying
(i),
set x of ~
[G,F]
one fashion
onto a well-founded
K[ x] E F.
= G e F.
2) every well-
of the surrounding hence
(iii),
for each
set theory x can be mapped
e. But obviously
sets of ~6~[G,F]
and F is a filter
relations
. Hence
sets x.
part of ~ [ G , F ] .
then by lemma
of the surrounding
(the well-founded
part
for well-founded
conditions
wellordering
set x of ~ [ G , F ]
one on an ordinal
founded~et
p(x)
If G < Aut(V,¢)
in the well-founded
If x is well-founded,
Since the one-to
+ (AC)):
of G satisfying
then ~ [ G , F ] holds
on the w e l l - f o u n d e d
of M.
can be mapped
and F is a filter
(it) and (iii)
in~[G,F]
then a
in a one-to-
set y of ~ [ G , F ]
iff
57
Proof.
a) Suppose that there
~[G,F]
is such a o n e - t o - o n e
from x e M onto a w e l l - f o u n d e d
f] e F it is sufficient
~((u,f(u)))
Hence,
:
take T e H[f]
~(f(u)))
: (T(u),
since f(u) e y, hence f(u) well--founded, lemma
2. But T e H[f] ÷ T(f)
(it) K[x]
= u. Thus T e K|x]
e f + T((u,f(u)))
f(u) ) e f. But f is one-to-one,
and H|f]
e F, H|f] < K|x]
and union hold
s of x w h i c h can be m a p p e d (e.g.
singletons).
~H(fs).
in ~
implies by
that for some
~ (M,E)
is a model of
in it, and there are subsets
one-to-one
For each
The set
of functions
suppose
e F. Since ~ [ G , F ]
ZF ° the a x i o m of p a i r i n g
theory)
: f(u) by
e F.
x e M we have that KIx]
of ~
f(u) )
hence T(f(u))
b) In order to prove the converse,
that K[x]
in order to
and u ¢ x. Then
= f, hence ( u , f ( u ) )
g f, w h i c h means by ('): (T(u), hence T(U)
f in
set y e M. Since f e M ÷
to show that H[f] ~ K[x]
verify that K[x] e F holds. (,)
mapping
onto w e l l - f o u n d e d
such map fs'
s C _ x, it holds
(in the sense of the s u r r o u n d i n g
fs' ordered
sets
by "is an e x t e n s i o n
set
of" is induc-
tively ordered and has t h e r e f o r e
by Zorn's
fo. The m a x i m a l i t y
that fo must be defined on the
whole
of fo implies
set x (this argument
ZF + (WF)
+ (AC)).
takes
place
lemma a maximal
in the s u r r o u n d i n g
But f, C f= ~ K[x] < H[f2]
Corollary
by (8), f o e
5: (In ZF + (WF) + (AC)). is w e l l - o r d e r a b l e
Proof.
If K[x|
ordered.Hence be w e l l o r d e r e d
iff K[x]
e F.
, then there
hence K[x]
C) THE I N D E P E N C E N C E
~ where
of x. If, conversely,
is i n ~
map-
a one-to-one
f and ~ are i n ~ .
x can
mapping
f
But e is well-
c F by lemma 4.
OF THE A X I O M OF CHOICE
In the set theory ZF ° + (WF) the r e f l e x i v e are all d i f f e r e n t
a one-to-one
set y. By lemma 3, y can be well-
f induced a w e l l o r e d e r i n g
from x onto an ordinal founded,
A set x of the m o d e l ~ [ G , F ]
E F, the there is by lemma 4 i n ~ L
in~
= x C M we
M, and lemma 4 is proved.
in~IG,F]
ping f from x onto a w e l l - f o u n d e d
set theory
< HIf, ]. Hence
K|x] < H[fo ] and H[f0 ] e F. Since Rg(fo) C M and Dom(fo) infer that fo ~ M. Hence,
element
sets in the base R0(A)
but not too much distinguished.
fact in the c o n s t r u c t i o n
of a Z F ° - m o d e l ~ i n
= A
We shall use this
which
(AC) does not hold.
58 But in order to ensure satisfies
that choice
some symmetries.
These
by the group G ~ Aut(V,c) Further
suitable
models
choices
fails
in~
symmetries
and the filter of groups
we require of ~
fragments
are determined
F of subgroups
G and filters
in which the "general ~' axiom of choice
that ~
of G.
F will give ZF °-
fails but certain
of (AC) are true.
The Model of A.Fraenkel. We shall
present
the model
with the modifications We work
constructed
due to E.P.Specker
in a set theory
Z~
+ (AC)
by Fraenkel
[SZ]
[~I]
but
(see [8~]p.197-198).
in which there
is a countable
set A of reflexive
sets such that the weak axiom of foundation
holds with respect
to A,
tive to ZF ° was
shown
id est V = U ~ R e ( A ) .
interchanges
rela-
in section A).
Let {a0,al ,a2 ,...} A transposition
The consistency
(WF)
= A be a fixed
of A is a one-to
just two elements
for all other elements.
enumeration
one mapping
of A.
~ from A onto A which
of A and is the identical
Call a transposition
mapping
~ of A "kind"
iff
interchanges a2k with a2k+l. If B = {{a2k,a2k+l}; k e ~}, then U B = A and every kind transposition ~ maps B onto itself. Now let G be the group
of those
permutations
duct of kind transpositions. set of subgroups
We shall
Obviously
H of G whose
F = {H; H < G ^ [G:H]
index
and
Lemma
(H.Poincar~):
are a fiDite
G is abelian.
Let F be the
in G is finite:
satisfying
conditions
(i),
The intersection
index has finite
of a finite
number of subgroups
index.
Proof.
Let G be any group and let H, and H2 be subgroups
finite
index.
Elements
iff ab -I e H~ N H2. Thus we obtain all non-empty
with right
of H2. Thus [G:HI
cosets
case follows
by induction
It is well-known and H ~ G, g e G filter,
background
all right
intersections ~ H2]
on the number
= [G:g-IHg].
conditions we refer
is finite.
(i),
(ii),
of
cosets
of H~
- The general
of subgroups.
Hence
the reader
cosets
of right
that H, < H2 ~ G, then [G:HI]
÷[G:H]
satisfying
of G of
a and b of G lie in the same right coset of
Ht ~ H2 by taking
oretieal
(ii),
(iv).
of finite
H, ~ H2
pro-
< ~}
show that F is a filter
(iii)
T of A which
= [G:H2]'[H2:HI],
our set F is in fact a
(iii).
(For the groupthe-
to "Group Theory"
by W.R.
59
SCOTT
(Prentice-Hall,
Inc.New Jersey
If a e A, then [G:H[a]] the groups H[a]
set B is a set of ~ [ G , F ] In ~ [ G , F ]
Proof.
hence
condition
for all a E A (iv). Thus the
sets is a set of the ZF°-model ~ [ G , F ]
Now B C M and since H[ B]
set
= 2 (obviously),
are in F and F satisfies
base A of r e f l e x i v e
Lemma.
1964)p.20).
= (M,E) .
= G e F we infer that also the c o u n t a b l e .
the set B = {{a2k,a2k+l} ; k e ~ } h a s
no choice
•
Suppose
there would be i n ~ I G , F ]
B, such that C contains {a2k,a2k+l}. T g G which
Then HIC]
precisely
a choice
one element
set, say C, of
from each set
would be in F. But the only p e r m u t a t i o n
fixes C is the identity.
Hence H[C]
subgroup E = {e} w h e n e is the identical
would be the trival
mapping
on A.
But E has
infinite index in G, a c o n t r a d i c t i o n G o H[C] e F.B is countable i n ~ , thus: Corollary: The weak axiom of choice (AC~) saying, that every countable set of u n o r d e r e d independent
pairs has a choice
function,
is
from the axioms of the system ZF ° + (WF).
A model of E.Specker. Again we choose reflexive
ZF ° + (AC)
+ "there
sets such that V = U ~ R e ( A ) "
is a countable
as s u r r o u n d i n g
set A of
set theory.
Let G be the group of those o n e - t o - o n e
mappings
w h i c h move only finitely many elements
of A. Define F" = { ~ tl ;
t is a finite
subset of A} and let F be the filter of subgroups
G such that F" is a f i l t e r - b a s i s that
in the m o d e l ~ [ G , F ]
(i) The powerset tains
of F . S p e c k e r
the following
of A is neither
no c o u n t a b l e
kind-finite (2) There
~ from A onto A
subset).
shows
of
in | 8 % ] p . 1 9 8 - 1 9 9
holds:
finite nor t r a n s f i n i t e The powerset
(i.e.
con-
of A is hence Dede-
but not finite.
is no o n e - t o - o n e
mapping
of A. Hence the statement
D) THE I N D E P E N D E N C E
f from A × A into the powerset
Am(m2
~ 2 m) fails
OF THE G E N E R A L I Z E D
in Specker' s model.
CONTINUUM-HYPOTHESIS
FROM THE
ALEPH-HYPOTHESIS
We shall obtain the result m e n t i o n e d Fraenkel's
model
in the heading by c o n s t r u c t i n g
in a set theory ZF ° + (GCH)
in w h i c h there exists
60
a countable
set A of reflexive
The following
lemma shows
If ZF is consistent, table
Proof.
chapter
is consistent
then ZF ° + (GCH)
+ "there
exists
Define
a permutation
ZF + (AC) ~
is consistent
F of the universe shown there
Rep(F,ZF ° ) A
by G6del's
as in t h e o r e m
(and in t h e o r e m
Rep(F,(AC)>
Now let x be any set of power M~ and let PF(X)
be the power
= {F(t);
has also
But then x can be mapped
in the F-sense
power M~.
one onto ~s-m0
= {8; ~ < 8 < ms}.
y e ~ ^ y # I}
is empty
the l-l-mapping
same holds ~+I-~0
for P(x).
(otherwise
No£ice
set of that x one-to-
that x N (w U {F(y);
map x onto such a set).
Then
is not moved by F. By (GCH)
in a l-l-fashion
But in the model
are equipotent
t C x}.
We may assume
f from x onto m~-~0
be mapped
4 of
2), that
x in the sense of F, id est PF(X)
can
a coun-
such that V = __~ R (A)" is con-
sets
then ZF + (GCH)
3, section A. It was
P(~e-m0)
and
too.
If ZF is consistent,
theorem.
theory
in it the model of Fraenkel.
set A of reflexive
sistent
such that V : --~JR (A) is true.
that the latter
can thus be used to construct Lemma.
sets
and P(x) has
onto m +I-~0,
determined
hence
the
by F,m~+ 1 and
in the F-sense
power Me+l"
Thus
ZF + (GCH) ~ Rep(F,(GCH)). Let A be the set {F(x); This
shows
that Z ~
sets ~' is consistent. restrict without
= {x;F(x)
e m'}F"
+ ~'there is a countable
But similar
the universe violating
F(x) c ~'}
+ (GCH)
to
v.Neumann's
of all sets to those
any Z ~ - a x i o m
nor the
procedure
which
(GCH).
set A of reflexive we may
lie in some Re(A)
The lemma
is thus
proved. Now we proceed
in the set theory whose
by the le~ma and define tion C). there
It was
is no wellordering
Fraenkel's
in it Fraenkel's
shown there
model.
consistency
model~[
G,F]
is assured as in sec-
that the set A is i n ~ I G , F ]
of A i n ~ [ G , F ]
The technique
. Hence
for showing
the
and that
(GCH)
fails
in
that the aleph-hypothe-
sis (AH): --/\e(2Me that the model Lemma.
= M ) holds in ~ [ [ G,F] is simply by proving ~+i inherits the (AH) from the surrounding set theory.
If the aleph-hypothesis then the aleph-hypothesis
holds
in the surrounding
holds
set theory,
also in Fraenkel's
model.
61
Proof. that
Let x be a set in Fraenkel's
inn,
x is w e l l o r d e r e d
a one-to-one
function
m e is a wellfounded tended
f from x o n t o
set. Hen~ne
to a one-to-one
mapping
= {f(z);
set.
Since the aleph-hypothesis
the initial
K|x]
from P(x)
in M, we obtain
o r d i n a l m e. But
P(m e) is a wellfounded (in the set theory),
onto me+ I. Since both sets are
that g C M. But g is wellfounded,
and therefore
by (8) (see section
can be mapped
one-to-one
is i n ~
onto P(m e) by defining
(AH) holds
g from P(~e)
such
e F by lemma 4. f can be ex-
z e y} for y C x. But again
is a 1-l-mapping
~ (M,e)
and of power M s. Then there
f(y)
there
model~[G,F]
hence H|g]
B):g is in the model.
onto me+l and has therefore
= G g F
Thus P(x)
in ~ p o w e r
Me+ l , q.e.d. Corollary.
The statement
(PW):
is w e l l o r d e r a b l e " fails Remark.
H.Rubin
is equivalent
proved
has shown,
Annalen (KA)
(AC)
(see:
the
(GCH)
H.Rubin,
Equivalents
By our corollary
from the result
E) THE INDEPENDENCE
G.Kurepa
in full ZF the axiom of choice
of the Ax.of
AMS, choice,
(AC)
be
is independent
in full ZF.
of H.Rubin,
that
(AH) are equivalent.
is independent
Notices
(PW) ÷ (AC) cannot
in ZF ° the axiom of choice
and the aleph-hypothesis
CHAIN
(AC)
though both are equivalent
It follows
set
though
(PW)
or H.+J.Rubin:
in ZF ° alone:
ZF ° alone
that
to the statement
1963,p.77-78).
from (PW),
of a well-ordered
in Fraenkel's-model,
in it.
vol.7(1960)p.381, Amsterdam
"The powerset
holds
from
in full ZF the
(GCH)
Our lemma says that
in
(AH).
OF THE AXIOM OF CHOICE
(AC) FROM KUREPA'S
ANTI-
PRINCIPLE
has considered
in his paper
126(1953)p.381-384,
"Every partially
Here a subset
the following
ordered
set ( s , ~ )
t of s is called
7 ( x ~ y v y ~ x). Kurepa has
"Ueber das Auswahlaxiom", statement has a maximal
an antichain
shown that
Math.
antichain".
iff for x,y e t ÷
in Z ~ , (KA)
in conjunction
with the (0) Orderin$-theorem: is equivalent
"Every
set can be linearely
to the axiom of choice:
ordered ~'
Z ~ ~ (AC) ~ [ (KA) ^ (0)]
and
62
has asked wether proved
(KA) alone
(U.Felgner,
in fact
(AC) and
is equivalent
(KA) are equivalent
used in the proof).In
contrast
was able
from (KA).
The proof
is contained
in Halpern's
thesis
(Berkeley
in a set theory
G consisting called
a support
Definition
("Tr~ger")
of G which
model.
(a permutation
subgroup.
conditions
(i),...,(iv).
follows in~
such tha~
with respect
Remark.
is a surjective
the axioms (WF),
of G satisfying as in section
of ZF ° and that A = Ro(A)
is
holds
of A, the supports
of G are uniquely
If H is a finite
grp{Hi,H2}
est: the smallest
support
determined
the supports
subgroup
support-subgroups
is also a finite = supp(Hi)
is the subgroup subgroup
= {T E G; T leaves grp{H l ,H 2 } < H3.
T ~ H3. Define
B. It
to A as base.
supp(grp{H1 ,HI})
Clearly
a fini-
(If
would
of G we let supp(H)
of H.
grp{Hi,H2}
H~
mapping).
the weak axiom of foundation,
I. If H,, H 2 are finite
Proof.
of subgroups
Define ~ = ~ I G , F ]
support-subgroups
not be unique).
Lemma
one-to-one
not be the full group of permutations
be the support
subset E of and E is
H of G which contain
Since G is the group of all permutations
of the finite G would
support
Let G be the group of all permutations
F is a filter
that ~ s a t i s f i e s
a set of ~
leave a finite
of
of H.
Let F be the set of all such subgroups te-support
is an
of A. If H is a subgroup
then we say that H has finite
of Halpern's
of A = R0(A)
and not yet published.
in which there
sets such that V = ~ J ~ R a ( A ) .
of those elements fixed,
1962)
ZF ° + (AC)
Let G be a group of permutations A pointwise
is
J.D.Halpern
(AC)
set A of reflexive
(the axiom of foundation
is independent
in ZF ° alone
We shall w o r k
that in full ZF
to this result
to show that
infinite
to (AC) of not. We have
Math. Zeitschr.1II(1969)p.221-232)
of G, then
support-subgroup
of G and
~ supp(H2).
of G generated
of G containing
by H I and H2,
H I and H 2 . Define
supp(H, ) N supp(H 2 ) pointwise
In order to prove
id
the converse
fixed} consider
a = supp(H l ), b = supp(H2 ) and let
a - (a N b) = {a, , .... ak} Let c I ~...,c k be distinct
elements
not in a U T'l(b)
U b. Let o, be
63 the permutation of A : R0(A) which perl~utes a i with c i (i ~ i < k) and is constant otherwise.
Then ~t s H2. Let
b - (a n b) : {b, ..... b m} N0£ice that g,T-l(bi ) # a and b i # a (by definition) the permutation of A which maps b i onto ~tT-J(bi)
Let c2 be
and is constant
for the other elements of A. Hence g2(b i) = g,~-l(b i). and ~2 e HI. Finally define g3 -- Tg[Ic2.
Then
O,(b i) : T~[l~2(b i) -. T~[l~t~-l(bi ) = b i for I ~< i ~ m. Further~ since T ,01 and oz leave a f% b pointwise fixed, ~ also leaves a ~ b pointwise fixed. Thus "3 leaves b pointwise fixed and we get ~ g H2. But ~3 : T ~ f ~ z - ~ r = ~3~[i~i, thus T g grp{H~ ,Hz} q.e.d. Lemma 2. If H[x] = {T e G; T*(x) subgroup,
: x} includes a finite-support
then H[ x] contains a finite-support
which includes all other fiDite-support are contained Proof.
subgroup
subgroups which
in H[x].
Remember that T* is the unique extension of T to an automor-
phism of the universe V (see section B). Let I be the intersection over the set of supports of the finite-support in Nix].
Since each support
subgroups
included
is finite, hence I is finite and can
be represented as an intersection of only finitely many of supports. The group generated by the union of subgroups
corresponding to the~ =
finitely many supports has support I (this follows from lemma I and an ordinary induction),
is contained
in H[x]and
is largest in the
sense stated, q.e.d. Remark and Definition.
If x is in the model ~ [
G,F], then H[ x] g F.
But by definition of the filter F, there is a finite support-subgroup H = contained gest finite-support
in H[x]. Thus, by ].emma 2, H[x] subgroup.This
and depends only on H[x] finite-support
includes a lar-
subgroup is uniquely determined
(if x is in ~ [ G , F ]
subgroup of H[x] by HoIx].
), and we denote this
Further we write F(x) =
supp(Ho [ x] ). We shall prove that Kurepa's Antichain ~[G,F]
. Since every "abstract"
Principle
(I
in
partial ordering < on a set s can
be represented by the inclusion relation _C, we may restrict ourself to the discussion of sets t, where _C is the partial ordering on t
64
(namely~ for x e s define [x] Then ( s,~ ) and (T~C_) Lemma 3: (KA) holds proof.
= {y e s; y < x} and t = {[x] ; x e s},
are isomorphic).
in Halpern's m o d e l ~ [ G ~ F ] .
Let t be a set of ~
Z = {y~ y _C t ^ y e ~
-~ ~ [ G , F ]
, and define
^ F(y) C_ F(t) ^ y is an anti-chain}.
Z is a subset of the model ~
but not necessarily
Z ~ ~.
We want
to prove that Z has maximal elements. If T is a subset of Z, totally ordered by C_, then U T is again an antichain.
Further HI U
y g Z ~ Ho[t]
~< He[y]
Since T C_ Z C ~ , therefore
~T
T] i> N
~ H[y]
hence
£ ~
{HI y] ; y £ T} and always:
w h e r e Ho[ t ]
~ j T _C ~
e F. Thus HI U T ]
by the Transitivity
e F.
of ~
(see (e) and (8) in section B). Finally:
Ho[ t] ~ H[ ~J T] ÷ Ho[t] ~ Ho[ U
T] ~ Thus
F(~T)
~
C F(t). This shows that
(by lemma I):
T e Z. Since Z is inductively
ordered by C there is in Z by Zorn's lemma a maximal element, yo,i~ Z
and
(no{Ice ~ha~ Z ~a ~
say
since F(~) = ~ C F(t), hence ~ e Z).
We want to show that yo is maximal among all antichains
of (t, C - ) in
Suppose Yo is not a maximal anZichain of (t,C_ } in ~ .
Then
there is an element y e t - Yo such that yo U {y} is an antichain. Let
Y,
:
Then Holt] ~ H[yo] H[y,]
Yo U { T * ( y ) ;
T e Ho[t]}.
implies Ho[t] ~ H[yl] • Since Holt]
e F. Since Yo C_ Yl C t _C ~
therefore
y, e ~
e F therefore (by (8) in
sect. B). On the other hand H0[ t] ~ H[y,] imDlies F(yl ) C F(t). yl cannot be an antichain, since otherwise yl e Z contradicting the maximality of Yo. Hence Yl must have Two comparable elements.
We
have two cases: Case I: There are z ¢ Y0 and T e He[t] T*(y) C z. Case 2: There are T1 ,T2 E Ho[t] ~l(y) c ~(y). If case I holds, we have T c Ho[t]
< Ho[yo]
contradicts
such that z C T*(y) or
such that T~(y) ~ T*(y) and
(T-l)*(z) C y or y C_ (T-1)*(z).
~ H|yo]
and z c yoimplies
the fact that y o u
(T-l)*(z)
{y} is an anti-chain.
But
e y. This
65
If case
2 holds,
put T = T[IT2.
T e H0[t]
and we o b t a i n
y # T*(y)
and y C T'(y).
y. H e n c e
the e x i s t e n c e We want
= {w e F(y);
D2
= {z e A = Ro(A); n-times
is f i n i t e
after
to w. Thus DI has Also,
only D2
tation
D2
Vn(0
finitely
is c l o s e d
of the p e r m u t a t i o n
numver
is f i n i t e
D~
images
since w e D,
iterations
is f i n i t e under
are e i t h e r
successive
cyclic
then
D2 ) = 0 } .
is n e v e r
n z such that m > n z implies
in D2
Tm(z)
~ F(y)
devide
the o r d e r of ~
N {Tm0(w); We d e f i n e
u = Tm°(x)
~
.
is a f i n i t e
F(y)
multiple
is f i n i t e of no
Then
Tm°(z)
by d e f i n i t i o n / F(y)
= ~m°(x)
= v otherwise, identity
for z e F(y)
Thus
of mo: if z e
= z.
o of A = Ro(A)
that q is the = ~m°(z)
~m°(z)
of the o r b i t s
which
maps X = F(y)
for x e X, o(u) o is w e l l - d e f i n e d
as was
just p r o v e d
onto
= x for since on above.
we o b t a i n
= (~)*(y)
(o-ITm°)*(y) Since
then
by o(x)
e Y and o(v)
since o ~ F(y)
similar way
. Further
hence
a permutation
F r o m o(z) o'(y)
w e F(y)}
z e F(y)}
X N y we h a v e
(2)
is h e n c e
(since
for some k e m and the c a r d i n a l i t y
-- D, + n z < n% ,
Y = {rm°(z);
(i)
- DI}.
cycles
= w if w e D2
since mo = k. no
z e F(y)
finite
(and there
z e F(y)
F(y)
of ~.
and f o r m o r b i t s
The g r o u p ~
strictly
of ~
w of
is a p e r m u -
on D 2 , t h e
Let mo be the first
Tm°(w)
back
groups. ULet no be the Uorder of
Tn(z)
than M a x { n z ;
that
element
T ~ D2
of d i s j o i n t
and ~n(z) # z for all n > 0). greater
implies
iteration
with
mapping
equal
is finite,
of T one comes
and e v e r y
T. Thus D2 t o g e t h e r
group (D2,T~
- D,,
Tm°*(Y) =
= z] }.
^ yn(w)
of T. Since F(y)
~ is a o n e - t o - o n e
sum of t h e s e
If z e F(y)
D2
many
S z = {z,$(z),T2(z),...}
the d i r e c t
application
since
under
Since
such that
e m such that
= w)}
E D,
of s u c c e s s i v e
is f i n i t e
group.
V w Vn[W
Further
number
is a group:
of Y e H0[t]
to find a mo
< n ^ ~n(w)
iterated
too.
a finite
Ho[t]
to look at:
D,
Tn means DI
it is n a t u r a l
Since
= T m ° ~ F(y), : y,
id est o*(y)
also ~T m°
is the
from o(Tm°(z))
~*(Tm°)*(y)
By h y p o t h e s i s
h e n c e o-IT m° = T m *o
identity
e Ho[y]
~ H|y],
thus
(y). on F(y) we o b t a i n
in a q u i t e
: z for z c F(y):
= y.
y C T*(y),
and
since
T* is an a u t o m o r p h i s m ,
we h a v e
66
y £ ~.(y) £ (~).(y) ~ . . . ~ (,~).(y).
(3)
Thus by (i); y C ~'(y). But (3) also yields o*(y) C o*(Ym°)*(y). Applying
(2), we have o*(y) C y. Thus y : o*(y).
deduce y : (Tm°)*(y).
Finally,
From (1) we
from (3), we arrive at the contra-
diction y : T'(Y) and lemma 3 is proved. Let (AC~) be the axiom of choice for families elements
are couples
the (unrestricted) model~,
(= unordered pairs).
axiom of choice
we ~ a l l
model ~ [ Proof:
V, V(u,v
A = Ro(A)
Thus Y _ C ~
(AC2) fails in ~ .
(AC2) does not hold in Halpern's
G,F] .
Let Y : {z;
(V~U)})}.
Instead of proving that
(AC) does not hold in Halpern's
show that already
Lemma 4: The weak axiom of choice
g A : Ro(A)
is a set of ~
^ u ~ v ^ Z : ((U,V > ,
as was noticed previously.
But Y is closed under G, thus H[Y]
= G e F. Hence,
by (8) of section B, Y is a set of the m o d e l ~ . = 2 and distinct elements of Y are disjoint. be a choice set C for Y i n ~ dinality subset
(: sets) whose
I] and C e ~ .
, id est
Suppose there would
/ \ w [w e Y ÷ w N C has car-
It follows H[C]
of the infinite set Ro(A).
Also z E Y implies
E F and F[C]
is a finite
Pick elements u~v e R0(A) - F(C)
such that u ~ v. Let T be the permutation of Ro (A) which interchanges u and v and is the identity otherwise. T*(C)
Then T s Ho[C] ~ H|C], hence
= C, and y ={(u,v ) ,(v,u )} g Y. Suppose(u~v ) e C, then
Y'((u,v )) =(v,u ) E T*(C), hence ( v , u ) (v,u)
E C then one concludes
tradicting
~ C: a contradiction.
similarly that ( u ~ v )
If
e C, again con-
the assumption on C. Thus Y has no choice set in ~ , q . e . d .
This finishes the proof, that in Halpern's model ~ [ G , F ] axioms of ZF°~ Kurepa's Antichain
Principle
(KA) and q(AC2)
all are true.
As a Corol!ary
(J.D.HALPERN):
The axiom of choice (AC) does not follow
from Kurepa~s A n t i c h a i n - P r i n c i p l e Remark. ciple
Since (AC2) fails in Halpern's m o d e l ~ ,
(0) fails in ~
(BPI) + (0)
too, since Z F ° ~ ( 0 )
(via compactness-theorem
lus, e.g.) where
(BPI) is the
p.37-IIi)
÷ (AC2). Further Z F ° ~
of the lower predicate calcu-
Stone
(Trans.AMS voi.40(1936)
has shown in ZF ° that (BPI) is equivalent to the "Repre-
sentation T h e o r e m for Boolean Algebras": ,•
the ordering prin-
Boolean Prime Ideal theorem "Every
Boolean algebra has a prime ideal".
(B,U
(KA) in ZF ° .
, I>
"Every Boolean Algebra
is isomorphic to a set-algebra (C, U, N, - >"
67
The statement (SPI):
"Every
follows
infinite
set algebra has a non-principle
from the (BPI).
Tarski
has asked, w h e T h e r ( S P I )
provable.
Halpern has shown,
that
provable.
Halpern
in the model above the
while
the
(BPI)
Lemma
5 (U.Felgner~M.Z.
shows that
fails
(KA) implies every Proof.
111(1969)):
erdered
such that there
onto K with the property
which
chains.
selects
function
theorem
(LW) holds
mot m t h e o r e m
from o~f
(LW).
mapping
potency
by abstraction.
p.442~
is
By Zermelo's
or 65(1908)p.107-
too. proved
Since
ZF ° ~ (LW) ÷
D: in ZF ° the that
(AC)
(LW)--~ (KA)
is
IN ZF ~
is a one-to-one number
(or equinumerous),
function
mapping
x of x is obtained
~ equipotent
(AC) with
we are still
x ~ la Frege-Russell-Scott:
A z(Z
~ x ~
of sets y of lowest by abstraction
x on
from equi-
of the axiom of choice
to be the least ordinal
p is the Mirimanoff-rank
and
- {~}.
(AC) but the axiom of foundation~
adequately
x : {y; y ~ x ^
Definitions
C of U K
just one element.
in chapter
Fel~er
In the presence
the term x can be defined
x consists
by Ks but K is
q.e.d.
OF CARDINALITY
of the cardinal
x. If we do not have
where
- {~}
e K is isomor-
Z? ° .
x = y, iff there
able to define
f from P(s)
antichain
on P(s)
model ~ L
our result
Further,
P(s) of
are pairwise
f(t)
isomorphically
One says that the sets x and y are equipotent in symbols
Principle
says that
The powerset
from each chain
in Halpern's
F) THE U N D E F I N A B I L I T Y
y. The notion
is not
(Math.Ann.65(1908)p.
(Math.Ann.59(1904)p.514-516,
(PW)~ we have strengthened is independent
set.
A maximal
g defined
128) the set s can be wellordered, Thus
Antichain
for ~ ~ t e P(s),
is represented
a choice
well-ordering
ordered
is a one-to-one
disjoint
function
is
(SPI) holds
is a set K whose'elements
that
a set of pairwise
Thus we get a choice
implication
(LW) which
a theorem of Zermelo
28) there
phic to t. Thus P(s)
÷ (BPI)
set can be well-ordered.
Let (s, ~ ) be a linearly
disjoint,
Kurepa's
in ZF ° the statement
linearely
theorem
in ZF ° this
ideal"
in it.
s is a set of chains.By 261-281,
prime
p(y) ~ P(Z))}
function
(see chapt. I,sect.E).
rank equinumerous
in axiomatic
Here
with x (see D.Scott:
set theory,
BulI.AMS
6--1(1955)
68
[~]
Dana SCOTT: The notion of rank in set-theory; Summer Institute
Summaries
for Symbolic Logic, Cornell Univ.1957,
p.267-269)~ We remark,
that even in the absence of both the axioms of choice and
regularity but in the presence of either the weak axiom of foundation in the form "there is a set A such that V = ~ R ~ ( A ) " (UoFg.,Arehiv d. Mat~.20):
or the axiom
~'the universe V can be covered by a well-
ordered sequence of sets s~, ~ an ordinal". ZF ° without any additional
We shall show that in
covering axiom (like foundation,
etc.)
there is no adequate definition of the term ~o This result was obtained first by Azriel L@vy
[50]
A.L£VY:
The Definability
of Cardinal Numbers;
of Mathematies~'~ GSdel-Festschrift~
in: "Foundations
Springer-Verlag
Berlin
1969~p.15-38. Also R.J.Gauntt has obtained this result [~Z] R.J.GAUNTT:
Undefinability
U.C.L.A,-set
(independently):
of Cardinality;
Theory Institute
Proceedings
of the
1967. To appear in 1970.
In the presentation of the proof we shall follow mainly R.J.Gauntt but in few details A.L~vy. When one considers the question of whether one can define in ZF ° --
the eardinality operation x~ the following possibilities (a) x is definable
turn up:
i n a set theory ST: there is a term t(x) of ST
with the only free variable x such that ST ~ % % [ t ( x )
= t(y) ~ x ~ y]
(b) x is relatively definable
in a set theory ST: there is a term t(x,
z) of ST with the only free variables ST ~ X Obviously
Ax ¢[t(x'z)=
(a) entails
(b) (L~vy [50]
If we take ZF ° + foundation ST, then (a) holds. namely,that Theorem
t(Y'Z)~
z and x such that x ~ y]"
eonsiders further possibilities).
(id est ZF) or ZF ° + (AC) as ~et theory
We shall prove a strong undefinability result,
even (b) does not hold for the set theory ZF ° .
(L@vy,Gauntt):
If ZF ° is consistent~
then so is ZF°plus the
schema
(*)
"VxAVy[@(y,a,x)^
Ab(a
~ b ~ ¢(y,b,x))] .
69
Proof.
If ZF ° is c o n s i s t e n t ,
"there
is a p r o p e r c l a s s A of r e f l e x i v e
c l a s s of all o r d i n a l s ) ,
there
sequel
Each ordinal
(the
of Chapt. III,
mapping
of ZF ° p l u s the s c h e m a
(').
(in a u n i q u e way)
and n £ ~ (this
Define e -
On o n e -
a Fraenkel- In the
0 iff n ~
follows
as
8 + n where
from Cantor's
0 (congruence modulo
normal-
2) f o r
^ n £ m, and d e f i n e ~ -= I iff n = I m o d u l o
for ~ = 8 + n ^ Lim(8)
^ n E e. The o r d i n a l s
0,2,4,... ~m,m+2,e+4,...
and the o r d i n a l s
m + l , ~ + 3 .... For e a c h o r d i n a l end
(see the r e s u l t s
now construct within this universe
e can be w r i t t e n
= 6 + n ^ Lim(8)
w i t h On
of A are c a l l e d atoms.
8 is a l i m i t o r d i n a l form theorem).
is c o n s i s t e n t
model ~L
the e l e m e n t s
sets e q u i n u m e r o u s
is a f u n c t i o n G (a c i a s s t e r m )
t o - o n e o n t o A. We w i l l Mostowski-Specker
( c a l l e d ZF V) ZF°+
s u c h t h a t for e v e r y x t h e r e e x i s t s y E x w i t h
e i t h e r y ~ x = ~ o T y ~ A" sect.A). ~ence
t h e n a l s o the t h e o r y
if ~ -- o t h e n A 8 1 8
~,
congruent
congruent
{G(~),G(e+I)}
2
o are thus
I are
1,3,5,...,
is a p a i r of a t o m s
{G(S),G(6+I)
--- 0 ^ a ~e 6 ÷ { G ( ~ ) , G ( e + I ) }
~|. Definition.
F(~)
The f o l l o w i n g (1917)p.33
= {G(6);
definition
(6 m 0 ^
8 < e) v
is due to D . M i r i m a n o f f
Ker(x)
(L'Ens.Math.vol.17
= C(x) ~ A = the set of atoms closure
t h e k e r n e l of x; M i r i m a n o f f
We n o w r e s t r i c t i.e.
8 4 e)}.
and p.211).
Definition.
(read:
(8 m I A
the u n i v e r s e
V = U ~ ( U y R y ( F ( e ~) ) ) . .
of all x for w h i c h
0
of x.
u s e d the t e r m
to e l e m e n t s
That
"
~( e )
"noyaux~').
of sets b u i l t up f r o m F ( e ) ' s ,
is, t h e r e s t r i c t e d
V ~ V..(x ~ y ^ Ker(y) L ~ y
-+
in the t r a n s i t i v e
universe
consists
c F(e)).
,,.,
- - ~Atoms
A
F(6) No%ice
that
class.
For each permutation
(restricted)
e a c h F(e)
universe
is s set and U ~ F ( s ) f on F(e),
as follows:
= A, w h e r e A is a p r o p e r
define
f(x)
o v e r the e n t i r e
70
f(x)
= x
for atoms
f(x)
= {f(y);
This de£inition universe,
y e x} for sets x.
is welldefined
then x C Ry(F(8))
definition assumed
x not in F(s),
of Ry(a)
if x is in the restricted
for some ordinals
see p.53].
that f is defined
since
By induction
~ and y [for the
hypothesis
for all y e R6(F(8))
it is
for ~ ~ y and
all 8. Definition.
A permutation
f on F(e)
id est,
that 8 ~ 0 there
exists
y < e such that y m 0 a n d
A permutation
6 ~ 0 it holds
is called
id est:
x is symmetric
~ there
fixed,
is a finite
symmetric
the transitive Digression.
of F(~) rence.
model
are called
In the definition
of the notion
Hence
"nice". support
we avoided
the filter formulate
F a collection
sets can be collected and in which to systems
classes
etc
But there
are already
proper classes. classes
and
It is possible theory
to
in which
(~ l a v . Neumann-Bernays-G6del) to totalities~ "set"~
totalities
"class",
"totality"
using an idea of !.L. Novak-Gal
In such a set theory
the filter
the use
and in the defini-
of proper
of ZF-set
(in which the predicates
K[~],
K[a]
diffe-
the use of a filter of subgroups.
can be collected
3_~7(1951)p.87-110).
the groups
permutations
set x we avoided
subgroup
extension
tO classes
y of
just defined
is one important
of those totalities.
"system" .... are primitiv) Math.
x and every element
C. The admissible
be totalities
a type theoretic
leaves
pointwisel).
sets x which are here-
of the m o d e l ~
is done since the permutations the groups K[a]would
T which
of x is symmetric).
of a symmetric
of a finite
tion of the m o d e l ~ This
(id est:
in section
there
permutation
are those
closure
Nofiice the similarity
with Fraenkel~s
set a of atoms
fixes x (not necessarily
Sets of the m o d e l ~ 6 ~ ditarily
iff
: {G(B),G(B+I)}.
such that each admissible
Definition.
admissible
for all 8 < ~ such that
that
f({G(6),G(8+I)})
pointwise
for all 8 < e such
= {G(y),G(~+I)}.
f on F(e)
it fixes pairs,
Definition.
semi-admissible
pairs~
f({G(~)~G(8+I)}) Definition.
is called
iff it preserves
F~ eZc.
But since
(Fund.
one can talk about in the discussion
7~
above r e f e r e n c e permutations
is made only with respect
f of the sets F(~) we could restrict ourself to men-
tion only p e r m u t a t i o n s "subgroup",
to one single class of
of certain type.
The use of the notions
"filter" would make only linguistieal
Further remark that a p e r m u t a t i o n
only elements w h i c h are in some F(e). of a p e r m u t a t i o n
on A is a set. This
tion of a symmetric class-variables The formulae:
Hence the "essential" explaines
set we have q u a n t i f i e d
are thus ZF-formulae.
Thus
(thus
on A are not needed).
and "x is h e r e d i t a r i l y
~ :
part
that in the defini-
only over sets
to range over p e r m u t a t i o n s
"x is symmetric"
differences.
of the class A of atoms moves
symmetric ~'
{x; x is h e r e d i t a r i l y
symmetric}
is a c l a s s - t e r m of ZF. The following to those of section
lemmata are easily proved.
The proofs are similar
B.
Lemma 1. (In ZF V ): If f is s e m i - a d m i s s i b l e
and g is admissibles
then f-lgf is admissible. Lemma 2. Lemma
(In ZF V ): x E ~
~ (x C ~
3. (In ZF V ): No two disjoint equinumerous
^ x is symmetric). infinite
sets of atoms are
in~.
~ucb Proof.
Suppose the lemma
and y and a one-to-one
is false.
Then there a r e ] i n f i n i t e
function g, m a p p i n g x onto y, i n ~
x,y and g are symmetric~
there are finite
that every a d m i s s i b l e
permutation
fixed~
fixes x (resp.
y,g).
G(~+I)
e x. Now pick ~ e 0 such that G(~)
g maps x onto y~ g(G(~)) (G(~),G(8)
interchanges
identity otherwise. tical m a p p i n g (G(e+I),G(8))
Lemma 4.
c ~(g)
= g
maps ~ onto itself.
Since
e y. Thus permu-
but is the
~ acts as the iden-
= (~(G(e))~G(8)) e pointwise
=
fixed.
g w o u l d not be one-to-one~
(In ZF v ): Any p e r m u t a t i o n
Let a be the finite
))
since ~ leaves
a contradiction~
Then every a d m i s s i b l e
= G(y)
e g. Take an admissible
Since x and y are disjognt,
finitely m a n y atoms, Proof.
e x - (a U b U c).
the atoms G(~) and G(e+l)
on y. Thus ~((G(~),G(8)
G(B),
such
b,c) pointwise
e x - a for e m 0, then
e y and g(G(~+l))
} e g and ( G ( ~ + I ) , G ( y ) )
tation ~ which
g(G(~+l))=
= G(B)
. Since
sets a~b,c of Atoms
~ leaving a (resp.
If G(~)
sets x
Hence q.e.d.
on F(~), which moves only
is i n ~ .
set of atoms moved by the p e r m u t a t i o n
permutation
T which
leaves
a pointwise
~.
fixed
72
Lemma 5. (In ZF V ): For each x and semi-admissible x
Proof.
~
~ ~
-~(x)
1 of chapt.
III,
Lemma
6o For each ZF-formula
(i)
2 and proceed
section
the following
quantifiers
are theorems
~ semi-admissible
Lem~a
The proof
÷ [~(xl ..... x n) ~ ~(~(xl)~..,~(Xn))] ÷ [Rel(~(xl
obtained
I, page lemma
is (with respect
mappings
set x in the restricted
These
universe
is essential
Len~a
~(xl,x2~x3)
8o For each ZF-formula the following
Proof.
hence
VxAaVy[~(y,a,x)
Suppose
~(xl~x~,x3)
is provable
that the lemma
and a set x i n ~
in the restricted
Ker(x)
C F(~), where
Cleary~
DI
e ~.
Suppose
C F(~).
Case 2. Ker(y)
~ F(e).
S~o
define
There
of ad-
are sets! segment
with three
free variables~
Ab(a
is false.
~ b ~ ~(y,b,x))]).
Then there
there
above.
is a ZF-formula Since x is i n ~ ,
is an ordinal
~ such that
Define
- F(~).
Case 1. Ker(y)
Then D2
groups
only an initial
as required
y is the
id est R e l ( ~ ] ~ ¢ ( y ~ D ,
If case 1 holds,
B, p.54)
(definite).
~ m 0 can be choosen.
DI = F(~+~)
y g~,
(in section
in zFV:
^
universes
is
"hyper-classes ~' of all
from A onto A, but take only the groups on the sets F(~).
all
5o
theorem
F(~) of the class A of atoms
Rel(~,7
14). The proof
to e) a model of ZF °.
2, 5 and 6. Do not take the
permutations
For every
from ~ by restricting
(see chapt.
is like the one of Specker~s
one-to-one
..... Xn)) ~
is the formula
on the length of ~, using
lemmata
missible
of zFV:
..... ~(Xn)))].
7. (In ZF V ): ~
using
as in the proof of
55-56.
Rel(~,~(~(xl)
to the c l a s s ~
by induction
B, page
~(x~ ~...,x n) with n free variables~
(ii) ~ semi-admissible
Here R e l ( ~ , ~ )
~,
.
Use lemma 1 and lemma
lemma
permutation
D2
(unique)
cardinal
of Dl, where
~x)).
= F(~+m.2)
- F(~+m),
is a semi-admissible
where
permutation
~.2 = m+~. w of the atoms:
73
w(G(~+n))
= G(~+~+n)
~(GC~+~+n)) ~(GCB))
= G(~+n)
= G(8)
for B < ~ or ~+~.2 < B.
Thus ~ fixes each element of F(e) and takes Di onto D2. Hence ~(Di ) = D2 ~ w(D2) Rel(~¢(y~Dl
= DI, ~(x)
= x and ~(y)
,x)) ° R e l ( ~ , ¢ ( ~ ( y ) ,
Rel(~,¢(y,Dx
= y. Then
~(D,),
~(x)))
°
,x))
Hence y is also the cardinal of D2. Thus R e I ( ~ , D , lemma
G(~)
~ D2) violating
3. If case
2 holds~
E Ker(y)
^ G(8)
define
there ~ F(~).
a permutation
8 ~ ~ such that
Pick an ordinal
• on DI U F(y+I)
G(y) and interchanges G(8-1) w i t h G(y-1)
is an ordinal
which
G(8+1) with G(Y+I)
Y,Y ~ 8, Y > B, and interchanges
6 < ~, T fixes all elements
of F(e).
and hence
T is semi-admissible.
Rel(~,¢(y,Dl,x)) Thus ~(y)
Thus
is the cardinal
and is a one-to-one
of ~(D~).
in~
= x. T moves Ker(y) Hence by lelmma 6:
Since y is the cardinal cardinality
of D~
and are there-
. But by lemma 4, • is a set of
function
would be equinumerous
T(x)
~ F(~) ÷
~ ReI(~,~(T(y)~T(DI)~x)).
and ~(y) ~ y, D~ and T(D~ ) have different fore not equinumerous
and
iff 8 m 0~ and interchanges
iff B ~ 1. Since e ~ 0, hence G(6)
T(y) ~ y. Clearly
G(~)
in the sense o f ~ .
in~,
Thus D~ and T(D~)
a contradiction.
Lemma
8 is thus
proved. The t h e o r e m of L @ v y - G a u n t t
follows d i r e c t l y
from lemmata
7 and 8.
G) A FINAL WORD
The main
idea behind GSdel's
construction
of the model ( L , ~ )
ZF + (AC) was to make all sets of the model definable by means
of a certain
complexe
defined)
wellordering
of the language
model-class ~in
language.
The natural
induced
choice
fails
is to guarantee
tely many sets of "indiscernible" why a function indiscernible
f defined elements
should choose
and not the other element.
that~
of ZF°-models
contains
Then there
of the
infini-
is no reason
set of sets of mutually
from each set just the one
This was made precise
the groups G of p e r m u t a t i o n s "atoms ~' (reflexive
sets.
on a infinite
(inductively
a wellordering
L. The main idea behind the c o n s t r u c t i o n
which
of
(or nameable)
on some infinite
by i n t r o d u c i n g
set A = Ro(A)
sets) and the filter F of subgroups
of G.
of
74
The symmetries
of the model ~
~ C ~ a n d , x = {T(y); discernibles
B of these
In Fraenkel's
{a2k,a2k+l}
are e.g.
sequence
model
(see this chapter,
sets of indiscernibles.
sets of indiscernibles
part to Russell's
by F. If x is in
T e G} for every y c x, then x is a set of in-
in~.
C) the sets
are determined
The set
is the set-theoretical
of pairs of (mutually
section
counteP-
indiscernible)
socks. The "classical" indiscernibles "reflexive
was to take an infinite
sets"
acts on them. right
filter
way for obtaining
and to take a certain
The choice
of the right
of subgroups
of choice holds atoms the
sequence
filter
method.
conditions
applications
In the next chapter This method pendence
applies
results
from the (GCH), further
results.
to full ZF-set
"below"
Again
sets by destroying to construct
which contain
indiscernib!es,
~--->(~)~ structible
cardinal~
satisfies that the
and then the model Thus~
in order
to get
the filter F has to
Cohen's
forcing
and yields
of (GCH)
the
ZF models ~
, see J . S i l v e r ' s universe,
theory
it is possible
indiscernible
l~rge
such that the
method.
not only inde-
th~ (AC) but also the independence
the independence
on
{i} of G.
we shall describe
even possible tence of
of sets.
subgroup
F defines
set theory the axiom
model ~ [ G , F ]
of the FMS-method,
never the trivial
group and the
(it is supposed
(i),...,(iv)),
with the whole universe
non-trivial contain
is discrete
or
group which
in all applications
The filter
and the weak axiom of foundation
F satisfies
coincides
of "urelements"
permutation
If in the surrounding
iff the topology
of sets of
nice permutation
form a set, then the corresponding
(AC)
families
is the alpha and omega
of the F r a e n k e l - M o s t o w s k i - S p e c k e r the group G a topology.
those
from (AC) and lots of
to introduce (AC).
in Cohen-models
We remark
that
in which V = L holds
but then one has to assume satisfying paper:
of V = L
the p a r t i t i o n
A large
cardinal
Fun~.Math.69(1970)p.93-100o
it is and the exis-
relation in the c o n -
~5
Additions
to chapter
1) The part K[xl
III
s F then there is a o n e - t o - o n e
some w e l l - f o u n d e d trivially
set, of lemma 4 in section
c F then x is w e l l o r d e r a b l e
be any w e l l o r d e r i n g
in~[G,F]
of x, then w C ~ I G , F ]
can be
thus w e ~ | G , F ] .
are w e l l - f o u n d e d 2) The corollary
; namely
Thus x is w e l l o r d e r a b l e
sets,
let w
. But o b v i o u s l y
are i--I-mappings from x onto some ordinals
in~L
i n ~
KIx]
and there
. But ordinals
Q.E.D.
on p.62 which says that
(PW) holds
mode! ~ q ~
can be s t r e g t h e n e d
by a s s e r t i n g
~while
(AC) fails.
Let ( s , < )
in~.
from x onto
B, p.57-58~
proved as follows°
If K[x]
H[w],
mapping
Proof.
in FraenkelVs
that even
(LW) holds
be a lineari!y
in
ordered
Define R = {(a,b ) ; a,b e s ^ a ~ b}; t h u s H [ R ]
set
e F and
H[ R] ~ HI s] . We claim that for each y s s it holds that HIR]
~ H[y].
Suppose not~ then there are y e s and a T e H [ R ] s u c h that T(y) ~ y. But T(y) {y~T(y)
e s and R is a linear ordering on s, thus either ) e R or ( T ( y ) ~ y )
T((y,T(y)
} ) = (T(y).T~(y))
But ( y , Y ( y ) )
H[R]
leaves
3) It holds
: (T(y),y)
e R ^ (T(y),y)
The same argument
relation
e R. If ( y ~ T ( y ) )
applies
s R yields
: R, since
s pointwise
fixed.
Thus,
< H[w]
and it follows
s R. Thus every • e
if w is any w e l l o r d e r i n g that w s ~
that ZF ° ~ (AC) ÷ (LW) + (PW), while ZF 2) that
(LW) + (AC) [6~]
in ZF ° . Let us indicate
ciple of choice of proper, ski's model,
since
(PW) holds
in it (see Mostowski:
note further
subsets
in ZF ° .
(PW) ÷ (LW) is not Kinna-Wagners
cannot hold
in it and otherwise
(AC) would be
has shown that in M o s t o w s k i ' s
ideal theorem
(BPI) holds
prin-
in Mostow-
Colloqu.Math.6(1958)p°207-208).
that J . D . H a l p e r n
the Boolean prime
is not provable
that o b v i o u s l y
, q.e.d.
~ (AC) ~ (LW) ~ (PW).
one shows that
non-empty
T 2= 1.
y = T(y), a contradiction[
on s, then H[R]
We have shown under
true
e T(R)
to the case ( T ( y ) , y )
Using the model of Mostowski provable
e R~ then
Let us model
(Fund.Math.55(1964)
p.57-66. 4) Finally we refer to some important is applied: A.Mostowski:
H.L~uchli:
E.Mendelson
[61]
papers
,[62]
On the Principle
of D e p e n d e n t
(1948)p.127-130:
[~81.
Auswahlaxiom
in which the F M S - m e t h o d
, and:
in der Algebra;
choices;Fund.Math. B5
Comment.Math.Helvetica
37
(1962/63)p.1-18. H.L~uehli:
The Independence tricted
of the Ordering principle
axiom of choice;
from a res-
Fund.Math.54(1964)p.31-43.
CHAPTER IV
COHEN EXTENSIONS OF ZF-MODELS In this chapter we study Cohen's forcing technique for constructing extensions
of ZF-models.
This technique was introduced
in 1963 by Paul J.Cohen. Using this method Cohen has solved the long outstanding problems of the independence of the Continuumhypothesis
from the axiom of choice and the independence of the
axiom of choice from the ZF-axioms [9]
P.JoCOHEN:
The Independence of the axiom of choice; mimeographed
notes(32 pages), [IO]
P.JoCOHEN:
(including foundation):
Stanford University
1963.
The Independence of the Continuum Hypothesis;
Proe.
Nat.Aead. Sci.USA, part 1 in vol.50(1963)p.1143-1148, part 2 in vol.51(1964)p.105-110. A sketch of the proofs is contained in: [Ill
P.JoCOHEN:
Independence results
of Models-Symposium,
in set theory;
In: The Theory
North Holland Publ.Comp.Amst.1965,
p.39-54. In these papers the constructible G6del's F(e)-hierarchy
closure is obtained by means of
(GSdel~s monograph [~S],of 1940). Dana Scott
has remarked that the constructible closure can be obtained in a much more elegant way using G~de!~s M~-hierarehy
(G6del's paper [2~]
of 1939). The presentation of the independence proofs in Cohen's monograph is based on these improvements: [I~]
P.J.COHEN:
Set Theory and the Continuum Hypothesis;
New York - Amsterdam 1966 (Benjamin,
Inc.).
Since the publication of Cohen's papers [9 ], [10]
and [ ~ ]
the
forcing technique has been modified in various ways by several authors.
Using modified
"GSdel-functions
F" W.Felscher and H.Schwarz
have studied systematically Cohen-generic models
(see Tagungsbe-
riehte Oberwolfaeh April 1965 and the dissertation of H.Schwarz: Ueber generische Modelle und ihre Anwendungen;
Freiburg i. Br.1966).
A topological approach to forcing has been developed by C.RyllNardzewsky and G.Takeuti:
77
[~]
G.TAKEUTI: Topological Space and forcing; Abstract in the J.S.L. vol.32(1967)p.568-569.
A detailed exposition of this approach is contained in: [66] A.MOSTOWSKI: Constructib!e Sets with applications; Amsterdam - Warszawa 1969(North Holland + PWN). That forcing can be understood as a boolean valuation of sentences V
has been discovered by D.Seott, R.M.Solovay and P.Vopenka
-see
the forthcoming paper by Scott-Solovay, or Scott's lecture notes of the UCLA set theory Institute (August 1967) and : [7~]
J.B.ROSSER:
Simplified Independence Proofs; Academic Press
1969. [86]
PoVOPENKA: General theory of V-models; Comment.Math.Univ. Carolinae (Prague) vol.8(1967)p.145-170.
For further litterature on V-models see the bibliography in [@6]. Some of Vop~nka's papers have been reviewed by K.Kunen in the J.S.L.
34(1969)p.515-516.
-We shall present here the forcing me-
thod in a way close to P.J.Cohen, using ideas which are due to D.Scott~ R.M.Solovay and others. The following basic publications will be useful:
[39]
R.B.JENSEN: Modelle der Mengenlehre;
Springer-Lecture Notes~
voi.37, 1967. [40]
R.B.JENSEN: Concrete Models of Set Theory; In Sets, Models and Recursion theory~ Leicester Proceedings 1965, North Holland PublComp.Amsterdam 1967~ p.44-74.
[80] J.SILVER: Forcing A la Solovay; unpublished lecture notes (28 pages). [51] A.L~VY: Definability in axiomatic Set Theory I; in: Logic, Methodology and Philosophy of Sci., Congress Jerusalem 1964, North Holland Publ.Comp.Amst.1965~ p.127-151. The main difficulties which arise when one wants to extend a given ZF-model~
by adjoining some new sets a0,al,.., to g?~, are that
the sets a i may contain undesired information encoded by the interior 6-structure of a i. For instance, the interior E-structure of a i may give rise to mappings which destroy the replacement axiom in the extension. These "new" sets a i which~ when added t o ~
,
$enerate a ZF-model are called "~eneri c sets". The forcing method
78
is a technique
to obtain
generic
sets.
that every
finite
be i n ~ ,
id est, a i has to fulfill
which
can be posed
in~
terior E-structure baum's
completing
a "complete
the main
E-structure
. Then a determination
process
idea is
of a i has to
finite amounts
of a i is obtained
sequeneo
of conditions
of the whole
in a way similar
in-
to Linden-
(see e.g. M e n d e l s o n [ 6 ~ p . 6 4 )
by choosing
of conditions".
In this chapter The extensions
Herein
part of the interior
~e shall not construct
soealled
"endextensions ~'
we are dealing with are those which contain
the
same ordinals!
A) THE FORCING The simplest
general
ZF is provided approach, work~
RELATION
IN A GENERAL
framework
by considering
a straightforward
of Solovay's
Let ~
partially
ni-ary relations
R i (i E I) defined
set in the sense of ~ . copy of 0 ~
introducing
V
limited
Eex%(x):
E is taken (read
system
of ~
in~
on A. We assume
I
~'happens" this
formal
shall in ~
ex-
~ we
to talk about
is done by
E e (intended
interpretation
less than e satisfying
~'Ensemble")
"there exists
in a certain
. Since
Formally
terms
set of sets x of rank as:
~
which has means
Ve separately. comprehension
to
which this copy has to ful-
way all that what language
that A is a
by adding
are expressed
from the french word
V x~(X)
gene-
with domain A and some
£ describes
as a ramified
every v.Neumann-Stufe of
~
in a very detailed £
here a slight
of ZF (see p.25 for the
We want to extend ~
. The language
construct
model
. The properties
fill in the extension press
This
original
e ! relational
language I
structures.
of Cohen~s
Let
be a first-order
a generic
of
approach.
of "stanCard").
(A;Ri)i
Cohen models
ordered
We shall present
~ (M,E M ) be a standard
definition ~:
for constructing
generalization
is due to R.M.Solovay.
ralization
SETTING
and limited
~; the
quantifiers
an x of rank less than ~ such
that ~(x)). Th~ A l p h a b e t h
oftheramifi~d
ianguage
i) One sort of set-variables: are used to stand 2) Set-constants
for these
v0,v~,v2,...,Vn,... variables.
x for each set x of ~ .
(n e ~). x,y,z,..
79
3) Constants
A. for each j e A. ] predicates ~i for each
4) ni-ary 5) logical
symbols:
6) limited
comprehension
V~
for each ordinal
It is possible the following
V
~
~, v
= ( 1,~
, V
operatirs ~ of~
to arrange
= ( 0,3
~i = { 5,i ) and
The formulae
of 2
tion as usual formulae
constitutes
Definition.
limited
) , E~
( = (6,0)
of ~
of a ranked
and
(for
no occurrence set-constant
£
formulae
are defined
comprehension
formulas,
terms,
and of a as follows: set-
then so are 7 %,
of~
such that
8 > ~, (iii)
~ contains
x for a set x of M i r i m a n o f f - r a n k
is limited
of a free variable
la without
free variables
comprehension
no occurrence
comprehension is defined
constants terms
of a limited
spect to the parameter X ~ . I n
no
> ~
(iv)
of a.3~ then
term.
as usual;
is said to be a limited
to the set-constants,
other
(i) % con-
of V 8 w i t h 8 > ~, (ii) ~ contains
of E B with
The notion
Definition.
of all
in
if ~ ~ I then ~ contains
respectively
by concatena-
formula with no free variables
tains no occurrence
above
symbols
(= limited)
than x~ and e is an ordinal
definition
) ,
A.] or variables, then ul g u~ are limited formulae.
(c) If ~ is a limited
the limited
= ( 3,X
that the collection
t e r m of
(b) If # and ~ are l~mited
refer
~
) , x
or constants
and ~i(u, ,...,Un.)
ESx~(x)
in
.
(a) If u, ,u2 ~... are limited
v
= (0,2) = ( 2,~
from these
comprehension
constants
are sets of ~
,) = ( 6 , 1 )
It follows
a class
The notions
symbols
= ( 0,4+i
are obtained
by reeursion.
quantifiers
, and finally brackets.
~ v : (O,l 7 , V
) , Vi
&j = ( 4 , j ) ,
exists).
E ~ and limited
that these
way: 7 = ( 0 , 0 )
) , e
i e I and e for membership.
(not, or, there
a limited sentence.
formuWe shall
of the form Aj (j e A) and
as constant
comprehension
terms.
Remark
that the
term is given with re-
most applications
we choose
k to be
~+1. Let p(x) be the M i r i m a n o f f - r a n k sense of ~
(see p.14).
term t is given by: (a) 6(x) (b)
= ~(x),
6(Aj)= I
(c) 6(Eax¢
(x))
:
of the set x in the
The degree
~(t) of,a constant
80
Abbreviations.
Let u and v be constant
terms or variables;
then
u = v stands
for A (x e u ~ x e v) where x is a variable distinct x For constant terms u and v~ u ~ v will stand for
from u,v.
A~x e u ~ x E v) where ~ = Max{~(u),6(v)}. x limited sentence. Next we define ring b e t w e e n
in ~
limited
a well-founded~
formulas
Read Ord(¢)
in
the the forcing relation ~
formulas
between
Obviously
"conditions"
m2 s + m°e + m we could define Ord(#)
For a limited 0rd(#) where
=
ordering
Ord(~)
to be
to these triples.
formula ~ define ~2.e
+
~.e
m
+
(i) e is the least ordinal
no q u a n t i f i e r
and limited
instead of defining
and then taking the l e x i c o g r a p h i c a l
Definition.
partial-orde-
to ~ an ordinal Ord(~)
as "the order of ~". This then allows to define
by induction on Ord(¢).
to be the ordinal (~,e,m)
localizable
~ by a s s i g n i n g
of 9 ~ . 9~
u ~ v is thus a
V 6 with
such that # contains
8 > e and no constant
term t of
degree ~ ~, (it) e = 3 iff ~ contains
at least one of the symbols
e = 2 iff ~ does not contain least one of the symbols
any ~i but ~ contains
A~, e = 1 iff ~ contains
symbol ~i and no symbol A~ but ~ contains
~i'
at no
a subformula
J
V S u where v is either a constant or a variable w h i c h stands quantifier
term with ~(v)
+ 1 =
in the scope of a limited
A ~ (for e defined
in (i)), e = 0 in all other
cases. (iii) m is the length of ~. Let S be an infinite sup{p(x);x
e S} where
0 ~ =
set i n ~
such that x e S ÷ 0(x) < ~ and ~ =
~ > ~. We want to find for each j e A (where
is the given first order relational
subset aj of S and generic relations
these aj's such that in the m e t a t h e o r y are isomorphic.
({aj;j
certain properties~
they have to satisfy certain conditions
instance,
a
e A},B i) ieI
Id est: we want to find a generic copy of 0~.
The sets aj have to fulfill words:
Sand
system)
B i for i 6 I b e t w e e n
or in d i f f e r e n t p (like "7 e a." for ]
or others) which can be posed i n ~ .
Instead of d e f i n i n g
the sets a. directly we first give a list ] saying that the sets aj and the r e l a t i o n B i have in ~ (the extension
of~)
only those properties
amount of information.
which are "forced" by some finite
81
Definition:
A condition into
Let P be the ordering
set of ~
The d e f i n i t i o n
of all c o n d i t i o n s
function
from S × A
ordinal
and
let ~ be the p a r t i a l
by p < q ~ p C q.
of f o r c i n g
on Ord(¢).
for a g i v e n
partial
2 = {0,1}.
in P d e f i n e d
induction
p is a f i n i t e
is given
Notice
first
B, all the r a n k e d
constitute
a set i n ~ .
Therefore
definition
of p IF # by i n d u c t i o n
Definition
of the
(strong)
for l i m i t e d
that p v a r i e s
sentences
¢ w i t h Ord(¢)
(by the r e c u r s i o n
on Ord(¢)
forcing
sentences
theorem)
relation
IF for l i m i t e d
takes
place
class
terms
and let u be a v a r i a b l e
over c o n s t a n t
< the
is p e r m i s s i b l e .
%. The d e f i n i t i o n of c o n s t a n t
¢ by
over the set P and that
in~.
sentences
Let T be the ~ ranging
terms.
(1) p I~ u e ! ~ ( ~ Y e x)(p I~ u = X). (2) p i~ u e
Eex~(x)
~ (~t
e T)(6(t)
< ~ & p I~ u ~ t &
p i~ ~(t))o (3) p l~u (4) p [ ~
e ~. ~ ( ~ x e S)(p IPx ~ u & p((x~j}) ] ¢ ~ ~(3q > p)(q I~ ¢).
(6) p IF V ~ ¢ ( x )
~ (~u
x (7) p I~ wi(u, ..... Uni) & p
!i-ul
=
&.
&
...
e T)(~(u) < ~ & p IF¢(u)). ~ (~j,,...,Jni
IR
& p
]~
strictly Further e =I .
smaller remark
occurring
=
by
~.
Ri &
).
In i
on Ord(¢)~
side of ~ have
occurring
in the d e f i n i t i o n
e A)(<j, .... j n ~ e
induction
on the right
that the f o r m u l a e
that
u
ni
To see that p I~ ~ is indeed d e f i n e d that the f o r m u l a e
= 1).
on the
of Ord(ul
notice
order
left side of ~.
~ u2) we h a v e
82
The definition given
of p I~¢ for arbitrary
in the Metalanguage
(ordinary)
(and not i n ~ )
by induction
length of ¢. This definition
over a set P and the collection
will be valid
of all formulae
a set in the sense of the meta-theomy sense of the meta-theory).
Z-sentences
Again
¢ will be on the since p ranges
of Z constitutes
(since ~ i s
a set in the
let u,v range over T and p,q
range over P. Definition
of p I~ ¢ for arbitrary (8)
p Ib u ~ v and p l ~ i ( u ,
(9)
p I~
,..,Uni)
< a & p I~ ¢(u)).
~ ( B u 6 T)(~(u)
to this definition
for limited
sentences
iff p I~ ¢ according
section
of the forcing
as above.
~ p)(q Ib ¢).
~ (3 u e T)(p I> ¢(u)).
that
¢.
are defined
(11) p I~V~¢(x)
The rest of this properties
¢ ~ ~(~q
Z-sentences
(12) p l ~ x ¢ ( X ) It is obvious
mata
(unlimited)
is devoted relation
¢ of £, p I~ ¢ according
to the former definition.
to the study of the formal I~. In the following
three
lem-
let @ be an~f Z-sentence.
Consistency-Lemma. Proof.
For no p e P do we have both p I~ @ and p I~7 ¢.
If p I~ @ and p I~ q ¢ for some p 6 p and some Z-formula
then by (9) p I~ ~ ¢ ÷ ~ p I~ ¢ and we get a contradiction metalanguage,
If p I~ ¢ and p < q, then q Ib ~.
Lemma.
Proof by induction tences
in the
q.e.d.
First Extension
on the complexity
# by induction
on the length
¢,
on Ord(¢)
of ¢ (i.e.
for limited
and for unlimited
sen-
¢ by induction
of ¢), see e.g. Jensen [~9]p.94-95.
Second Extension
Lemma.
For every p 6 p there
is a q 6 p, p ~< q,
such that either q I~ ~ or q I~-] ~.
83 Proof.
Suppose that for no q ~ p we do have q i~ ¢. Then p I~7 ¢
by (9). Suppose now that for no q > p we do have q I~7 ¢. Then by (9): p I~ 7(7 #). But applying p I~77
# * ~(3q
(Vq
(9) twice one gets
> p)[~(3 q' > q)(q' I> ¢)] > p)(3 q' > q)(q' I~ ~)
Thus there exists q' > p such that q' I> ~, q.e.d. Remark that forcing does not obey some simple rules of the propositional Furthermore, (I0),
calculus.
Exempl~ gratia,
p may force 7 7 ~ but not ~.
the forcing relation }~ has by definition
(12)) a homomorphism
(v,%) and existential
property with respect %o
quantification
junction ^ and universal
(V,3).
quantification
A
(^ ,&) or for universal p I~¢^ holds.
~ ~ (~q,
quantification ~ p)(~q2
We shall introduce
If we introduce property
(A,~).
~ p)[ql
the homomorphism
for conjunction
For example only
I~ ¢ & q~ I ~ ]
a relation I~f (called weak forcing), which
and universal
property
con-
then one no-
has the property that p I~*¢ ~ p I~7 -I ¢ and the homomorphism ty for conjunction
(5),
disjunction
as usual,
tices that I~ does not have the homomorphism
(clauses
quantification,
for disjunction
proper-
l~'does not have
and existential
quantifi-
cation and is, as we may say, dual to the strong forcing relation I~. Definition.
p I~*¢ ~ p I~-7(7 ¢) "p weakly forces p H ¢ ~ (p I ~
Qp
I~7 ~) "p decides
¢" ~"
p II*¢ ~ (p I~¢
~"
~ P)(p, ~ q & p= ~ q).
Lemma A: The weak forcing relation has the following properties (u,v are variables (i) p IZ*$ * ~ ( ~ q ) [ p
for terms and ¢,~ are any £-formulae)
< q & q Ib-] $],
(it) p ~-~ ~ p [ ~ , (iii) p Ib*-] ~ * p I~-'] ~, (iv) If # is of the form W , ^ W2,Wl * ~=,/\x ~, /\$W,.. u = v or u =, v, then p I~-$ * p l~-*O, (v) p I~¢ A ~ ~ [p IC¢ & p IC~] , (vi) p I~ A x ¢ "
(~u
6 T)[p l~*¢(u)l ,
84
(vii)
p ICAx~¢ ~ ( V u
(viii)
p I~-*¢ ~" ~ " [p I~-'¢ ~ p I~-*~],
(ix)
(~p'
(x) Proof. ~(~q
E T)[ 6(u) < ~ " p l~¢(u)],
i> p ) ( ~ q
(~q
> p')[q
Ib*~ ~ q I~'~] ~" p l~f¢ ~" ~
I> p)[q I~-*¢ ~ q I~*~] =~ p [~- ¢ *~ ~.
Ad(i):
Let ~ be ~ ~. By (9) of the forcing
~> p)[q I~-~] is equivalent
by definition Ad(ii):
of IF* this
follows
Ad(iii):
to p IbM ~ which
is equivalent
from (it).
This
Ad(iv):
is in contradiction
Notice
introduced
from
by definition
p I~
(9).
¢ and suppose
for some q > p. Thus by (it)
with p I~] ~ by the consistency
that all the forms
thus the claim follows
lemma and
Now assume
that - p N--] @. Then by (9): q i ~ q I~¢.
to p l~#.
from the first extension
"~' follows
definition is p Ib-] ~ ¢ ;
of ~ listed
(iii).
are of the form-IF,
The symbols
for longer
lemma.
^ , ~ , A , A e are
expressions
in terms
of
-]~V , V ~ V ~ only. Ad(v):
p I~
^
~ is by
by definition:
p IbT(q } v ] ~). This
(I0) of the forcing Using
again
Ad(vi):
Again
Ad(viii):
is similar
~< q' =~--(q'
suppose
extension
Lemma
By (i) there
(9) of the foreing
lemma.
follows
is in contradiction definition
In this way one immediately from
definition
(ix).
B. Let p and q be elements
this gives
of ÷. Using with p I~@ ~ ~.
does not hold and proceed
(10) of the forcing
directly
definition
q Ib-](¢ ÷ Y) by definition
that the conclusion
(9)~
(9) of the forcing
is exten-
i{--] ¢ ~ q' J ~ ) ] .
(v) one sees that this
(i) and
Ad(x):
p I~-] -] ~ & p I~ ~ -] ~.
to (v) or (vi).
q Ib-]-] ¢. Thus by
us q I~ ~(-]. @ v T). Thus:
using
(9) and then
to the proof of (v)
similar
(I0) and then again
Ad(ix):
first
to
p [b*~ ~ ~ and p I~¢ but -p !<~.
(~q')[q
and
is~ using
is
q of p suc~ that q I~-] ~. Since p ~< q the first
sion lemma yields
(iii)
with p Ib ¢ m ~, which
equivalent
equivalently
the proof
Assume
an extension
Using
definition
(9) one gets
The proof
Ad(vii):
(iv) equivalent
and the second
gets a contradiction.
This proves
lemma A.
of the set P of conditions
and
let ~ and ~ be Z-sentences. (i) If pl and p2 are compatible q {b ~ ^ (it)
and pl IF ~ and p2 IF ~
~ for every q greater
than both,
then
p, and p2.
If p II ¢ and p II ~ then p II ¢ v ~, p II ¢ ÷ ~, p li ¢ A and p Zl ~ ~ ~.
85
(iii)
If p II ¢ and p II ~, t h e n
p I~ ~ A
(iv)
If p II ¢ and p II ~, then
p I~¢ ~ ~ ~ (p I ~
Proof
by d i r e c t
Lemma
(p IF¢
& p I~?). ~ p I~).
computation.
C. Let p, ¢ and ~ be as in lemma (i)
~ ~
If p II*¢ and p H'~,
then
B.
p ~]'7 ~, p H'# /k ~, p II*¢ v
~,
p II*¢ ÷ ~ and p I]*¢ ~ ~. (ii) (iii)
If p II'~ and p I]'~, then
p lJ'¢ v ~ ~ (p I~*¢
If p I]*¢ and p ll*~, t h e n p i~*~ ~ ~ ~ (p I~¢ ~ p t~'~).
Proof
by d i r e c t
computation
Lemma
D. If p II ¢i (i = l,...,n), id est:
(use
an o p e r a t i o n
connectives
sentential
connective
of ~ and Q
(viii)
C is an n-ary
which
sentential
analog
lemma A,
and
sentential
is an i t e r a t i o n
7 and v)
(ix)). connective
of the p r i m i t i v e
and ~ is the c o r r e s p o n d i n g
of the m e t a - l a n g u a g e
(id est:
the
), then:
(i) p II C(¢, ..... #n ), and (ii) Proof
p I~ C(¢, .... ,¢n ) ~ t(p IF-C, .... ,p II- Cn ).
by i n d u c t i o n
(use lemma Lemma
on the n u m b e r
of times
7 and
v
are used
in C
B).
E. Let C be a n-ary
sentential
connective.
a tautology
for all ¢, ,...,~n,
then
Z-sentences
¢, ,...,~n
that
it holds
If C ( ~ , , . . . , ¢ n ) is
f'or all p and
for all
p I~C(¢, .... ,¢n ). Proof. Lemma
Use
lemma A and C (see
F. Let u, v and w stand
e.g.A.L@vy
[5~]p.141).
for c o n s t a n t
terms;
then
for every
p:
(i) p I~-u = u, (ii) (iii)
p l~u [p l~u
Lemma G. A g a i n
= v ~p
l~v
= u,
: v & p I~ v : w] ~ p I~ u : w, let u, v and w be c o n s t a n t
terms,
then
for every
p:
(i) p I~ u --~ v ~ p ]~ u = v, (ii) (iii)
[p l#u e w & p l~u
[p i#w e u & p ll-u : v] ~ p l~*w e v.
For a proof of Math.
: v] = p I~ v e w,
of l e m m a t a
Logic,
F and G see L~vy
voi.1(1970):[~].
|$I]p.141
or Easton,
Annals
86
Corollar~ .H: If p I ~ i ( u l , .... Uni) and p I~ul
= v, ,..p l~Uni= Vni,
then p I~ ~i(vl ..... Vni). This follows easily from the definition of forcing, clause (7), and lemmata F and G. Digression:
The forcing definition p l> ~ between elements p of
the set of conditions
(of 9 ~ )
in t h ~ " g r o u n d m o d e l " ~
and limited £-formulae ~ was given
while the definition of p I ~
was given in the underlying meta-theory.
for unlimited
We shall show in the
sequel that for each specific £-sentence ~ the forcing relation can be defined i n ~ ,
because # is finite and the construction of the
class K~ of p's forcing ~ can be done in finitely many steps. For each specific ~ the mechanism of constructing within ~
K~ can be implemented
but the mechanism is not universally applicable for all
sentences ¢ of £, so that within ~
we do not have the whole rela-
tion I>. This is not too much surprising,
since the definition of
forcing resembles very much the definition of truth, and by the Epimen~des-Tarski ~-language)
paradox we cannot define in ZF (or within the
the notion of truth for £-sentences
p.138, A.Tarski:
Logim, Semantics, Metamathematies
p. 248, Fraenkel-BarHillel: 1958) p.306 and Kleene: Groningen 1967) p.39,42,
[see L4vy [51] (Oxford 1956)
Foundation of Set Theory (Amsterdam
Introduction to Meta-Mathematics
(Amsterdam-
501, see also Mendelson [60]p.151].
However we can define forcing for a ~ingle given sentence # or f ~ some particular family of senten-ees w i t h i n ~ . Lemma I: Let ~(x,,...,x n) be an unlimited formula of £. There is a class K# of the model ~J'~ whose elements are the (n+l)tuples (p,ul,...,u n) such that p l ~ ( u l , .... Un) , where the ui(l ~ i ~ n) are constant terms. According to our remark on page 79 the constant terms u i are considered as certain special finite sequences of symbols which are in ~ -
for more details see Easton's thesis, Annals of math. Logic,
vol 1(1970). Proof by induction on the length of the formula ~. Since the atomic formulae are all limited formulae,
the lemma is true for
atomic ~. If ~(xl ~...,x n) is ~,(xl .... ,xn) v ~2(x,,..,x n) and the classes K~, and K~2 satisfy the lemma for ~i and ~2 respectively~
87
then K@ : K~I U K~2 is the required class for ~. If @ is V y ~ ( y , x l , .... x n) and if K~ satisfies the lemma for ~(x0,xl,...Xn) , then {(p,z); ~y(p,y,z) is V ~ ( y , x ,
E K~} is the required class. The case that
.... x n) is similar to the previous
7 ~(xl ,...,x n) and if K~ satisfies
{(p,z)
; p c P ^ ~Vqep(p
one. If @ is
the lemma for ~, then
~ q ^
E K~)}
is the required class K@. This proves lemma I. Definition:
A set ~ of conditions
is dense (cofinal)
set of all conditions~
in ( P , ~ k
the
iff for every p ~ P there is a
q e ~ such that p ~ q. Definition:
A sequence ~ of conditions
is complete
ordered by C and of ordertype ~, ~ = p(k),...},
iff ~ is well-
{p(0),p(1)
...,
such that ~ N ~ ~ ~ for every dense set
of eonditions. Remark.
Both definitions
not i n ~ ) .
above are given in the meta-language
(and
The notion of a dense subset of a partially ordered
set is due to F.Hausdorff who used the name "cofinal". definition of "completeness" was ~ bit m 0 ~ W.B.Easton
restrictive.
(Thesis,
for sets ~ of conditions
The definition
Princeton
The original of P.J.Cohen
given above is due to
1964, the main part appeared
in the
Annals of math. Logic, voi.1(1970)). Lemma J: if ~ is a complete sequence of conditions, £-sentence Proof.
then for every
@ there exists p(k) 6 ~ such that p(k)il
Let ~ be given.
By lemma I there is a set K in ~
elements are just those conditions
~.
whose
p for which p II @ holds.
By ~he
second extension lemma K is a dense subset qf P, thus K ~ ~ ~ 9, and there are conditions
p ~ ~ such that p E K and hence p 11 ~,
q.e.d. The following of~.
!emma is the only place where we need the countability
Notice that the weaker assumption,
~-subsets
of P is countable,
used e.g. by R.Solovay contains
in the construction
a non-constructible
namely the ~ - s e t
is already sufficient.
of
This was
of a model ~
which
A~-set of reals.
Lemma K: Ther~ a~e complete sequences
of conditions.
Moreover,
for
every condition p there is a complete sequence B in which p occurs as first element.
88
Proof.
Since ~
is countable
(in the meta-theory),
enumeration of all sets of ~
there is an
and in particular an enumeration
of the set of all subsets of P which are i n ~ be an enumeration
. Let {s ;n e m} n of the powerset of P in the sense of ~ .
Take any condition p and define p(0)= p. If p(n) is defined~let p(n+l) be any condition
in Sn which extends p(n) if such an element
exists, etherwise put p(n+l)
= p(n). We show that the so-defined
sequence ~ = {p(0),...,p(n),°..} conditions.
intersects
every dense set of
If ~ is a dense set, then it has a number, say n, in
the enumeration~ thus X = Sn. By definition p(n+l) e s n and p(n+l) e ~, q.e.d. Definition:
Let ~ be any collection age) of conditions
(in the sense of the meta-langu-
and let ¢ be an £-sentence.
I~ ~ for (3 P E ~)(p 15 @) and similarly ~ [ ~ (~ p E ~)(p [ ~ ) .
Notice,
quence of conditions,
We write for
that if B is a complete se-
then ~ [> # and ~ ~
are equiva-
lent. Lemma L: Let ~ be a complete sequence of conditions an Z-formula.
and ~(x,,...,x n)
If ~ [Q ul = vl ,...,~ I~ u n = v n for constant
terms u, ,... ,Un,V , ,...,Vn, then [~ #(u, .... ,u n) ~ ~ [~ ~(v I ,... ,v n).
This follows by induction on the length of ¢ from lemmata F~ G and corollary H. So far we have investigated forcing relation.
complete sequence of conditions predicates
several useful properties
gives raise to a valuation of the
~j so that the resulting
B) COHEN - GENERIC
sets are generic.
SETS
We shall use the terminology and formalism introduced Definition:
of the
In the next section we shall show that every
in section A.
Let ~ be a complete sequence of conditions.
Define
the function val~ (valuation or interpretation with respect to ~) on the set T of all constant terms of the language £ by induction on their degree as follows: valB(u)
= {val~(v);
Finally define:
v e T & 6(v) < 6(u) & ~ t~ v e u}
89 : {val~ (u);u 6 T}
~
(we shall usually omit the subscript ~ from val~ and
~).
Le~ma M: Let u and v be constant terms. If p [~ u e v then there is a constant term w such that ~(w) ~ ~(u), 6(w) < 6(v) and p Ibu = w, p I~ w e v. (for a proof see e . g . A . L ~ v y
[511p.141).
Lemma N: ~ I~ u = v ~ ~ i~ u ~ v ~ val~(u) ~ val~(v). Lemma O:
~
is a transitive set. For each x 6 ~ ,
hence
~
Proof: The transitivity of of val~ and
~.
val~(x) = x,
C ~. ~
follows directly from the definitions
val(x) = x follows easily by induction on ~(~)
using the definition of val(x), the forcing-definition lemma N. Thus the witnessing tained in ~
constants x ensure that ~
and the is con-
as a transitive submodel.
The semantics of £. For each x ~ ~
let r(x) be the least 6(w)
for which val(w) = x. Thus x,y e ~
& x 6 y ~ r(x) < r(y) by
lemma N. Now the formulae Qf £ can be interpreted
in ~
in the
following way: (i)
A term u is interpreted
(it)
u e v holds i n ~
in ~
(iii) The sentential connectives 7, v rare
by val(u).
iff val(u) 6 val(v). and the existential quantifier
interpreted as usual by -, ~ and ~ .
(iv)
V~(x)
holds in ~
iff there exists y E ~
(v)
y, ..... Yni satisfy ~i(x
with r(y) <
such that ~(y). .... ,Xni) in ~
jl,...,jn. ~ A such that (j,,...,Jni) i .... Yn i = va!(ajni).
iff there are
~ R i and yl : val(ajl),
One of the most important steps in showing that ~ ZF is by proving that ~
is a model of
can be described to a large extent
within~
. When one is dealing with inner m o d e l s ~ 1
structure
~2
in the verification of the axioms in ~ I
ally uses the fact that
~,
Cohen-extensions
similar features.
Though ~
~
one usu-
can be described entirely within ~ 2 ,
thus reducing the validity of some axioms in ~ 4
in ~ 2 .
of some
to the validity
of countable ZF-models ~ extends ~ ,
~
have
can be described to
90
a good extend i n ~ , of statements
so that again questions about the validity
in ~
can be reduced to questions which can be
posed (and "answered")
in~.
~his is the content of the following lemma: Lemma P: Let ~ be an £-sentence. For a proof see e . g . A . L ~ v y Lena
Q: p l ~
iff
~
iff ~ I~ ~.
[~I]p.144 or Easton's thesis [ ~ ] .
~ ¢ for all complete sequences ~ containing p.
Proof (see L~vy or Easton, Lemma R: v a i ( [ ~ x ¢ ( x ) )
loc.cit.).
is the set of all members y of ~
r(y) < e and y satisfies set of all y e ~ in ~
Then ~ holds in ~
%(x) in ~
. val(aj)
such that = aj is the
such that r(y) < ~ and y e a~ holds
.
Lemma S: For every j 6 A, val(aj) ~ S and val(aj) ~ , sets val(aj)
thus the
= aj are "new".
Proof.
a.] -C S follows easily from clause (3) of the forcing definition and lemma R. We have to prove that aj is "new". Suppose aj is not new. Then a 3• 6 ~
val(~)
and since aj = x for some x e ~ ,
= x by lemma O. Hence val(~)
yields:
= x = aj = val(aj)
and lemma N
~ I~ Aj = ~. Therefore p t~ aj = ! for some p 6 ~. Since p
is finite there are sets s e S such that p is not defined for (s,j). Now define q0 = p U {((s~j)~0) } and q, = p U {((s,j),1) }.q0 and q, are extensions of p. Proceed by cases. If s ~ x then s £ val~(~) for every complete sequence ~. Also~ hence s 6 val~(aj)
then ~ i~s e aj and
= aj by lemma R (since s e S ÷ p(s) < I and
6(aj) = l). Thus s 6 &j holds s ~ ! fails in
if q, g ~
= x
~.
in
Hence ~ l ~ j
~
for ~ containing q, while
# x for every complete sequence
containing q, (by lemma Q). Lemma A (iii) yields:
q, I~ aj ~ ~.
If s e x then proceed as above and obtain q0 I~ aj ~ !. Thus we have shown that every condition p has an extension q such that q I~ a.] ~ -x. By lemma A (i),(iii) this implies that every condition p forces a.] ~ -x. It follows now from lemmata 0 and P that aj = val(&j) ~
~,q.e.d.
Lemma T:
have the same ordinals.
~and
O~
91
Proof.
Since
~
dinal of ~
S ~)~
(notice
by lemma 0 every ordinal
that since 9~Z
the notion of "being an ordinal" ordinal of ~
of ~
s 6 ~.
x E y ~ r(x) < r(y), But by definition for all x 6 ~
E-models
Now let a be an
6(e) 6 ~ ,
Here O(s) g 6(e) holds thus x 6 ~
is an or-
are transitive
is absolut).
. Since e = p(s) g 6(e),
implies
Having
and ~
of ~
the transitivity
since s,y E ~
and
~ ~(x) g r(x).
of r(x) we have r(x) g 6(x),
thus p(x) ( ~ ( x )
, q.e.d.
proved
the m a i n - t h e o r e m
all these various
of forcing manely
lemmata we are able to prove
that the structure
~
is a
model of ZF.
HAUPTSATZ
of the forcing
Let ~ 0~=
be a countable (A'Ri)ieI
If the forcing
table,
relation
transitive
That
~
transitive
be a (iSt-order)
for every complete
Proof:
technique
E-model
I~ is defined
sequence 6-model
is transitive
of ZF and let
relational
system
in~
.
as in section A, then
~ the structure
of ZF which
~
extends
and includes
~
is a coun~
.
has been shown
in
lemma 0. Ad.axiom(0): in ~
Since val(~)
= ~ 6 ~
, the axiom of null-set
holds
.
Ad(I):
Extensionality
Ad(II):
follows
If a and b are sets of 7L
for terms
holds
The power-set
Cohen's
which avoids
constant
e such that
Since val(~)
= ~ by lemma
:
(Se~:
axiom holds
0, the
COHEN
instead,
G~del's
an elegant
argument
that for any constant if val(s)
term s), then val(s)
C val(t) = val(s')
that V = L proof due to
(see Easton's t e r m t there
holds
[1~]p.~&-~)
in the extension
the proof of G~del [ ~ ]
We follow,
We shall show,
s and
.
axiom h o l d s i n ~
iI) follows
the GCH.
R.M.Solovay ordinal
in ~
proof that the power-set
(see [~0]part implies
b = val(t2)
= ~2 and e = max{s, ,
pair of a and b.
Similar to (II). Ad(IV):
axiom of infinity
.
~ t, v x =~t2) is a term of degree
is the unordered
Ad(III):
of ~
, then a = val(tl),
tl , t2 of 2. Let 6(ti ) = el ~ ~(t2)
e2} + 1, then t3 = E ~ x ( x val(t3)
Ad(V):
from the transitivity
in ~
thesis ~ ] ) . exists
(for some
for some constant
term
an
92 s" of degree
less
sense
is given
of ~
than ~. Then
val( E ~ x ( x
the p o w e r - s e t
of val(t)
in the
by C t)).
Let t be a c o n s t a n t
term.
For each
constant
term
s define
(p is any condition): ¢(s) By lemma
: {
;6(w) < 6(t) ^ p l~w
I each ¢(s)
is a f u n c t i o n classterm
. Let this
G. We c l a i m
val(t)
N val(s2).
Proof.
Suppose
¢(s,)
by the d e f i n i t i o n p(n)
is a set of ~
of ~
¢(s,)
We c o n c l u d e
that val(w)
l~ w e s2 implies ~(w*)
< ~(s2),
= va!(w*), since
above]. we have
inverse
¢(si)
by lemma val(w*)
6 val(s2)
shown
that
val(t)
in the
implies
same way,
p(n)
precisely
val(w)
set
of terms
N:
E val(s2).
~ val(s2)
val(t)
and we have
and val(s2)
such
By lemma
6 val(t)
that val(s{)
I~ w e s~. :
of a term w"
implies
A val(s,)C
w
e ¢(s, ). Since
w* e s2.
we get val(w)
Then
~. F u r t h e r
entails
[more
existence
=
p (n) I~ w e t and
(p(n),w)
N val(s2)
M the
6 val(t)
D follows
= ¢(s2)
Thus
N val(s,)
N val(s,).
sequence
e ¢(s2 ) and this
6 val(t)
thus
Thus
val ~ val~:
s ~ ¢(s)
by the
val(t)
e val(t)
p ( n ) I P w ~ w* and p ( n ) i ~
val(w)
as stated
implies
in the c o m p l e t e
~(w) < 6(t).
p(n)
Further,
and val(w)
we get (p(n),w)
that
val(w)
p(n)
be r e p r e s e n t e d
= ¢(s2)
of the v a l u a t i o n
so that
= ¢(s2)
and the c o r r e s p o n d e n c e
function
¢(s,)
= ¢(s2)
IF w e st for some
may be t a k e n
The
that
~ s}
N val(s2).
proved
are equal
that modulo
val(t). Let T~(t) ~(t).
Then
For every ~(k)
be the
~(s)
C P × T6(t) , where
k C P × T6(t)
let
: {s; s e T ^ ~(s)
(~(k)
may be empty
axioms
of p o w e r s e t
s e ~(k)
Now
but
of this
suppose
and x C y holds
: k ^ /\s [s, in any case
set and define
that in ~
. Then Thus
~(u)
empty,
k is r e p r e s e n t e d
less
, {y;
than
conditions.
of all c o n s t a n t = k÷~(s)
is a set of ~
. Let
V Vk
8o be the
terms) 6(s,)] }.
). By the
s (k ~ P x T6(t ) ^ supremum
of
81 = 8o + i and e = 81 + 1.
x = val(u)
~ val(t).
since
~(k)
x and y are sets
u and val(u)
class
e T ^ ~(s~)
in~
is a set of ~
of d e g r e e
P is the set of all
(T is the
and r e p l a c e m e n t
^ 7 = ~(s))}
the o r d i n a l s
(in~)
of ~
suchlthat
for some
y = val(t)
constant
= k C P x T~(t ) and ~(k)
as a ~(u)
for some
constant
term is not t e r m u and
93
~(k) contains those terms of minimal degree. Hence let So be some element of ~(k) = ~(~(u)). Then val(u)
= val(t) N val(u) = val(t) N val(s0).
Define s, = E~Zx(x e t ^ x E so). Since ~(t) ~ 8o < 81 and ~(so) < 8,, sl is a constant term of degree 8, and val(u) true in ~
= val(sl ) is
(by lemmata M}N and R). Moreover 6(s, ) = 8, < BI + I =~.
Thus we have obtained an ordinal e (of O ~ )
with the required pro-
perties, q.e.d. Notice that the proof given above is similar to the proof that the powersetaxioms
holds in L - see page 28. Also the proof that
the replacementaxiom holds i n ~
will be inspired by the correspon-
ding proof for L. We need two lemmata. Lemma U: Let O(x0,xl ,...,x n) be an unlimited formula of £. Then for every ordinal e of ~ t h e r e
is an ordinal 8 of ~
such
that for every condition p and constant terms t, ~...,t n of rank less that e~
p Proof.
Vxo (Xo
,.. ,t n)
p J o (Xo ,t, ,.. ,tn).
Let @ be given and suppose that ~ has no free variables other
than xo,...,x n. By lemma I (see section A) there exists in ~
a
class K whose elements are the n+2-tuples (p,to,t, ,...,t n) such that p I~ ¢(t0,...,tn).
Hence
C = {(p,to,...~t n) ; p I~ ~(t0,...,t n) ^ ~(t i) < e for I ~ i ~ n} is also a class of ~ . By the axiom of foundation i n n the following collection C* is a set of ~ C* : {(p,to,..,t n) ;(p,to, ÷A(p,t;,t,
:
,tn) e C ^ [p I~
Vx ~(Xo,tl,..,t n) O
÷
,...,t n) e C (~(t°) < 6(t;))]}
Thus C" contains only those n+2-tuples (p,t0,.. ,t~ which to has minimal degree whenever
from C for
Vx0¢(x0,t, ,...,t n) is forced
by p. By the replacement axiom i n ~ , D : {y;
V ( p , t 0 , . . . , t n) 6 C *(Y : ~(to))}
is again a set of ~
~ and using again the replacement axiom i n ~
there exists an ordinal 8 such that y 6 D * y < 8. Then it is easily seen that the equivalence
stated in the lemma holds for this 8, q.e.d.
Lemma V: Let ©(xl,...,s n) be an unlimited formula of ~ and s be an ordinal of ~
. Then there exists a limited formula
@V(xz,...~Xn)
such that
94'
I (x , . . , X n) '~ ~ 7 ( x l , . . . , X n ) ]
Xn holds in ~
..
Proof by induction on the length of @. Case 1. If @ is atomic, then we can let #V be ~. Case 2. ~ is -] ~(xl ,...,Xn).
By the induction hypothesis there is
a limited formula ~V(x, , ... ~xn) such that A ex4 ... A ~ n [ ~(x, , . • • ,Xn) ~* ~V(X, ,...,X )] is true in ~
. Hence, we can let #V be -](~V).
Case 3. 9 is V y ~ ( X , ,... ,Xn,Y). Let Y = max{~,6}. hypothesis
By the induction
there is a limited formula TV(x, ,... ,Xn,Y) corresponding
to ~ and y such that: ""
holds in % .
A Xl Y
"
AYX n
.... ,Xn,Y) ~ ~V(x, ....
,Xn,Y)]
It follows that .. A Y X n [ ~y'{'(xl ~ "
is also true in ~
" " 'Xn
'"
" "
'xn ,Y)]
. This shows that we can define @V to be
Case 4. ~ is V y ~ ( X , ,...,Xn,Y). and ~ an ordinal
,y) ~ ~JSy~V(x*
By lemma U there is for given ¢
B such that for each condition p and constant
terms t, ,... ,tn of degree less than ~,
p l~ Vy~(t, , .... tn,Y) ~ p l~ VyS~(tl ,.... tn,Y). This means by lemma A, (vii) of section A. that every condition p weakly forces
A X~1 . . .
Xn[ V y * ( X , , . . . . X n , Y ) ~ \/B~'x A c~ y y ~. ~ . . . .
,Xn,Yl]
Hence also every p in the complete sequence ~ weakly forces this formula,
and therefore ~ also strongly forces the formula.
By lemma P of this section, this formula holds in ~ y = max{~B}
. Let
and proceed as in ease 3 (using B rather than 6),
q.e.d. Using lemma V we are able to prove~ that the replacementaxiom (VI) holds
in the structure ~
. Notice that our proof ressembles
very much the proof t h a t (VI) holds
in GSdel's model L.
Continuation of the proof for the Hauptsa%z Ad(Vl):
The replacement-schema
holds
in ~
Let t, be a constant
term of degree ~ and let ~(x,y) be a formula of £ such that it holds in ~
that for every x e tl there is precisely one y such
that ¢(x,y).
By lemma U there is an ordinal
8 such that
95
By lemma V, there
where
is a limited
~ = max{~,8}.
It follows
co) Let
formula
#V(x,y)
such that
from (+) that:
-
s = Ely( V x~ A z8
t e r m of 2, where
{V(x,y)
y = z ^ x e t ^
I = max{~,8}
is the image of val(t)
), then s is a constant
+ I = ~ + I. It follows
under the function
~ in ~
that va!(s)
, q.e.d.
Thus we have proved the Hauptsatz. Digression.
What have we done so fax? Our main q u e s t i o n was w h e t h e r
the i n d e p e n d e n c e
results we have obtained by means of the Fraenkei-
Mostowski-Specker-method
for the system ZF e (without the a x i o m of
foundation,
the e x i s t e n c e
but assuming
of r e f l e x i v e
are also true for ZF = ZF ° + foundation.
Obviously
dence results
results
carry over to i n d e p e n d e n c e
(AC) and (PW) - see p.62 independent
from (PW)
construction relations
are equivalent
is
in the
(i.e. r e f l e x i v e
model~
sets)
over the structure
(A,Ri) i 6 I, In order to obtain
ZF-set theory
independence
(including
foundation)
to a given countable model ~
results
which apply to full
our general
of ZF a generic
= (A'Ri) i 6 I" Obviously, meta-theory
we cannot
a contradiction
do is to construct
c o n s t r u c t w i t h i n our
to GSdel's
since this would
theorem.
from some given model ~
of ZF in w h i c h some i n t e r e s t i n g
idea was to add
copy of a structure
(which is ZF + (AC)) a model ab ovo,
give otherwise
~
(AC)
procedure
of ZF ° was to define some
a set A of atoms
the p e r m u t a t i o n
in ZF, since e.g.
in ZF w h i l e
The general
of a p e r m u t a t i o n model ~
R i (i 6 I) between
and then to construct C~=
-
in ZF ° alone.
sets x = {x}) not all indepen-
But what we can
of ZF another m o d e l
statements
~ are true while
others ~ fail~ thus proving that ~ ÷ ~ is not d e r i v a b l e A g a i n bv G~del's
theorem,
ZF-model w h e n p r o v i n g
that the e x t e n s i o n
we have d o n e by r e d u c i n g questions questions fact,
which can be posed
that the extension
(see iemmata determine
I and P).
in ~ .
these sets a. have. 3
~
about ~ This
can be d e s c r i b e d
is a
is a ZF-model.
This
by means of lepta P to is the most a s t o n i s h i n g in the g r o u n d m o d e l
It was the a i m of the forcing d e f i n i t i o n
the interior E - s t r u c t u r e
in such a way that
from ZF.
we have to use the fact that ~
in ~ These
of the "new" generic
to
sets aj(j6A)
we have evough
i n f o r m a t i o n what p r o p e r t i e s
finite amounts
of informations
were called
96
"conditions".
No%ice that since we are dealing with finite condi-
tions, the sets aj generic over O ~ are called Cohen-generic o v e r ~ .
determined by these conditions, The name "Cohen-generic" was
chosen in honor of the man who first invented forcing with finite conditions. Forcing with perfect-closed subsets of the real-line is usually called Sacks-forcing and the generic sets obtained by this way of forcing are called Sacks-generic.
Forcing with
Borel-sets is called Solovay-forcing and the corresponding generic sets Solovay-generie - see Silver's Seminar notes [$0] and the articles of Sacks and Solovay. We have developed Cohen-forcing in a general setting and have obtained for infinite sets S Cohen-generic subsets a. C S. In many cases we shall\~ as S. The Cohen-generic subsets of m will
be called simply
Cohen-generic reals, since every subset of
determines a real number. Instead of proving one independence result after the other we shall first collect some additional informations about the generic extensions ~
. We have shown (see lem~a T) that
and the extension ~
have the same ordinals. We ask: do they
have the same cardinals? id est: are the ordinals k which are initial ordinals in the sense of ~ of~?
just the initial ordinals
or better: under which conditions is this true? Another
question: Under what conditions on C ~ =
of choice (AC) true in the extension? Further, what are the conditions C ~ h a s order to ensure that the extension ~
to fullfill i n ~
in
satisfies the ordering
principle? In the following section we discuss these questions and give some solutions. C) O R D E R I N G S AND W E L L O R D E R I N G S
IN GENERIC
EXTENSIONS
We start with the presentation of a theorem which says that, if is a countable standard mndel of ZF + (AC) and O~ is finite, then the model ~ ~,
obtained from ~
by adding a generic copy of ~ b t o
satisfies ZF + Axiom of choice.
A necessary remark. In section B we have shown, that the model extends ~
(see lemma 0)~ but ~
subclass of ~
need not to be a ~ - d e f i n a b l e
. This, however, can be attained by adding to the for-
cing language £ a further unary predicate symbol g. The intended interpretation of g(v) is ~'v is in the g r o u n d m o d e l ~ " .
97
The forcing-definition
(')
has to be enriched
p I~ ~(t) ~ ( ~ x e 9 T C ) ( t
for condition~p 0~,
and constant
id est the semantics "y satisfies
It follows x 6 ~}
that g defines
the fact that ~ e.g.
is
proof that
Theorem.
If ~
~
that
~
standard
standard
are
model
model
usual)
constant
~(~)
= a, ~(~)
(defined
p.79),
it follows
= {(t~(t)) constant ~JaNa
: x
is possible
limited
in
quantifiers ~ of ~
V ~, , so
} are classes
of ~
by recursion
an interpretation
(for x e 9 ~ terms
here as on p.79). ~-definable
(as
of all
~ for the
of £ by recursion
~-class
N
theorem,
every
set x can be well-ordered. exists
that the function
(T is the class of all
of all sets of ~
there
the conventions
) of constant
= {~(t);t 6 Ta}.
remarks,
that
.
let us prove
that in
Let x be any set of ~
in ~
terms t
It follows
an ordinal ~ such that x C N a.
We claim that T e can be wellordered
on 6,
the correspondence
(see e.g.
Let T a be the set (of 9 ~
these preparatory in ~
),
Since
from the recursion
less than a. Define
~s the After
~[a]
language
so that ~, the collection
Define
;t 6 T) is ~ - d e f i n a b l e
terms).
of degree
(this
of ZF. Obtain
to all constant
x ~ x was by definition on
formulae,
~
of ~ by setting
and then extending the degree
to
of ZF + (AC) and a
a ramified
{(a,E a) ;a @ On ~
is a class o f ~ .
terms
is close
E ~ foz all ordinals
all the symbols
the w e l l - f o r m e d
these wff's,
is made
of ZF + (AC).
~-definable),
} and
that forcing
then the extension
Set up in ~
operators
{(a, V ~) ;a e On 9~
has furthermore
{val(~);
we shall need
the proof of which
over ~ ,
[~0]p.69).
comprehension
precisely,
(~). This assumption
with a name a for a, names x for x in ~ , since a and
by:
in L.
is a countable
is a countable
limited
of £ in
term t such
we shall assume
clause
theorem,
, more
of g. Whenever
0~-definable
(AC) holds
(R.B.Jensen
is a constant
in 0~
by means
is Cohen-generic
Proof.
t. The interpretation
iff there
in a way including
in the following
G~del's
terms
and ~ IP g(t)".
is ~ - d e f l n a b l e
was done
~ x).
for £~ then has to be enriched
g in 0 ~
that y ~ val(t)
by the clause:
In fact
; then
.
98
K s = {!,x e ~
^ 6(x) < ~} is a set of ~
Since M s = {~(t),t 6 Ke} is a set of ~ well-ordered
in ~
and induces hence
of K s . By definition
and included in T~
, this set M~ can be (via ~-1) a well-ordering
(see p.79) constant terms t of degree less
than ~ are constructed as finite sequences S = Ka U
{ES;B < ~} u
The set of ZF-symbo~s the set S (of ~ ordered,
{V8;6
of symbols taken from
~ a } ~ {the ZF-symbols}.
is countable,
hence wellorderable.
) is we!lorderable,
e.g. lexicographically.
Thus
and the set T~ can be well-
Let W
be a well-ordering
of T e.
For y 6 N~ let ty by the first t 6 T~ (in the ordering W~) such that y = ~(t). The funetion {(y,ty);y e x} is ~/~-definable and hence so is the Well-ordering {{Y,,Y2) ;Yl ,Y2 e x ^
be a countable,
standard model of the NBG-set
theory Z + (E) and let a be Cohen-generic over ~ % Then +
9~
e ~[a]
.
is a countable standard model of
(E).
Here Z is the set of axioms of groups A,B,C,D in GSdei's orange monograph[2~] Corollary
and (E) is the global version of the axiom of ehoice.
2. Let ~
be a countable standard model of ZF + (AC) and
a,,...,a n be finitely many sets which are Cohen-generic over ~
. Then
~
~ ~[
a, .... ~a n] is a countable
standard model of ZF + (AC). Corollary
2 was obtained by S.Feferman using ideas of GSdel and Cohen:
6] S.FEFERMAN:
Some applications
of the notions of forcing and
generic sets. Fund.Math. also Feferman's
article
56(1965)p.325-345.
See
(with the same title)
in
the "Theory of M o d e l s " - S y m p o s i u m volume, North Holland Publ. Comp. A m s t e r d a m 1965. Symmetry Properties
of Generic Extensions.
standard model of ZF and let in ~ .
Let
the ramified
9
C~=
be the group in ~
Let ~
be a countable
(A'Ri)iEl be a relational of automorphisms
language having constants
aj for each j 6 A and ni-ary predicate
of C~.
x for each x E ~
system Let £ be
, constants
symbols w i for each i e I
99 in section A). Let I~ be the forcing relation as defined
(as defined
in section A. For a 6 ~
and £-formulae # let a(#) be the formula
<7
obtained from ~ by replacing every occurenee of aj in ~ by ao(j)" For conditions
p (id est: finite partial functions
from S × A into
2 = {0,i}, see p.81) define o(p) by: (<s,j) ,0) e p ~ <<~,o(j))~0)
e o(p)
((s,j> ,1) e p ~ <<s,o(j)> ,1> e o(p) By definition a formula ~ of £ may contain some particular terms like x or Eex~(x) but contains never variables
for terms. Thus if
Eex~(x) occurs in ~, then E~xa(~(x)) occurs in a(¢). According to the forcing-relation Symmetry-Lemma
defined on p.81-82 the following holds:
(P.J.Cohen).Let condition.
~ be any £-sentence
Then for every a e ~
we have p J~ ~ ~
tJ
a(p) Ik a(¢). Proof by induction on Ord(#)
and let p be any
for limited formulae ~ and then for
unlimited # by induction on the length of 4. Exempla gratia, suppose the lemma is true for limited formulae # of order < e. If ¢ has Order ~, proceed by cases.
If ~ is u e i, then
p I~ u e ~ ~ (~y e x)(p I~ u z Z) ~ (~y e x)(o(p)I~ a(u)~ Z)) since y 6 x ~ p(y) < p(y), hence Ord(u - y) < Ord(u e ~). The latter is equivalent to (~y e x)(a(p)
J~ a(u) - [) which in turn
is by (1) of the forcing definition equivalent to a(p)Ik o(u) c ~, id est a(p) J~ o(u g x). One proceeds.similar in all the other cases. > The symmetry-lem~ma has the following consequence, if p i~ # and p is in the complete sequence ~, which d e f i n e s ~ holds in ~
p I~ o(~), hence o(~) holds i n ~ can handle symmetry-properties We shall use the s y ~ t r y - l e m m a models ~ L theorem,
, then by lemma P,
. If a is an automorphism of C~L as well. of ~
and o(p) = p~ then
Since
~
is in ~ w e
in the ground model ~ .
in order to prove that there are
of ZF in which choice fails, thus proving Cohen~s that the axiom of choice
(AC) is not deducible from ZF.
However, we shall not present Cohen's original proof [9 ], | ~ ] . In the proof given here a Cohen-extension standard model ~ but Dedekind-finite A.L@vy:
is constructed
~
of a countable
in whiqh there exists an infinite,
set A. This construction
is due to J.D.Halpern-
100
[~5]
J.D.HALPERN-A.L~VY:
The B o o l e a n
Prime
i m p l y the a x i o m of c h o i e e ~ To a p p e a r
in the P r o c e e d i n g s
p o s i u m at U C L A (see also J e n s e n ' s Definitions.
is c a l l e d
lecture
notes
A set x is c a l l e d
Dedekind-finite~
x one-to-one
of f i n i t e n e s s
was
every finite
onto
used
"Was s i n d und was
finite,
of c h o i c e
turns
~ x i o m of choice:
~ Every
Proof.
Let x be i n f i n i t e .
f defined
This
1 < n • m}
ZF + (AC ~)
t h ~ n f"y whole
x has
,...}.
By
f
f(y) • y. a denumerable
S n = {y C x; ~
subset.
= n} for' n • ~.
(ACW)°ther~
Define
g(n)
exists
a
= f(S n) for 1 < n < m.
is an i n f i n i t e
using
x would
(AC ~)
one o b t a i n s
D e f i n e h*(n)
countable
and t h e r e f o r e
=
s u b s e t of x.
every
S
n
iff it is D e d e k i n d - f i n i t e .
and s u p p o s e x is not finite.
a countable
infinite
Extend
f to a f u n c t i o n
= u iff u ~ y and f*(u)
is a o n e - t o - o n e Thus
set x has
a function
mapping
from x onto
be D e d e k i n d - i n f i n i t e ,
= f(u)
Then
subset y =
D e f i n e a f u n c t i o n f on y into y by:
= {z2 ,z3 ~
x - {zl}.
that
set x of n o n - e m p t y
exists
that
~ A set x is f i n i t e
lena
of x by f*(u)
T h e n f*
obviously
q.e.d.
Let x be D e d e k i n d - f i n i t e
{z, ,z2 ,z3 ~...}.
famous monograph
f r a g m e n t of the a x i o m
For e v e r y there
s i n c e x is i n f i n i t e
for 1 < n < m n o n - e m p t y ,
Proof.
f
In o r d e r to p r o v e the c o n v e r s e
= G is c o u n t a b l e . T h u s
t h e n { h * ( n ) ; 1 ~ n < m}
by the p r e c e e d i n g
in his
on G s u c h that h ( g ( n ) ) • g(n).
set is i n f i n i t e
Corollary.
a function
In ZF it h o l d s
is d e n u m e r a b l e .
n elements.
a function h defined h(g(n)),
not e x i s t
A set x
on T s u c h t h a t f(S n) • S n for all n, 1 ~ n < ~.
f(S n) c o n t a i n s
The set {g(n);
infinite Define
finite.
to some
s u b s e t of x (This d e f i n i t i o n
for all y • x it h o l d s
T h e n T = {Sn;0 < n < ~}
Sym-
sufficient:
such t h a t x is c o u n t a b l e ,
ZF + (AC ~)
function
does
the f o l l o w i n g
out to be a l r e a d y
Lemma:
Hence
x is e q u i p o t e n t
in 1888 by R . D e d e k i n d
set is D e d e k i n d - f i n i t e .
such that
iff
iff it is not
iff t h e r e
the a x i o m of c h o i c e ;
sets
93 pages.
of the 1 9 6 7 - S e t T h e o r y
s o l l e n die Z a h l e n " ) .
The c o u n t a b l e
Notes~
[Bg]p.164-167).
some p r o p e r
one n e e d s
(AC~):
t h e o r e m does not
(AMS-Publications).
m e m b e r n of ~. A set x is i n f i n i t e ,
mapping
Ideal
Mimeographed
f(z n)
= Zn+ 1,
f* d e f i n e d
on the
iff u • y.
(the p r o p e r
subset)
a contradiction,
q.e.d.
101
Now we shall construct a Cohen-extension model ~
in which
Theorem.
[~]
of some ZF-
(AC ~) fails by showing that in ~
iflfinite sets which are Dedekind-finite. to Halpern-L@vy
~
there are
The model used is due
as indicated above.
If ZF is consistent,
then
"ZF + there exists an infinite set which is Dedekind-finite" is consistent too. Thus Proof.
Let ~
following
be a countable
structure
0~
(AC m) is not provable
in ZF.
standard model of ZF. Consider the
: (A,R)
inn
, where A is m and R is
the unary predicate which holds for every x 6 A; thus ~ = Define a ramified language £ i n ~ ZF-symbols
which has besides the usual
also constants ~ for every set x of ~ ,
constants ai
for every i 6 m = A, a constant $ and limited %uantifiers limited comprehension operators that the sequences are i n ~ ( s e e
V ~ and
E ~ for all ordinals e of ~ ,
{ ( ~ , V ~ ) ; ~ 6 On 7Y% }
section A for
(m~m).
details).
so
and {(~,E ~) ; ~ E 0n9Tt } Define a condition p
to be
a partial finite function from e × ~ into 2 : {0,1} and define the forcing relation I~ as in section A. Thus the key-clauses and (7) read in the present context
(3)
(t is a constant term):
p I~ t e a. ~ (3 n s ~)[p Ib t = n & p((n,j)) 3
= 1]
Obtain a complete sequence ~ of conditions and thereby an interpreta%ion val~ of the constant terms of the language Z, which defines the m o d e l ~ .
Write aj = val(&j), b : val(i)~ then aj C ~ for all
j e ~ and b = {aj~ j 6 ~}. By our "Hauptsatz", ZF. We
want fo prove that in ~
finite.
This is done in several steps.
~
is a model of
, b is infinite while Dedekind-
1 Step. a i ~ aj if and only if i # j. Proof.
Suppose there are integers
a i = aj holds in ~
i and j such that i ~ j and
. Then.(by lemma P) a i = a~ is forced by some
p in the complete sequence ~, which defines ~
J. Hence p i P a i = aj.
Since p is finite there exists a natural number n such that (n,i)
~ p and (n,j)
~ p. Since i ~ j, we can extend p to a condition
q by defining: q = p U {((n,i),1),((n,j),0)}. By the forcing definition q I~ n e ai and q I~7 n e aj where p ~ q.
I02
Thus,
by the
2.Step. Proof.
It holds
ping b onto
Proof.
there
first
It holds Let
to show,
would
j e m} is infinite.
n of ~. This
in ~
function
is i m p o s s i b l e
f in ~
map-
by the r e s u l t
that b is D e d e k i n d - f i n i t e .
there
exists
function
that
in ~
(f"b
a number
which
--)c = b.
m 6 ~ such
maps
b onto
It is s u f f i c i e n t
that
f(aj)
-" aj for
j > m. By d e f i n i t i o n
E~x~(x)
= tf.
of ~
~ f is a limited
By our a s s u m p t i o n
thus by lemma
p J~ Fnc(tf) of numbers
^ Fnc(tf -I)
j such that
be any
(sufficient
Dom(p)
C ~ × k. This
that
Otherwise
in ~
that
term
f is one-to-oNe;
p 1~(gj,aj)
all
number
such
j e oec(~)
C
be the
~-set
tf : Eex#(x). that
Let k
occ(¢)
are smaller
C k and
than k and
i 6 m, e E 2, then n < k. e tf for
exist
and n a t u r a l
= b ^ Range(tf)
6~. Let occ(¢)
in ~, where
natural
means:
there w o u l d
of forcing)
^ Dom(tf)
sequence
aj occurs
large)
((i,n) ,e) 6 p for some
We c l a i m
it holds
comprehension
P:
for some p in the complete
if
a contradiction.
step.
c of b. We claim
that
obtained
be a o n e - t o - o n e
f be any o n e - t o - o n e
subset
we have
that b = {aj;
some m e m b e r
in the
3.Step.
all
in ~
Otherwise
proved
some
I st E x t e n s i o n - l e m m a ,
j > m = k + 1.
an e x t e n s i o n
numbers
n, ,n2
q of p (by the d e f i n i t i o n
such that
n~ ~ n2 , m ~ nl ,
m ~ n2 and (0)
q l~ ( an, " 'an2 " ) e tf
Choose
h 6 ~ such that (< n,j) ,e) 6 q implies
that n, ~ h, n2 # h. Define ~(n2)
= h, ~(i)
lemma
to
a permutation
n < h, j < h and
~ on ~ by ~(h)
= i for i 6 ~-(h,n2 }. An a p p l i c a t i o n
such
= n2 ,
of the s y m m e t r y -
(0) yields:
~(q) Ib < an, ,Ah) c tf since occ(~) q(Eex#(x)) q U a(q) ~(q)
is a f u n c t i o n
.Therefore
q U o(q) Hence
C_ k < m ~ nl ,n2 , hence
= Eexs(~(x))
and hence
by the first
[~ (an, ~an 2)
q u o(q)
: Eex~(x)
I~ &n2
o(#)
= tf.
= ~, thus q(tf) By d e f i n i t i o n
a condition
extension
lemma
extending
both
and lemma
B:
e tf ~ (&n, '&h ) e tf ^ Fne(tf) = tf(an, ) = ah'
since
=
of o, q and
^ Fnc(tf-l).
tf is a function.
This
103 contradicts the result proved in the first step. Thus in fact p ~tf(aj)
= aj for all j > m and f must be surjectiv. This
finishes the proof the theorem. D) THE POWER OF THE CONTINUUM IN GENERIC EXTENSIONS If we assume the axiom of choice (AC), then every set x is equipotent with precfsely one aleph M s. If reals then
~
= ~
~
is the set of all
for a certain ordinal ~. Is it possible to
determine this ordinal? It follows from Cantors theorem A x(~ < that ~ > I. G.Cantor has spent many years in order to solve this problem without arriving at the determination of the value for ~. The natural approach to this problem is to determine the cardinalities of various subsets of perfect set has cardinality 2M° compact subset of
~.
Cantor showed that every
( a set is perfect iff it is a
~, non-void and every element of it is an
accumulation point of it). Moreover, the Cantor-Bendixion-theorem asserts that every closed subset of
~
is either countable or the
union of a perfect set and a eountaS%e set. Thus no closed subset of
~ has a cardina$ strictly between Ro and 2 M° . Some further
results of classical descriptive set theory read as follows: (a) Every uncountable ~ - s e t
of reals contains a perfect subset.
(b) Every ~ - s e t
of reals is the disjoint union of MI many Borel sets. It follows that every ~ - s e t has power < M0 or = 2~0 Since Borel-sets are ~ every ~ - s e t notion ~ ,
(Souslin's theorem), hence ~ ,
of reals has cardinality < M, or = 2~
(For the
etc,.., see chapter II, page 44). For a treatment of
these results see: [~]
it follows that
[~8] an~:
A.A.LJAPUNOW: Arbeiten zur deskriptiven Mengenlehre; V.E.B.-Deutscher Verlag der Wissenschaften, Berlin 1955.
Since it was impossible to exhibit a subset of strictly between M1 and 2 ~ (CH)
2~
~
of cardinality
, Cantor conjectured in 1878 that
= ~,,
called the "Continuum-Hypothesis".
David Hilbert listed this
problem as the first problem in his famous list of unsolved problems at the first international congress of Mathematicians in 1900 in Paris. Despite many ~ttempts this problem remained for a long time unsolved. It was however used freely in proofs since it turned
104
out to be a powerful
assertion
W.Sierpi~ski
a large number of propositions
deduced
C1 -
C82)
[~91
W.SIERPI~SKI:
from
Hypoth&se
In the lit erature
du continu.
New York
there are many papers
to other
concerning
H.Rubin has shown, some r it holds
a series Logic
(GCH)
in which
of notes
published around
is discussed
on the
(GCH)
What
Kurt G6del
II of these
it fellows,
(GCH)
lecture
Hence
cannot be decided including Theorem
Proof.
has published of formal
K.GSdei
a survey
has on results
the more philosophic
Problem?
(GCH)
notes.
Since the
provable
question
Amer.Math. vol. SS(1948)p.151.
is consistent
Thus the
here a proof~ (GCH)
that also the
is neither
[ 9 ]-[I~]
B.Soboci~ski
Corrections
in 1938 that the
(AC) to the axioms
P.J.Cohen
Continuum
54(1947)p.515-525,
(GCH)
that
implies
(GCH)
has shown that the
(GCH)
cannot
(AC) cannot the
(AC) -
in the system
if we are
willing
set theory
(CH) or even the (GCH)
the truth or the falsity
with
is not a theorem
nor refutable
arises:
of Zermelo-Fraenkel
then the continuum-hypothesis
ZF + (AC).
in ZF to:
q, then for
of the (GCH):
ZF. But now the following add the
is equivalent
in which he gives
from the ZF-axioms.
of ZF. Thus
contributed
in the Notre Dame Journa!
in ZF. We have presented
24 -
and proved
°
3 and 4 (1962,63).
is Cantor's
showed
ZF, see chapter be proved
(GCH)
and in which he discusses
Monthly
be refuted
•
p and q, if p covers
(GCH)
I~ II, I!I~ vol.
K.GODEL:
•
W.S1erplnskl
AMS.65(1959)p.282-283).
in 1947 an article
the
see page
that the
cardinals
problem of the "truth" [Z6]
statements.
(CH) or the
that p = 2 r".
Bull.
(parts
1934
the (GCH).
e.g.,
"For all transfinite (see H.Rubin,
(there called
Warszawa-Lw6w
•
papers
situations.
1956).
continuum-hypothesis
to be equivalent many
symplified
(CH),
(2nd edition,
generalized
and also often
ZF, is
derivable?
is not provable
of the continuum
on the basis of the usual axioms
to
in
hypothesis
of set theory,
the axiom of choice•
(P.J.Cohen).
If ZF is consistent,
is consistent
too.
not a theorem
of ZF + (AC).
Let ~
Thus
be a countable
then ZF + (AC) + 2 M° > ~2
the continuum
standard
hypothesis
(CH)
model of ZF + (AC).
is
We shall
105
construct
an extension
new subsets
(though
that we will
Since
Define
find
a ramified
x for x 6 ~
hence has only eoun-
~ are all classes on p.79).
Define
from m × ~2 into
of is hope, since
£ which has besides
existential
quantifiers
E ~ (for ordinals
predicate
of ~ t ~ (use e.g. a condition
Z = {0,1}.
(see page
81-82)
the
symbol
a.
x ~ x~ ~ ~ E ~
the standard
p to be a finite
Define
the
V e~ the
~ in ~ ) ,
so that the correspondences
presented
{~ in the usual way
subsets
) M,), there
of m not yet i n ~
, a further binary
and ~ ~ V
function
language
operators
Define £ in such a way~
. We shall
of~.
the limited
comprehension
constants
many subsets
outside
in ~
so many
to add B~-in the sense o f ~ - n e w
~q~ is countable,
(M2)~-
in ~
ZF-symbols,
limited
generically
= HI is violated
these sets have cardina!ity
is countable
usual
2~
adding
(in the sense of the mete-language)
in ~
(~)~
~by
it is sufficient
of m to ~ .
tably many
of
of m, such that
show below that subsets
~
trick
partial
the forcing relation
containing
the following
key-clause:
p((n,9)) This means system
~
in terms = (A,R)
introduced
a generic
a complete
sequence
function
= ~2 y~u
copy of
and defining
B, our Hauptsatz
tells
~
the valuation-
us, that the model
of ZF which contains
a = val(Ea(x,y) a(x,y))
(the superscript
indicates
C ~ x MI
~as
a
~ for
that the concepts
are
in the sense of ~ ) .
By a theorem proved
in section
we have added t o ~ , set. Thus
it remains
by the t r a n s i t i o n from
if we define a~
C, ~C~ is also a model
of (AC),
since
a model of ZF + (AC), only one new Cohento show,
hypothesis is wrong. Since ~TT~ is the ordinal
for ~ < ~ = {n;
n e ~
then a~ ~ ~ and ~, ~ ~ theorem)
case A : 1 = {0} and R = ~,
in S = ~ × ~2. Thus by choosing
in this way is a model
and contains
understood
generic
~
~ of conditions
as in section
~obtained submodel
in section A: We take as relational
the very special
and choose
= 1).
and we get
m29T%
that in ~
in ~ a n d
to
the continuum ordinals
is an ordinal of
are preserved Thus
= y: ^ (n,~)
e
+ ag, ~ a ~
a}
(as in the proof of the preceeding
106
=7%
2 ~° > y y is in ~
(in ~
).
the second infinite cardinal:
~ : m2 7[b ; We shall show
that cardinals are preserved in the extension, which is a cardinal in ~
is a cardinal
i.e. an ordinal
in ~
and vice versa.
Then it will follow that y : m29~ is also the second in£inite cardinal in ~ n
:
y : m2 yt
, and hence 2 M° > M2 in ~
as desired.
To
this end we need some lemmata. Lemma 1. If B i~ in ~
a set of conditions
are pairwise incompatible, (~B
Proof.
C Cond)[B 6 ~ h
such that its elements
then B is ( i n ~ )
countable.
& ( ~ p l ,p2 e B)(p, ~ p2 ÷ p, u p2 ~ ¢tnd)
Cond is The ~'~-set of all conditions.
false, and let B be a set of ~ ,
Suppose the le~ma is
such that p,,p2 E B ÷ (pl = p~ v
p, U P2 ~ Cond) and ~7~L > m. Define B n : {p 6 B; { < n} n_L~_~lBn=
: B and B is uncountable,
B n is ( i n ~ )
Since
number n 6 m such that
there is
uncountable.
There are conditions
q E Cond such that (p 6 Bn; q C p} is in
~still
uncountable,
namely the empty condition q = ~ has this
property.
On the other hand the cardinality of all such conditions
q is bounded by n, since q C p. Thus we may define m to be the greatest natural number such that there exists a condition q such that ~ = m and {p e Bn ; p ~ q} is in ~ a~ondition
uncountable.
of cardinality m having this property.
Let q0 be such
Now choose in
{p e Bn; p ~ q0} any condition pl • Since in B all conditions pairwise
incompatible,
are
the elements of {p 6 B; p D q0} are also
pairwise incompatible. pl-q0
is not empty, since otherwise p, : qo and p, would be
included in all conditions
in {p E B; p D q0}, and hence compatible
with them. Thus we can find ((k,9),e)
6 pl-qe
is contained
uncountabl~ many conditions
in (in the sense o f ~ )
from B* : {p 6 Bn; p ~ qo}. This follows,
such that ((k,~),l-e)
since p, is incompatible
with every p e B*. It follows that {p e Bn; p ~ qo U { ( ( k ~ ) ,l-e)}} is uncountable
iN the sense of ~
and q0 U {((k,~),l-e)}
has
cardinality m+l, a contradiction to the choice of q 0 ~ a x i m a l dinality having this property.
car-
Thus lemma 1 is p r o v e d . ~ ;}~
Lemma 2: If f is a function in ~
, such that Dom(f) 6 ~
Range(f) C x for some x e ~ ,
and
then there exists a function
107 g in ~
such that Dom(f)
x, and g(s) is i n ~ Proof.
Since f 6 ~
= Dom(g),
countable
, there
Range(f) ~ R a n g e ( g )
for every s 6 Dom(f).
is by definition
of ~
a term tf
of the forcing language ~ such that holds in ~ (for x,z E ~ ) :
f e vai(tf).
(*)
~ if + v = w] ^ Dom(tf) = ~ ^ Range(f) C x.
Since
~u 4~w
~
complete
|(u'v)
is a generic
e tf ~ (u,w>
extension,
Thus the following
there is a condition P0 in the
sequence ~ (which d e f i n e s ~
) such that pc forces
(*)
-see lemma P in section B. Using weak forcing and lemma A of section A, this entails: ('')
(Vu,v,w
6~)(~q
> p0)|q l~'(u,v) e tf & q IC(~,w3
Further,
for every u 6 Dom(f)
there is a condition
sequence ~ such that p' i~*(~,f(u)) (~,f(u))
E tf holds
in ~
e tf = V : W] .
p' in the complete
~ tf (thiS follows
since
). Since both p~ and p' are in ~ and
is totally ordered by ~ we obtain that P0 U p' is a condition. Hence,defining g(s) = {y; y 6 x & (9 p' > Po)(P';~
<:,~
e tf)}
for s E z = Dom(f), we obtain that f(s) ~ g(s). The function g:z~
x is in ~
by lemma I of section A, and Dom(g)
and Range(f) C ~ J R a n g e ( g )
We claim that g(s) is countable choose in ~
= Dom(f)
in ~
for every y 6 g(s) a condition
for s 6 z. For s e z py ~ p0 such that
py i~(s,[)
e tf. We claim~ that {py; y ~ g(s)} satisfies
hypothesis
of lemma 1. In fact~
a condition,
then PYl
= z
C x.
the
if y, , Y2 6 g(s) and p.,~ U py~ is
e tf & Pyl U Py2 ~*(s'z2) e tf. But (**) entails y, = y~. Hence PYl = Py2 , since for every• y e g(s) we have chosen one py. Now lemma 1 yields that {py;y E g(s)} U
n~y2 IF*(s,v,> - ~
is countable. This in turn implies, that g(s) is in ~ g--~--~TT%~ ~, quod erat demonstrandum. Notice,
that we could interpolate
only a "multivalued" is not i n ~
function
between Range(f)
sequence
, and could thus not be used in order to find the function g (if ~ would be available
show
, hence f e ~
~
and x
g, since the whole complete
interpolating ~=
countable:
in ~
, we could
, but this is contradictory).
108
But this defect countable, Lemma 3. Proof.
is not too heavy
since g(s)
is for s 6 z always
as we have shown.
Cardinals
are absolut
in the extension
from ~
to ~
.
Let ~ be an ordinal of 7 i (and hence of ~ , by lemma T =rt B), and let e be the cardinal of ~ in ~ and let
of section
=7%
be the cardinal
with e).
Since ~
of ~ in ~
C ~
w h i c h is in ~ -TL =~ Hence ~ ~ ~
We shall in~L
then f e ~
from 60 = ~ "
c ~, (hence
By lemma
: Dom(f)
= Dom(f)
2 there exists
: ?.)"~"ange(g) ~ is infinite
trivially.
If Hence
a~d
let
in ~J~ a
= 6o and 6, = Range(f)
=), and s E 6, ÷ g(s)
= ~, = Range(f) ~
(since Dom(g)
equipotent
8 ~ e onto
onto ~, = e
and 60 = 61 follows
function g such that Dom(g)
= 9Ti
from ordinals
show that also ~ holds.
us assume that ~ is infinite.
URange(g)
the least ordinals
function
is also in
Let f be a function is finite,
(i.e.
, every
C
is eoun,table in 9@L . Hence:
< Dom(g) (AC) holds
× ~0 7~6 = - - 9 ~ Z in 7 ~
),
=9% : =')'Y'(,
=')'C
Hence ~
-- ~
to ~
= (~'~
=
(~
and the len~na is proved.
The lermma implies (the second
(So
=
infinite
thnt in p a r t i c u l a r cardinal)
~-- the notion of being
is absolute
. Thus the c o n t i n u u m has
in 9~
in the extension
from
power > ~2. This proves
the theorem. Theorem
(P.J.Cohen). (GCH)
then so is ZF + (AC) +
+ V ~ L. Thus the axiom of c o n s t r u c t i b i l i t y
provable Proof.
If ZF is consistent,
Let O ~
in ZF + (GCH)
be a countable
standard modQl of ZF + V = L and let
~f~ be the model obtained by adding to ~ = ~[a].
Then V ~ L holds
tible sets of ~ in~
. But ~
is not
+ (AC).
in ~
one C o h e n - g e n e r i c
since the class of
real a,
construc-
depends only on the class of ordinals which are and ~
have the same ordinals~
the class of constructible
sets.
Since a ~
not c o n s t r u c t i b l e
in ~
first t h e o r e m of
ection C. Further the
a C ~, and V = L(a) ding proof of L ~
. On the other hand
holds
(GCH)
in ~
hence ~
is in
, we infer that a is (AC) holds
(GCH) holds
in~
in ~
by the since a
. ~o see this, use the correspon-
in chapter
II~
q.e.l~. I
109 The continuum hypothesis can be violated in generic extensions in various ways. R.M.Solovay has extended the result of Cohen and has shown that 2 ~
can be anything it ought to be. The only values
excluded for 2~° are th~ose excluded by K~nig's theorem, which asserts that 2 M° is of cofinality greater than No. [81] R.M.SOLOVAY:
2~
can be anything it ought to be; In: The Theory
of Models, 1963 Symposium at Berkely; North Holland Publ. Comp. Amsterdam 1965~ p.435. Theorem (Solovay [61]): Let Me be an infinite ~ardinal in the countable standard model ~ is an exteNSion ~ of ~
of ~
with M0 < cf(M~). Then there such that the ordinals
are precisely the ordinals
and 2 ~
= M~ in ~
(cardinals)
(cardinals, resp.) of
.
The (GCH) can also be violated in various other forms. E.g. 2 Me= for ordinals e < y and 2MY ~
e+l
~Y+l (Solovay [8~], Derrick-Drake,
H.Schwarz et al.), thus answering a problem of Hajnal (Zeitschr.math. Logik u.Gr.Math.vol.2(1956)p.131-136). see e.g. Jensen [~0]p.68-74,
the thesis of Schwarz (cited on p.76)
and the article of D e r r ~ D r a k e Solovay has furthermore
For a proof of this result
in the same volume as Jensen [%0].
shown that there are generic extensions
in which 2Ni=Nni is consistent where i < n i and no < nl < ... < n k (k E m) is any sequence of natural numbers. W.B.Easton has extended this result.
By means of forcing with a proper class of conditions
he constructs model ~ 2~
a Cohen-generic
extension
~
of a-countable
of NBG-set theory (with global choice) = ~G(~)
for every regular cardinal ~
where ~ is any function i n ~ the following two requirements: (2) ~ ( ~ )
standard
in which: holds,
from ordinals to ordinals satisfying (1) ~ < B implies G(~) ~ G(8) and
is not cofinal with any cardinal less than or equal to N~.
This result is contained in Easton's thesis
(Princeton 1964),
published partly in: [ ~ ] :W.B.EASTON:
Powers of regular cardinals.
Annals of math. Logic,
vol.1 (1970), J.R.Shoenfield
has developed a method for obtaining generic exten-
sions of countable standard models of ZF without using ramified languages.
Dana Scott told us, that Shoenfield's
valent with the Boolean-valued
approach is equi-
model approach (see the forthcoming
110
article of S c o t t - S o i o v a y
in the UCLA-set
theory S y m p o s i u m proceed-
ings, vol.2). Shoenfield [~]
presents
J.R.SHOENFIELD:
the result of Eas~on Unramified
in his article:
Forcing;
set theory symposium at UCLA,
Proceedings
of the 1967-
to appear in the A M S - p u b l i -
cations.
E) THE INDEPENDENCE
0T THE
We consider the following (BPI) Boolean
Prime
(BPI) FROM THE ORDERING
- THEOREM
statement:
Ideal Theorem:
Every Boolean algebra has a prime
ideal. A Boolean algebra (B,U meet,
B is a distributive,
, m , - ) , where x •
B
=
y the
of x. B can be partially ordered by
y = y. Then x L.D z is the least upper bound
for x and z in B~ and x ~ y. The maximal
lattice
y is the join of x and y~ x n
and -x the complement
defining x ~ y ~ x ~
complemented
element
by 0 B. An ideal I in
y is the greateS~
lower bound for x and
in B is denoted by I B and the minimal ~
element
is a subset of B satisfying the following
three conditions: (i)
0 B e I,
(it)
x e I ^ y ~ x ÷ y 6 I,
(iii) x 6 I ^ y E I ÷ x U A prime (iv)
y e I.
ideal is an ideal with the additional
property:
x e I * (-x) 9 I.
In a Bool@an
algebra the prime
ideals are just the maximal
proper
ideals. The Boolean Prime of equivalent
forms
ideal theorem
(BPI) has a c o n s i d e r a b l e
in several branches
of m a t h e m a t i c s
although by far not as many as the axiom of choice is thus an i n t e r e s t i n g Lemma. The following
and natural p r i n c i p l e
statements
Boolean-Prime-Ideal
theorem:
(in ZF) with the
Every Boolean algebra
to a field of sets.
(b) The T y c h o n 0 f f - t h e o r e m ' for T~-spaces: pact H a u s d o r f f - s p a e e s (c) In every commutative is included
(BPI)
(BPI):
(a) The Stone r e p r e s e n t a t i o n is isomorphic
(AC). The
of set theory.
are all equivalent
theorem
number
and in logic,
is compact
in the product
ring with unit,
in some prime
ideal.
The product of comtopology.
every proper ideal
111
(d) The Stone-~ech (e) Alao$1u's
compactification
theorem:
Banach-space
(f) In every Boolean
algebra, theorem
exists
for ISt-order
sentences
Z is consistent,
constants.
then For a proof
If
theorem
measure.
languages:
Let
with arbitrary
for 1St-order
If every finite
E
many non-
then it has a model. languages:
subset of
Let
Z
Z has a model,
Z has a model.
see the following
38 (1951)
a 2-valued
choices.
logical
be as in (h).
H.Rubin
there
of a
space.
be a set of Ist-order
(i) The compactness
60 (1954)
Hausdorff
of consistent
(h) The completeness
Math.
The unit sphere of the adjoint
is a compact
(g) The principle
theorem.
and Fund.
p. 390~ L.Henkin:
- D.Scott:
papers: ~ o s - R y l l
Math.
vol.
BuI!.AMS.
Bull. AMS.
Nardzewski:
41 (1954);
D.Scott:
60 (1954)
60 (~95u)
p.389;
Fund. BulI.AMS
p. 390; R.Sikorski:
Boolean Algebras ($pminger-Veriag Berlin 1964), AppendiX. We axe interested namely
here
in one of the consequences
of the
(BPI),
the ordering , principle:
(OP) Every
set x can be totally
We shall use the notions synonymously
(i.e.
ordered.
"totally
ordered"
equivalently).
and " l i n e a r l y
A somehow
stronger
ordered"
principle
is
the following: (OE) Order-Extension-Principle: on x, then there The
exists
(OE) has been discovered
W.Sierpi~ski:
Zarys
(Fund.
Math.
L.Henkin (proof
by Banach,
terji mnogosei,
first proof which appeared 16 (1930)
Zorn-Kuratowski
If x is a set and r a partial
a linear ordering
in print
p. 386-389).
in order to deduce
observed,
that
(OE)
ultrafilter ZF
theorem).
implications
also hold.
in ZF or not.
Marczewski (OE). ~os~
The (Szpilrajn)
used the lemma of Ryll-Nardzewski
a consequence
theorem,
and
of the (BPI)
or directly
whether
using the
It is not known~ In this
which
the converses whether
says~
that
(BPI)
of these
(OE) ÷ (BPI)
section we shall present
is
a result
(OP) ÷ (OE) is not a theorem
of ZF. In the next section we shall present that
p. 158).
is due to E.Marczewski
(see:
Thus we have:
in the problem,
of Adrian R.Mathias, A.L~vy~
and Tarski
19~8,
ordering
~ such that r C t.
~ (AC) ÷ (BPI) ÷ (0E) ÷ (OP).
We are interested provable
Kuratowski
Warszawa
is already
either via the compactness
ten
the proof of J.D.Halpern-
does not imply the axiom of choice
(AC),
112
Preparatory remarks. Mostowski has constructed in his paper [6~] a model~
containing urelements
(atoms) in which the ordering
principle (0P) holds while (AC) fails in it. Mostowski takes a countable set o9 atoms l i n e a r l y
ordered of type q0 (i.e. the order
type of the rationals). A set is called symmetric iff it is mapped onto itself by some finite-support subgroup of G, the group of all order-preserving mappings from qo onto no. A set x is in the model ~iff ~is
x is hereditarily symmetric. The proof that (0P) holds in based on the fact, that every set x of ~ h a s
a unique
minimal support, supp(x), where supp(x) is a finite subset of no (the set of atoms). The correspondence x ~ supp(x) is in ~
and
hence the lexicographic ordering of supp(y) for y 6 x together with wellordePings of the sets K e = {y 6 x; supp(y) = e} can be used to obtain a (symmetric) totaiordering of x. This idea can be carried over to Cohen-generic extensions. The r~le which was played by the urelements in Mostowski's model ~ , will be played by generic reals in +he Cohen-extension. However~ instead of adding a
generic copy of no (the rationals) to some
countable standard model ~
of ZF + V = L (in this case we would
not know, how to destroy (OE) in the e x t e n s i o n ~
) we shall add a
generic copy of a certain partially ordered set ( I , ~ ) t o ~ . If ( I , ~ ) h a s
sufficiently enough automorphisms, then the generic
Copy of ( I , ~ ) w i l l
not have in the extension ~
a total ordering,
which extends ~ (the symmetry-lemma will be used here). What are the properties, ( I , ~ ) h a s extension ~
to fulfill, so that in the
the ordering principle (OP) remains true. The typical
property of no, which was used by Mostowski in |6~]
in order to
prove the existence of a unique, minimal, finite support of every set x
of his m o d e l ~
(the supports are sets of urelements!) was
the homogeneous ordering of q0. We shall show, that, if we require that ( I , ~ ) i s
a countable, homogeneous, M0-universal partially
ordered set i n ~ , of ~ )
and ~
~ ZF + V : L, then in ~
(the extension
every set x = val(E~x~(x)) has a unique, minimal, finite
support supp(x), such that the correspondence x ~ supp(x) is ~-definable.
The verification of (OP) in ~
We need here, that ~
is then standard.
is a model of V = L for two reasons, first
in order to establish (using a theorem of B.J6nsson) that there are in ~
homogeneous, universal partially ordered sets, and second
in order to ensure that K e
:
{x; supp(x) C e},
of I, has a definable wellordering i n ~ b
e a finite subset
. Finally let us notice
113
that, in contrast to Mostowski's
permutation m o d e l ~ ,
the fact,
that we have chosen a partially ordered set ( I , ~ ) a n d
not a line-
arly
ordered set (like Mostowski's
q0) will not cause any troubles
when we want to linear order lexicographically
the supports~
since
the generic sets ai (for i e I) are subsets of m and {ai; i 6 I} has thus in ~
a definable ordering
(namely the ordering of the
real-line). Having clarified the basic ideas behind the construction of a model ~
of ZF + (OP) + 7 (OE) + ] (BPI)~ we start to present the
details of the proof. universal relational Hausdorff's
First we define the notion of a homogeneous, system.
The notion is a generalization of
notion of an De-set.
For more information we refer our
reader to the following publications: [41]
B~JONSSON: vol.8
Homogeneous
universal relational
[3 ] J.L.BELL - A.B.SLOMSON:
Models and Ultraproducts;
publ. Comp. Amsterdam 1969. Definition.
systems; Math.
Scand.
(1960) p. 137--142.
The t$pe • of a relational is a finite ordinal) of natural numbers
North-Holland
(chapter 10). system ~ :
(A~R i) i6~ where
is a sequence (n0,nl ~... n~_ 1)
such that for 0 ~ i ~ {-1 the relation
R i is ni-ary. Definition.
The relational ~=
(A'Ri)ie~
system ~ : (B,Si) ie ~ is a subsystc__m of iff B -c A and Si = Ri N Bni (restriction
of R i to B). Definitioh.
Let X be a class of relational type T. A system
systems all of the same
~9~ is ~-homoseneous,
iff the follo-
wing holds: (2) If ~ = (B,Si)ie~" D e ~, and if ~ is a subsystem of C ~ =
~,
then $ can be extended to an
automorphism of ~ . Definition.
Let ~ be a class of relational type and let e be an ordinal, (~,~)-universal
systems all of the same A system
O ~ 6 ~ is called
iff every system ~ = ( B ~ S ~ i E ~
that ~ ~ M~, is isomorphic
to a subsystem of ~ ,
such and A
114
has itself cardinality M~. Bjarni Jdnsson has proved in [9~] (GCH), that under certain conditions universal,
homogeneous
relational
under the assupmtion of the
on X and H e there are (Ms,X)-
systems.
that there are countable Mo-universal,
It follows in particular
homogeneous
partially
ordered sets (here ~ is the class of all partially ordered sets). Theorem
(A.R.D.Mathias):
Let ~
ZF + V = L; then ~
be a countable,
standard model of
can be extended to a countable standard
model ~
of ZF + (0P) + ] (0E). Thus the orderextension
principle
(0E) and a f o r t i o r i
the Boolean prime ideal
theorem (BPI) is independent
from the ordering principle
(0P) in the system ZF. This result is contained [~8] A.R.D.MATHIAS:
in
Tha. Order Extension Prineipie;
Proceedings
of
the 1967-set theory symposium at UCLA. To appear. Proof.
By the theorem of B.J6nsson [Wi]
countable,
M0-universal,
homogeneous
• We shall extend ~
there exists in ~ a
p~rially ordered set ( I , ~ )
by adding t o ~
a generic copy of ( I ~ )
.
We emphasize that if we write i < j then i ~ j and i ~ j, Thus < is irreflexive,
while ~ is reflexive:
We construct quantifiers in~),
in ~
a ramified
i ~ j ~ (i < j V i = j).
language 2 with the usual limited
V ~ and limited comprehension operators the ZF-symbols,
constants
E ~ (for ordinals
x for eaclh set x o f ~
, indivi-
dual constants ai for each i 6 I and two further constants A and < . The wellformed
formulae and limited comprehension terms are defined
as usuai~ with the restriction that if A or ~ occurs in the X-formula then Eex~(x)
is a limited comprehension term only if e > m ÷ 1.
A condition p is a finite partial function fron! ~ × I into 2 = {0,1}• Define the (strong) of this chapter.
forcing relation I~ as in section A
In our present case clause
(3) reads:
p I~ t e &i ~ (~ n e ~)(p I~ ~ = t & p((n,i)) where t is any constant term of ~. Clause
= ~)
(7) reads:
p ~ t e A ~ (~ i e I)(p I~ t = &i ) P }~t, < t 2
~ (~ i, e I ) ( ~ i 2
e I)(i~ < i2 & p l~t,
= ai &
p ~ t~ = a. ) 12
115 Let again l~denote
(2)
(Vi
Write
forcing relation.
sequence
of the constant
a 1. = val(ai),
~ of conditions
terms
and thereby
t of £, which defines
A = val(A),
~ : val(~
of A. By our Hauptsatz,
~L
is a model
show is, that the ordering in ~
principle
parial
in section
- lemma which we have proved
C (see page
the model ~
ordeming
.
(in ~
of ZF. All what remains
(OP) holds
in ~ L
. To this end we need the following
and the symmetry
a valuation
), then a i _C ~
A ={ai;i e I} and V is an irreflexive
fails
It follows
6 I)(~ I~*a i ~ i).
Obtain a complete val~(t)
the weak
while
restriction
)
to
(OE) len~a
in its full generality
99), so that it is available
in the present
situation. Restriction
lemma:
Suppose
that p I~ # and let occ(#)
set of elements
Proof.
i ~ e~},
let p/oec(#)
then p/occ(#)
I~.
c0 = o e o ( ~ ) ,
and q = p/occ(~).
section A) there mentions finite
names
set c, c,
By defintion
exists
of reals
Suppose
;WE ~ V
~ q I~.
an extension
p ^
Then by lemma A (i) (see
in co, and also others,
e0 N d = co n ci ~
say those in the
= ~. By the univerality
on co.
to an automorphism
I~ o ( q ~ ) .
exists
By the homogeneity ~ of ( I , ( )
we
But ~ is the identity coincides
where both have the same values, Hence p U o(q')
is a function
first extension
lemma
an isomomphism
of ( I , ~ )
. Hence:
o(i)
lemma,
~ T can be
= i for i 6 c0
q'I~] }
on co : oco(~),
and we obtain a ( q ' ) I ~ ] #. By construction p and o(q')
of ( I , ~ )
of I, such that (~o U d,~ ) and (c0 U e2 ~( )
c2n (eo U d) = ~. and there
is identical
functions
)E
q' of q such that q7 t ~ 7 ~. q'
and o(9) ~ e2 for j 6 ~ . By the symmetry ~(q')
E ~(n~i)~e
= {i 6 I;Vn~ m V e 6 2 (((n'i) ~e ) 6 q' " i ~ co)}.
are isomorphic,
extended
d = {i E I
i 6 occ(~) iff a. occurs 1 = {((n,i) ,e ) e p;i 6 oce(#)},
:
can find a subset which
of I such that
in ~. Further
Define
be the finite
implies
hence ~(#) ~ ~,
of o, the domain
only on a (finite
of the
subset of m x co,
since q C p and q C q~, q C ~(q').
and therefore
a condition.
(see section A): p U a ( q ' ) I ~ 7
By the
%. On the other
116
hand p I~ ~ entails also p U ~(q,) {p $, a contradiction to the consistency - lemma (see section A). This proves q I~*~, q.e.d. Lemma: The 0rderextension
Principle
(0E) does not hold in ~
particular V = v a l ( < ) cannot be extended in ~
. In
to a total
ordering of A = val(A). Proof•
$uppose V can be extended in ~
Let R be such an orderextension exists a limited conprehension val(t).
to a linear ordering of A•
of V. By the definition of
term t e Eax~(x) of £ such that R =
Let ~(t) be the £-sentence
+'t is a tctalordering
ding < " Let c = occ(t) be the set of indices occurs
~there
in t. Hence c : occ(~(t)).
of A exten-
i E I such that ai
Let S(c) : {i C I; V jec (i ~ j V
j ~ i)} be the "shadow of c in ( I ~ ) ~ " . By the universality
of ( I , < )
we may embed the following partially
ordered set into I-S(¢):
I
I. ii
12
I i°
I.
. 14
13
[all elements
@. 15
are pairwise
incomparable,
only i4
is smaller than i+I. Since R = va!(t) {ai,,ai2,ai3} (ai2~ai3) such that:
is a linear ordering of A in, it holds in + ~
are ordered by R. Assume e.g. that (ail,ai2>
6 R. Hence we have obtained • -
%~.
that
E R~
six generic terms Ai, ~ .... ai6
(i)
~ S~ai4
16
(ii)
~ 1~(~im.. < ~ik v ~ik ~ aim ) for m,k = 1,2,...6 with ( m , k )
(iii) ~ l{n(~im ~ aj V ~
# (4,6) ,
< aim)
for m = 1,2, .... 6 and j ~ c = occ(t) (iv)
3~u b (ai, ,ai ) 6 R ~ (al2. ,a13.) E R.
Since everything which holds weakly)
by some condition
in ~
must be forced (strongly or
in the corresponding
complete sequence ~,
we obtain, that there is a condition p in ~ such that p l~(t) By the restriction
^ (ail,ai2) e t ^ (ai2,ai3) e t lemma, we may assume that p contains
many ordered pairs
(in,i> ,e > (with n 6 m, e E 2) only with i6occ(t)
U {i,,i2,i3}
Define
finitely
117
p1(oec(t),i, ,i2) = P/oee(t)
U {il,i2}
p~(oec(t),i2,i3)
U {i2,i3}.
Then by the restriction (+)
= P/oce(t)
lemma:
pl(oce(t),i, ,i2) I~*T(t) ^
(++) Ft(occ(t),i=,i,)I~*~(t) Now define mappings
^ (ai~ 'at, ) z t.
T1 and T~ on occ(t) U {il ,i2 ,is ,is } and on
occ(t) ~J {is ,i3 ,i4 ,is }, respectively, T1
~ t
by:
• i, m is, i2 ~ is, T1 identical on oct(t),
r~ : is ~ is, i3 ~ i4, T z identical on oct(t); Henoe T, maps (oct(t)
<J {ii ,i2 },~ ) isomorphically
i6},~ ) and similar T2 maps (occ(t) (oec(t)
U {14 ,is},~ ) . By the homogeneity
by extended to automorphisms
on (occ(t)
U{is
U {i2 ,i3},~ ) isomorphieally
on
of (I,~<) , T, and T2 can
~I and ~2 ,respectively,
of ( I ~ )
.
Using s, we obtain from (¢) by the symmetry-lemma: (0) q, = p, (occ(t),is,is) and using ~Ifrom (++):
(00)
q~
(notice,
l~'T(t) ^ ( at6 =at ) e t
= pI(occ(t),is,i4) I ~ ( t )
That the symmetry-lemma
eing I~).
^ (&is,~i4)
~ t
also holds with respect to weak for-
Since q, U q2 is a con4ition
and ~ l~'ai4 < at6 we obtain:
q, U q2 J~*T(t) ^ &i4 ~ ai~ ^ (at6 'at,)
e t ^ (ai~
'at4) g
t
But t extends < , hence (at4 "ai~ ) e t. But ~(t) says, that t is a linear omdering the following:
on A, a contradiction' p is in 6{, the complete
Define 6{* to be the sequence second element,
since what we have shown is sequence,
and containing
then all conditions
extend p. Then ~* defines obviously
P/occ(t)
having p as its
q of 6{ which
the same model ~ .
P/oec(t)l~*~(t) we infer that every complete P/oce(t) model ~ 0
which defines ~ .
starting with p/oct(t))
Since
sequence 6~ starting with
must force ~(t), and ~(t) has to hold, hence also in the defined by some complete
sequence 6~ which starts with
and has q, <J q~ as second element.
But we have just shown~
that in this model val8 (t) cannot define a total ordering on v a l ~ (A) which extends val6{° ( < ) . a contradiction.
Next we want to prove, ordered.
In this way we have thus obtained
This proves the lemma. that in 0 ~
every set can be totally
The idea behind the proof is the following.
that for different /-formula
It can happen
~ (x) and ¢~ (x) we have that
118
val(Eex#1(x)) ~ val(E~x#2(x))
where
occ(¢~)
show that for every x in ~
there
is a 2-formula
u = val(Eex~(x))
it follows,
limited
{occ(#);
that every
comprehension
term E~x~(x)
with minimal
u ~ support
of u is definable
(OP) holds
in~
in ~ L
of a
set occ(~),
called
that the correspondence
. Then the rest of the proof
is standard.
If t : E~x¢(x)
c, then we shall write reals
such that
is the valuation
of u. ~e then have to show,
that
~(x)
u = val(Eex¢(x))}.
set u of ~
the support
Notation:
We want to
and
oct(T) C ~ Then
# occ(¢2).
is a limited
t ~ t(c)
comprehension
in order
term and occ(@)
to indicate
-
that t mentions
ai if and only if i e c.
Lemma.
Let t(c,dl) tioning where
and t'(c,d2)
only reals
disjoint
subsets
is a limited comprehension
m e n t i o n i n ~ reals
£. Hence
terms men-
of I. Suppose
that
term t" e t"(e)
a. for i e c such that i
~t(e,d~
) = t"(c).
in ~
is forced
All what holds
sequence
comprehension
ai for i ~ c U dl ~ i E c U d2 respectively
c,dl ,d2 are finite
then there
Proof.
be limited
there
exists
by some p in the complete
a condition
p E ~ such that
p J~t(c,dl ) = t'(c,d~). By the restriction
lemma we may assume
that ((n,i) ,e) E p implies
i 6 c U d~ U d2. [more precisely, not to be in ~
the restricted a priori,
has p as its second which assume,
extend
P/c U d, U d2
but the sequence
element
p defines
condition
and contains
obviously
that we have already
~
which
: po need
starts with po,
then all conditions
the same model ~
q of
. Thus we may
chosen £~ as the sequence
which
defines~]. Define Hence
Po = Po(¢)
= P/c'
Po U p, U P2 I~t(c,d,)
term t"(c) will be found It will be enough element~
say i 0 . The general
for the Z-formula
: t'(c,d2).
: P/d,
and p~ = ;2(d2)
A limited
=P/d2.
comprehension
for which p, U P2 U p31~t(c,d, ) = t"(c).
to consider
We shall write
p* = p' (dl)
the case when d2 contains case follows
only one
by induction.
i # y for ](i < j V j ~ i) and similarly
7(x ~ y v y < x v x = y).
O
x # y
119
Let ii,i2,i3
be elements of I, but not in c U dl U {i0},
such that iI < io < i3 ^ i2 I io and such that for v = 1,2,3 there are automorphisms i 6 c U d I + Tv(i) 1,2,3);
TI .~2,T3
of < I , ~ ) w i t h , ~ ( i o) = i v and
= i. Then by the symmetry-lemma
Po(C) U p1(d) U p2(i v) l~t(c,d,) where Tv(P2(i o)) : p~(iv) for notational
(for v =
: t'(c,i v)
simplicity.
with Po(C) U P1(d) L] p2(i°) l~*t(c,dl) = t'(c,i0)
T h i s together
implies
(using
the ~-estriction lemma): (*) for v
po(c) U P2(io) U p2(i v) i~*t'(c,io) -- t'(c,i v) = 1,2,3. We introduce the following notations.
For a condi-
tion q define Cq ("the content of q") to be the following £-sentence:
E aj; <(n,j> ,1 > e q} ^ ~ {
n m e aj; <
where
denotes conjunction.
Let Cq(-]/w)__ be the result of replacing
the generic constant a. in C
by the variable x. q For a finite subset s of I let D s be the diagram of ({aj ;jes),< >
i ° e.: °
D s - - ~ { a j i < a j2; jl < j 2 ^ jl ,j~ e s} ^ ~ {
-] a.]i < a.]2 ;
j, ,j2 e S ^ n jl < j2 }. Let Ds(J/x) be the result of replacing the constant a. in D by the ] s variable x. We now claim that the following continuity-property holds:
(*.) p0(c) u p~(i0) l:Ax[x ~ ~ ^ Cp2(i°/x) ^ %u{i0}(i%). t'(c,i°)
÷
= t'(c,X)} ,
where t'(e,x) results from t'(c,i0) by replacing a.
in t' by x.
i0
Let k be a limited comprehension term. We have to show that, if q is any extension of p0(e) U P2(i0) and q l~k e i A C (i0~) ^ D U"~i0J " ~ ( i ~ ~ then there is a q' D q with q ' ~ t ' ( c ~ i o ) p2= t'(e,k). Assume that k and q are given such that the just mentioned hypothesis
is fulifilled.
Since in particular q l~*k E A we know by
the definition of forcing, that there exists a q' _D q and a j E I such that q'I~* k = a.. Pick v = 1,2 or 3, s~ that ({i0 ,j},< ) and 3 < i 0 ,iv},< > are isomorphic. We claim that there is an automorphism T of ( I , < )
so that for this choice of v, T maps (e U {i0~iv},~ >
isomorphically oN (c U {io,j}~< > in such a way that T restricted to c U {io}
is the identical mapping.
In fact, since q' extends q the first ex%ension lemma (see
120
page 82) yields: q' I~'k = aj ^ Cp=(iYj) which means
in particular,
^ DeU{i0}(iy]),
that q' (weakly)
forces that c U {i}
and c U {j} are isomorphic. Since already ~ i~*Dcu{i0}(l~j) ly) we obtain that in fact
such ~hat there is an isomorphism
(obviousare
al leaving c pointwise
But by the choice of v there is also an isomorphism o2 from
(c U {i~},~< > onto (c U {i0),~< > leaving c pointwise To = o.Io2 is an isomorphism leaving c pointwise
fixed.
mapped isomorphieally pointwise
fixed.
from (c U {iv},~< )
fixed• Hence
onto (c U {j},< )
It follows that (c U {io,j},< ) can be
on (e U {i0,iv},< ) so that c U {io} is left
Let T be such an isomorphism.
Since q' I~*Cp2 (i~j)
it must hold that p=(j)
=~(p=(io))
C q,.
Hence p0(c) U p~(i0) U p=(j) C q' and applying
T to (*) the symmetry
lemma and the i st extension
lemma
entail : q' I~t'(c,i0)
: t'(c,j).
Since q' I~ k = aj, this gives us q' I~ t'(c,io)
= t' (c~k) as desired
and (**) is proved. The limited
comprehension
Suppose that t'(c,i0) the constants obtained
a.
!0
term t"(c)
and ~. for i E c. Let %*(x,y) i
from }(y) by replacing
of occurrence)
can now be constructed.
is E~y~(y) where ~ is a formula of ~ containing
- it is assumed,
be the £-formula
ai0 by the variable
x (at all places
that x does not occur in ~(y).
Then define t"(c)
= Eey(Vx[x
e A ^ Cp2(i°/x) ^ Dcu{i0}(iyx)
By (**)~ p0(c) U p=(io) p E ~ and everything
l~*t'(c,io) = t"(e).
Since po(e) U p=(i0) _C
forced by p is true in ~
Lemma:
The ordering principle
Proof.
Set up in ~
(OP) holds in ~
a ramified
^ ~'(x,y)]).
, the lemma is proved. •
language £* with a name a* for each
a E A, names A* (for A), V* (for V = val(~ )), names x* x E ~,
limited quantifiers
E ~ (for all ordinals
~ of ~ )
V ~, limited comprehension and the usual ZF-symbols.
in a way so that {(a~a* ) ; a E A} is a set of ~ E 0n 9%
}
and {(~,E~) ; ~ E 0n TL }
for each operators Do this
and {(~, V ~ ;
are classes of ~
. Notice,
121
that we assume here that ~ be made by the remarks
is
~
-definable
of page 96-97.
Notice
-this assumption
furthermore
can
that we
could not use symbols
like a~ for i E I as names for the elements l of A since the c o r r e s p o n d e n c e i ~ a i (i E I) is not in ~ (for each i E I we added a generic not add the c o r r e s p o n d e n c e Define an i n t e r p r e t a t i o n
real a i = val(a i) to 7 ~
, but we did
{(i,~ i) ; i E I} g e n e r i c a l l y
~* for the constant
to ~ ) .
terms of £* i n d u c t i v e l y
by setting: ~*(a*)
: a
~*(A*9
=
~*(V*)
: V , and
g*(x*)
= x
and then e x t e n d i n g that g° is
to all limited
comprehension
Now define
is a term t*(d)
of £* m e n t i o n i n g
= u. Read supp*(u)
lemma,
supp*
For each finite
V d = {x e ~ Each class
supp*(g*(t*)) constant
of £'used
; supp*(x)
wel!ordering [2S]p.36).
: val(~).
Notice,
Further,
constant
the symbols
wellordering
supp*(B*(t*))
the lexicographic
t of is
ordering
-deflnable ~ " by G~del
an induced well-
ordered,
b e i n g a subset of the real line. ordered
and thus the
as modified
~* one obtains
from
with
= d have an ~
of V d. Each d i~ a finite sub~et of A~ which
[For more details
terms ~upp*
terms t*
ordering
method.
that there
constant
wellordering:
Using the i n t e r p r e t a t i o n
d of A can be l i n e a r 1 ~
a* for a E d
= d}
: d have an ~ L - d e f i n a b l e
(e.g.
such that
omly the names
* between
to construct
terms t* of £* with
, as the
subset d of A let
V d has an ~ - d e f i n a b l e
the alphabet
cardinality
is always well defined.
t* of £* so that g*(t*)
~-definable.
terms of £*~ so for u ~ ~
as: the Support of u. By the
is a clear o n e - t o - o n e - c o r r e s p o n d e n c e £ and terms
supp*(u),
call it d, of A of minimal
with ~*(t*(d)) previous
for each x ~ ~ ,
~-definable.
finite subset, there
for each a E A,
A
is linearly
The set of finite
subsets
e.g. by the usual ! e x i c o g r a p h i c
see the proof of the t h e o r e m on page 97-98].
Hence let Lex be the l e x i c o g r a p h i c
o r d e r i n g of finite
subsets
of A and for each finite subset d of A let Wd be the w e l l o r d e r i n g of V d. Now if u is any set of ~ ,
then
{(xl,x2) ; xl ~x~ e u ^ [(supp*(xl),supp*(x2)) v (supp*(xl)
: d : supp*(x2)
is a linear o r d e r i n g of u. This proves of A.R.D.
Mathias
is proved.
E Lex
^ (X, ,x~} E
v
Wd)]}
the lemma and hence the t h e o r e m
122
Let (ACwo)
(in the notation
consequence
of the usual
CACwo): AxE Ay y Vf(Fnc(f) Theorem
(A.R.D.
Proof
([5~]):
~
yexf(y)
from (ACwo)
Take the model iff there
, then the ~ - d e f i n a b l e Thus,
a function
+ Ordering
~
f in ~
which
principle
constructed
above.
subset
C d. Hence,
turns
in preceeding
This proves
CHOICE
An interesting
[~]
Fund.
element,
(KW-AC):
then there
to each z 6 x a wellordering. holds
in ~ .
is Hence
As
(0P) and 7 (AC) are true in ~
Math.
version
and K.Wagner
W.KINNA-K.WAGNER:
it
.
Ueber
of the axiom of choice has been
in their paper:
eine A b s c h w l c h u n g
des Auswahlpostulates;
42(1955)p.75-82.
to the usual
the functions
but proper
out to be a well-
PRINCIPLE
weakened
by W.Kinna
In contrast
in
the theorem.
F) THE KINNA-WAGNER
considered
lemmata,
in ZF.
A set z is in
sets,
(ACwo) was shown
is
d of A such that for
well-orderable
assigns
(0P)
(AC)
if z is wellorderable
linear-ordering
if x is a set o
then ZF + (0P) + (ACwo)
Thus the Axiom of choice
is a finite
all y, if y 6 z, then supp*(y) ordering.
too.
÷
e y)l.
If ZF is consistent,
is consistent
wellorderable
be the following
^ y can be wellordered)
= x ^
Mathias):
independent
[3g]p.223)
axiom of choice:
x + y
^ Dom(f)
+ 7 (AC)
of A.L6vy
(AC) where
considered
the choice
function
by Kinna-Wagner
select
selects
one
non-empty,
subsets:
The Kinna-Wagner whose
elements
exists
choice
principle:
If x is a set all of
have at least two elements,
a function
f, defined
then there
on x, such that for all y 6 x,
~ f(y) ~ Y"
Ax[ Ay(y e x
+ 2 < ~)-~
Vf(Fnc(f)
^ Dora(f)
= x ~
Ayex(~ ~ f
Obviously
(KW-AC)
have shown, "Every
that
is a consequence (KW-AC)
set can be totally
implies
of (AC).
W.Kinna
the ordering
ordered":
and K.Wagner
principle
(OP)
123
Lemma (Kinna-Wagner~%]): Sketch of Proof.
ZF ~(KW-AC)
+ (OP).
Let M be a given set and let f be a function,
defined on P(M), the power set of M, such that ~ # f(y) # y, f(y) C y whenever y 6 P(M) and y contains at least two elements. Following the N - ~ e t h o d ordering-theorem"
of Zermelo's
second proof for (AC) ÷"Well-
(see: Math. AI~. 65 (1908)p. I07-128) one starts
the proof with the following definition: Let F be the least family of subsets of M satisfying: (i) M 6 F,
(2) (x 6 F ^ 2 ~ ~) ÷ (f(x) E F ^ x - f(x) 6 F), (3) t c r ÷ ~ t
e r.
Define an element e of F to be normal iff the following holds:
AxeF[e c
x ^ e * x + e c f(x) v
e i x - f(x)].
There are normal elements in F, e.g. M is normal.
Fere
normal,
define: ge = {x E F; x N e = ~ V Ix A e # ~ ÷ (e C x v x C f(e) v x C e-f(e)]}. One proves that'g e satisfies the conditions therefore
(1), (2), (3) and is
(by the minimality of F) equal to F. This shows that if
e is normal, then (°) A x e F [ X N e : ~ V (x A e ~ 0 A [e C x v x C f(e) V x ! e -f(e)])] holds. Now, let N be the set of normal elements of F, then N satisfies conditions ~is
(i), (2)~ (3) and thus N : F by the minimality of F.
shows that every element of F is normal.
means the following:
Intuitively this
by means of an iterated application of f to M,
then to f(M) and M-f(M), then to f(f(M)), (M-f(M)) - f(M-f(M)),
etc ....
f(M)-f(f(M)),
f(M-f(M)),
one obtains by transfinite
induction
the following binary tree: M
level O
level 1
~
f ( M ) ~
~
(M-f(M))-f(M-f(M)) lev.2/~.
124
where
at e a c h p o i n t
where
f(x)
x the tree
is the s u c c e s s o r
splits
into two b r a n c h e s ,
of x at the
the s u c c e s s o r of x on the r i g h t branch. every
element
either going
is n o r m a l
means
that
e and x are on d i f f e r e n t
difficult
~o prove,
The p r o o f
a chain
C in P(M).
this
that a binary,
The tree
of C, t h e n D(C)
(Kinna-Wagner
on the b r a n c h in mind,
than
after
it is not
p, d e f i n e d
on M by:
the
le~ma.
wellfounded
tree can be
of M, s u c h t h a t the image forms
itself
is e q u i p o t e n t w i t h if D(C)
is a m a x i m a l proves
[~])
is a b r a n c h
q ~ f ( ~ {x e F; p,q e x}).
proves
the p o w e r s e t
H e n c e M ~ 2 ~. This
Theorem
M
if t h e r e
level of x is s m a l l e r
P(M) ~ [. On the o t h e r hand,
completion and M.
of M. This
shows,
into P(M),
~, h e n c e
Or~
t h a t the b i n a r y r e l a t i o n
is a l i n e a r o r d e r i n g
embedded
the
of e. W i t h
p p q ~ p e f ( ~ {x e F; p,q e x}
that
if e E F and x E F, t h e n
e C x) or x a p p e a r s
one of the two s u c c e s s o r s
if x ~ 2,
and x - f(x)
The i n f o r m a t i o n ,
branches,
t h r o u g h x and e then e i t h e r
the l e v e l of e (case:
left b r a n c h
the
chain
some o r d i n a l
is the D e d e k i n d in P(M),
linking
following:
In ZF the
statement
(KW-AC)
is e q u i v a l e n t
to the s t a t e m e n t : (KW-O): there
For e v e r y
exists
powerset (KW-O) that
every
set M has
M i n { ( x U y)
first
function
a linear ordering
- (x N y)} e y. the
that
P which
for some o r d i n a l
Since
ZF
~ (AC) ÷
first or the last
t h a t b o t h c a n n o t be r e v e r s e d . shown,
~ such t h a t
m a p p i n g M into P(~)~
f o r m of the O r d e r i n g - p r i n c i p l e ,
o r d e r i n g ~ on P(~),
we ask, w h e t h e r known,
a one-to-one
is an o r d i n a l
the
of ~.
is a s t r o n g
canonical
set M t h e r e
(AC) does not
since
it a s s e r t s
is a s u b s e t of the ~ , d e f i n e d by x ( y ~ (KW-AC) ~
(KW-O)
a r r o w can be r e v e r s e d . J.D.Halpern
follow
from
÷ (OP)
It is
and A . L 6 v y h a v e
(KW-AC)
in ZF (see [ZS]
and: [$~]
J.D.HALPERN-A.LEVY:
the O r d e r i n g
a x i o m of ehoice~ (January The p r o b l e m , some y e a r s
(OP) + (KW-O)
though Mostowki
in ZF ° (id est ZF but w i t h o u t not i m p l y
(KW-O),
T h e o r e m does
not
of the 1~mer. Math.
i m p l y the Soc.
11
1964)p.56.
whether open,
Notices
see:
can be p r o v e d has
in ZF r e m a i n e d
shown already
in 1958,
the a x i o m of f o u n d a t i o n )
for
that
(OP) does
125
[6~] A.MOSTOWSKI: On a problem of W.Kinna and K.Wagner; Coll. Math. vol.6(1958)p.207-288. The independence of (KW-O) from (OP) in ZF 0 follows more directly already from the fact, that (PW) holds in every Fraenkel-MostowskiSpecker model of [b~] there,
(see p,62), hence in particular
in which
(OP) holds, while
then (AC) would also hold.
(AC) fails.
Das Ordnungstheorem
nersche Auswahlprinzip.
If (KW-O) would hold
In 1969 U.Felgner has shown, that
also in full ZF, (KW-O) is independent
[18] U.FELGNER:
in Mostowski's model
from (OP).
impliziert nieht das Kinna-Wag-
To appear.
Here we shall sketch the proofs for both results. prove that (AC) is independent model ~ [ a 0 , a
I ,...,A]
But~in order to
from (KW-AC) we shall not use the
(a n generic reals for n E ~, A = {an;n E ~})
which was used originally by Halpern and LEvy (for a more detailed description of this model~
see sections
model of Mathias,described
in section E, since this will give us the
additional
information,
sion principle Theorem.
C and G). We shall use the
that (KW-AC) does not imply the order-exten-
(OE).
The choice principle the model of Mathias.
(KW-AC) of Kinna and Wagner holds in Hence,
if ZF is consistent,
(KW-AC) does not imply the orderextension though
(KW-AC)
implies the ordering principle
Proof. We have shown in section E, that Mathias' following
features:
for every set u of ~
then
principle
model
(OE),
(OP). ~
has the
, the following relation
is a linear ordering of u: {(x,,x 2) ; x,,x 2 ~ u ^ [ ( s u p p * ( x , ) , s u p p * ( x 2 ) ) 6
Lex
v(supp*(x,)
= supp*(x 2) : d ^ (xl ,x2} e Wd) ]} Here Lex was the lexicographic ordering of finite subsets of A~ the set of generic reals ai, and W d was the wellordering supp*(x)
= d}. Since A C 2 ~°, every set u of ~
many wellordered
subsets.
that u can be embedded
(i.e. mapped into, by a one-to-one
large ordinal e. Thus
follows from results, Theorem.(U.Fel~ner):
splits into ~ 2 ~°
Thus we can compute an ordinal ~ such
into Nk × 2~°" Thus u can be embedded for a sufficient
of V d = {x;
(KW-O) holds
proved in section E, q.e.d. If ZF is consistent~
from the ordering principle
function)
into 2 N~, the powerset of ~
(OP).
in ~
. The rest
(i~ka ¢.~, ~ = ~ % )
then (KW-O) is independent
=
126
Sketch of proof. Let ~ Let Q be the set in ~ ordering
be a countable standard model of ZF + V = L. of rational numbers and let ~ be the usual
(of type no ) of Q. Define
which contains
besides
in ~
a ramified
the usual ZF-symbols
quantifiers V ~ and limited comprehension constants
and the useful
terms E e
ai,j' hi' for i 6 Q and j 6 ~, a constant
predicate ~ . Let 2 further contain constants a unary predicate
language £ limited
(e e On 9 ~
),
6 and a binary
x for each x 6 ~
and
g, which will be used to make 0 ~ an ~ - d e f i n a b l e
class. A condition
p is a finite partial function from ~ × Q × ~ into 2.
A forcing relation I~ is defined as usual containing key-clauses
the following
(here let t be any term of Z):
p l~t E ~i,j ~ ( ~ n
e ~)(p ]l-n -- t &?((n,i,j))
= I);
p I~ t ~ i i ~ (~ j ~ ~)(p l~t "- ai,j) ; p l~t e 6 ~ ( ~ i
e Q)(p I>t = bi);
p I~ tq ~ t2 ~ ( ~ i ~ , i 2 Obtain a complete
e Q)(i, ~ i2 & p I~ t,-- bi & p I~ t2-" hi2).
sequence ~ of conditions
val of the constant
and thereby a valuation
terms of £, which gives the model ~
It holds
that ai, j = val(~ i,j) _C m, b i = val(~i) = {ai,j; j e ~}, e = val(6) = {bi; i e Q} and v a l ( E ~ + 3 X ( V y , V y 2 ( X : (Yl,Y2) ^ Yl ~ Y2))) : ~ is a linear ordering of c. Notice, that inside of ~ there is no isomorphism between (Q,~< > and (e,~ > , but outside of ~
both are
isomorphic. Symmetries.
Let ~
be the group in ~
one-to-one mappings
of all orderpreserving
from Q onto Q and let ~ be in ~
the group of
all permutations
of ~, which move only finitely many elements
(1) ~'.~ operates
on £ by setting:
~(bi)
if ~ 6 ~
= b~(i)" ~ acts as the identity
(2) ~ operates Ti = (Y,i>
on £ by setting: and then:
w(p) = {<(n,~(i),j • i(p) = {(
of ~.
= a~(i),j'
on all other symbols
of 2.
E ~ × Q, then define
Ti(ai, j) = ai, T (j), Ti(Ak, j) = ak,j if
k ~ i, T i acts+ identical (3) If p is a condition,
if (T,i)
, then ~(ai,j)
on all other symbols of £.
z 6 ~
) ,e > ;
, T i = <(n,i,j
6 ~ ×. Q, then define:
> ~e ) e p}
(j)) ,~,<
We have two symmetry-
lemmata:
First s y m m e t r y - l e n a :
If ~ e ~ ,
then p IP # ~ ~(p) I~ ~(¢).
127
Second symmetrx-lemma: Restriction
lemmla:
If ~i e ~ × Q, then p 15 @ ~ Ti(p) I~ Ti(~).
If p J~-~ and P0 = {<(m,i,j) ,e ) 6 p; Ai, j occurs
in ~}, then p0 I~*#.
For a £-formula ~ define oct, (~) to be the set of ordered pairs (i,j)
6 0 × m suoh that ai,j occurs
in ~, and let occ2 (~) be the
set of rationals i such that ~. oo~urs in ~. For a limited eompreheni sion term t = E e × ~ write t = t(A1 ,A2) in order to indicate that a, = occl (¢) and A2 = occ2 (~). Further, (u,j)
let pr, (&1) -- {i e Q ; V j E ~
6 AI } be the projection of AI to the first coordinate
following support-lemma
is a generalization
support-lemmata of Mostowski [6~] Support lemma.
The
of the corresponding
and Mathias [58].
Let t(A, ,A2 ) and t*(A~ ,A*2 ) be limited comprehension terms of £ such that
T[ ~ t(A~ ,A2) = t*(a;,A'~); then there exists a limited comprehension term t o = t °(At°,A~) (A2 N A;) U
such that A ° = &1 f] &~
pr,(A~), and
~
and a~ =
~t(A~,A2)
= t°(A~,A~).
The proof is along the lines of the support lemma proved in section E, but in the present case slightly more complicated. be verified, define V(A *
that the ordering principle A ) : {x; supp~(x)
1 ,
2
(OP) holds
= AI ^ supp2(x)
Now, it can
in ~
:
= A2}, where
,
supp, (u), and supp2 (u) are the finite sets such that supp~(u) minimal cardinality, supp1(u),
supp~(u)
is of
is of minimal eardinality modulo
such that there is a term t = E e ~ ( x )
with occ1(~)
=
supp, (u), occ2(~)
= supp2 (u) and t is interpreted by u. Each class • Tk~S V
(see the corresponding
In a next lemma one shows that in ~
proof in section E). every subset of c = val(e)
is a finite union of open, closed or at one side open, at the other side closed intervals the restriction
from ( c , ~ )
(use the 2nd-symmetry-lemma
lemma to see this).
It follows~
that in ~
and
every
subset of c is definable by a formula #(x) of £ in which none of the symbols ai,j occurs, since only ~ and names bi for the endpoints of the intervals
are used. Now the argument of Mostowski
be used to show, that (KW-AC) no function f in ~
selecting
fails in ~
[6~]
can
by shewing, that there is
from each proper, closed interval ef
128
(c,~)
a non-empty,
proper
that in a transscription the restriction problems.
This proves
Notice, sions,
lemma,
that
of Mostowski's
the forcing
generic
D0 as a subset neither
that
shall prove,
is contained
but a copy of shown that in ZF.
(AC) FROM THE BOOLEAN
Prime
Ideal theorem
is a consequence
[35].
The Boolean
(BPI)
of (AC). Here we
namely
that
proviously,
A short outline
Prime
Axiomatic
Set Theory
Halpern
(BPI)
+ (AC)
this result
is
of this prove
first
(0P) holds
at UCLA,
showed
while
[6~]
(AC)
the (BPI) that
is violated
on
4, 1967
), 7 pages.
the (BPI) holds,
in [6~]
Notes
Institute
in 1962 in his doctoral
foundation)
in 1939
Lecture
July 10 - August
manuscripts
Mostowski
ZF without
Ideal Theorem;
with the Summer
distributed
had shown
in~
in~%
does
dissertation thus proving
not imply
the ordering . Halpern's
result
in print:
J.D.HALPERN: Boolean Cohen's
The independence
of the axiom of choice
prime
ideal theorem;
F~nd.
invention
of the generic
ZF-model's
in collaboration
that also [35]
holds)~
is not true,
in connection
Mostowski
proved
(BPI)
prepared
that in ZF Q (i.e.
After
We did not
in:
that in the m o d e l ~ o f
appeared
the Boolean
- A.L~vy
(informally
principle
approach.
exten-
(BPI)
of ZF. As we mentioned
[3 I] J.D.RALPERN:
Historically
by
of generic
is provable
OF THE AXIOM OF CHOICE
in ZF the
due to J.D.Halpern
argument,
do not pose additional
of 2 ~°. Thus we have
that the converse
is not a theorem
(KW-AC)
(OP) ~ (KW-AC)
E we considered
and noticed
different
in which
÷ (AC) nor
IDEAL THEOREM
In section
show,
Copy of n o as a subset of 2 M° to the groundmodel
a model,
G) THE INDEPENDENCE PRIME
continuity
conditions
to all our examples
of the powerset
(KW-AC)
given above
the theorem.
in contrast
(this would yield
[3Z]
The arguments
here we have used a somehow
add a Cohen
(AC).
subset.
in full ZF the
with A-L4vy (AC)
Math.
and H.L~uchli
is independent
from the
55(1964) in 1963,
p.57-66. Halpern
(via generic
from the (BPI)
models),
- see
and [ 3 1 ] . According
t~ explain
to the tradition
the ideas behind
of our lecture
the construction
notes we shall
of a model
start
of set theory
129
in w h i c h [~],
(BPI)
that
we need Lemma
+ 7
(AC) holds
in M o s t o w s k i ' s
two
Let
be the group subgroup
~
I and
of
I and
~{%(-b); %(-b)
{b}~
{b} and
I(b)
ideal
= {z e B;
closed
closed
there
ideal
generated
by J0,
subset
e = {xl .... ,x n} of J0 such that
•
=
By a s s u m p t i o n
i=l~i(Yi u
by t a k i n g
under
H,
H which
is a finite
sub-
lower
bound
of the set of
be the
ideal
generated
b)}.
Close
id est:
by
I(b)
under
J :{ ~ e; e is a
1 B 6 J, hence
the~e
1 : ~ e. Since
b)
~i(Yi ) L]
is a finite
x i E J0,
are Yi 6 I and %i E H (i = 1,2 .... ,n) such that
X"
under
(3 y 6 I(b))( ~ % E H)[z ~ %(y)]}
subset
i=1
Aut(~)
let H be any
~,
= {x e B ; ( 9 y E I)(x ( y U
4 of H: J0
of Jo}.
and
algebra,
that
of B for } e S. Let I(b)
let J be the
and hence
proof
But first of all
of
if i B E J, then
@ 6 S} is the g r e a t e s t
then
~
If I is an ideal
finite
there Hence
original
, - ) be a Boolean
if b E B and J is the s m a l l e s t
automorphisms and
holds.
of all a u t o m o r p h i s m s
set S of H such
elements
=
of A u t ( ~ ) .
includes
Proof.
model ~ ( B P ! )
Halpern~s
lemmata.
I([~]p.62):
and
by d i s c u s s i n g
x i ~ %i(Yi ~
b).
: ~i(b)
the complements:
- ( ~i=l~i(Yi) M
= ~i(b))
Since
I is closed
Since
u n v = o implies
under
: i=1
-yi ) n
H~ it follows
= ~i(-b)
that ~
v ~ -u in a b o o l e a n
= 0B E I
{%i(Yi); algebra,
i = l~...,n}
~ I.
put
u :~ {~i(-Yi); 1 ( i ~ n} and v = ~ {~i(-b); 1 ~ i ~ n}, then it follows v ~ -u e I~ hence v E I and if we define S = {~i; 1 ( i ( n}, then
the
Lemma
lemma
is proved.
2 ([3Z]p.62).
If ~
B and P is the {x,-x},
is a B o o l e a n
The
by
(BPI)
induction holds
(urelemants the
group
X a finite
subset
f on X such that
of
f(x) E
then
U ( N fx~X (x)) fEP Proof
algebra,
set of all functions
on the
: iB
cardinality
in M o s t o w s k i ~ s
of r e f l e x i v e
sets
of o r d e r p r e s e r v i n g
of X.
model ~ L . x = {x})
mappings
Take
ordered
countably of type
many
atoms
n o . Let G be
and F be the filter
of subgroups
130
generated
by the finite support
that in ~ =
~[G,F]~
of this chapter pern extended (BPI) holds. ~, there
in [3~]
support
~ 6 K[e]
that among the e-symmetric In fact consider
(outside
on p.65),
ordered
and has hence
say I0. We claim, Suppose
not,
that
K[e]
= {4 e G;
is an automorphism
ideals of B there of ~ )
includes
I there
are finite
(1)
~
in ~ ,
subsets
on
shows~
ideal
I.
~ H[J]}. situation element,
is e-symmetric). (-b) ~ I0.
I0 and b and is
ideal of B which
they cannot
of I0. Hence
{¢(-b);
Halpern
under K[e] . Since both,
IB, since otherwise
the maximality
of ~.
b 6 B such that b ~ I0 and
I0 and -b and is closed hence
of F
is e-
lemma a maximal
under K[e] , and let I2 be the smallest
dicting
Thus B
is a prime
ideal of B (I0
eludes
ideals,
in
~ is identical
^ 1B ~ J ^ K[e]
by Zorn-Kuratowski's
ideal of B which
not to include
even the algebra
the set ^ J E ~
closed
thesis
J.D.Hal-
by C (see the proof of a similar
then there exists
are e-symmetric
that in ~
Let ~ be a boolean
I0 is a prime
Let Il be the smallest
in [6~],
(see section E
and notation).
such that K[e] ~ H[B].
Z = {J; J is an ideal of B Z is inductively
showed
= B} e F and by the definition
subgroup
set of atoms)
and every
(AC) fails
by showiDg
as follows.
= {¢ e G; ~(B)
is a finite
Mostowski
III for details
this result
He proceeds
e} (e a finite symmetric
(OP) holds while
and chapter
Then H[B]
subgroups.
in-
I, and I2,
satisfy
the hypo-
Ii 6 Z and I2 6 Z, contra-
1 B 6 I, and 1 B 6 I2 and by lemma
S,, $2 of K[e],
such that
% 6 S, } e I0
and (2)
~{¢(b);
¢ 6 S,} e I0.
Let r = e, then r determines
r + i open intervals
A (in the orderin Z of type of the rationals). lemma finite
2) -the contradiction
subset X of B such that ~
Since one wa~ts from
to derive (1) and
theorem of R.Rado) and takes where
{f(x);
(2), Halpern
a certain
(see p,53)~
gives only in depend finite
~
{f(x);
cardinal
finite
extension
noe of
2 entails
that 1 B E I0.
x C X} 6 I0 by some permutation finds
(using a combinatorial
subset W of the set A of atoms ~*(b)
: x ^ ~(g) - e C W},
of ~ to an automorphism
and g is the support
number
(via
x 6 X} E I0 for all fuctions
then lemma
then as X = {x 6 B; ~ } E K[e]:
~* is the unique
universe
We want to get
that 1 B 6 I0. To do so~ we need a certain
f on X such that f(x) E {x,-x}; arguments
K i (0 ~ i ~ r) of
and K? n-7
q. The property,
of the whole
of b 6 B. Rado's
(i e r + I)
that the atoms
are
theorem
certain
131
totally ordered of type No is essentially used to conclude, between all points of W w e
that
can embed q points. W has a certain
partition property which is used to find an automorphism ~ of B such that either all elements of {~(b); # e $2} or {~(-b); ~ e S,} can be mapped into {f(x); x e X}. Then
~ {f(x); x e X } ~ 4 ( ~ {~(-b);
e S,}) or ~ {f(x); x e X} ~ ~ ( ~ { ~ ( b ) ;
~ e S~}) for ~ e K[e].
Since I0 is closed under K[e] , it follows from (I) and (2) that ~ ( ~ {¢(-b); ¢ e $I}) e I0 and ~ ( ~ {¢(b); since Io is ~n ideal, ~ { f ( x ) ;
~ e $2}) e i0. Hence,
x e X} e I0 for all f under considera-
tion. As indicated above, this yields 1 B 6 Io, a contradiction! Hence I0 is prime.
For all details of the proof, the reader must
be referred to Halpern's paper [$Z] . It is possible to carry over these ideas to the construction of a Cohen-generic model ~
of ZF + (BPI) + ~ (AC). This has been
done in collaboration by Halpern,
L4vy and Liuchli. The construction
of the model has been carried out by Halpern-L4vy construction
a combinatorial
[~5].
In their
argument was used (different from Rado's
theorem)~ which has been established by Halpern and L~uchli in: [~]
J.D.HALPERN Math.
- H.LAUCHLI:
A partition theorem; Transactions
Soc. vol.124(1966)p.360-367.
The model constructed by Halpern-L~vy shall, forcing.
however,
Amer.
construct
is a boolean valued model.
a Cohen-generic
We
extension by means of
Our remark, that in the case of permutation models the atoms
have to be l i n e a r l ~
ordered of type ~0, suggests that in the case
of Cohen generic models
~
the generic reals ai (for i e ~) have to
form a dense subset of the real line of ~ construct ~
by adding to some countable
. Following this idea we standard model ~
infinitely
many Cohen-generic reals ai (i E ~) and generically a set A which just collects these reals at" We shall use the notation ~ al,...,A].
This model ~
has been described
~ ~[a0,
in section C of this
chapter on pages 101 - 103. Theorem
(Halpern-L~vy): (AC). Hence, not provable
~[a0,a,
..... A] is a model of ZF + (BPI) +
if ZF is consistent,
then (BPI) ~ (AC) is
in ZF.
Proof. A detailed presentation
of ~ [ a 0 , a l
,...,A]
has been given
in section C, where it was shown, that in this model A is an infinite, but Dedekind-finite
set. Hence
(AC) does not hold in hold in this
132
model.
The proof,
that
First we r e m i n d Symmetry-lemma:
(BPI)
holds
our r e a d e r
Let G be
will
require
to the f o l l o w i n g
in ~
the g r o u p
Restriction
Lemma.
the finite
Suppose
in ~, and let p0
Now
If p Ibm,
p I~% and ~P0
such that q I ~ 7 ~. D e f i n e
I~.
co
= {i E ~ ; V n E ~ V e E 2 ( ( ( n , i ) , e
Since both, of t h e i r Since
c leaves
lemma tells
that ~(p)
~(p)
i
= P/occ(~)
be
{{{n,i) ,e ) E p{
is a c o n d i t i o n
q D P0,
) E q ^ i ~ Co)}
of w w h i c h
leaves
T h e n Dom(q)
extend o(p)
co p o i n t w i s e
us,
lemma y i e l d s
Hence
i such that a.
V e E 2 (((n'i) ,e ) E p ^ i ~ co)}
U d2).
q and o(p),
domain.
of £, occ(~)
~ occ(%),
d2
in u-(c0
numbers
T h e n there
= {i E ~ ; < E ~
dl
of ~.
then P0 I ~ .
d~
let ~ be any p e r m u t a t i o n
and maps
and 102):
p: ~ × ~ ~ 2 and any
% a sentence
set of n a t u r a l
i e occ(~)}. Proof.
(see p.99
~, p II-~ ~ ~ ( p ) I ~ - ~ ( ~ ) .
Let p be a c o n d i t i o n ,
occurs
lemmata.
of all p e r m u t a t i o n s
for any ~ 6 G, any c o n d i t i o n £-sentence
several
P0,
co pointwise
N Dom(~(D))
they c o i n c i d e
U q is a f u n c t i o n , fixed:
on the c o m m o n
hence
~(~)
= ~. Thus
11- %. T o g e t h e r
with q J~
fixed,
C ~ × c0.
also
part
a condition•
the s y m m e t r y ~, the e x t e n s i o n
U q il- % ^ 7 ~, a c o n t r a d i c t i o n .
Hence
Po Ib*~ must
be true. The next of
lemma w i l l
2 ~ in the p r o d u c t
the u s u a l
linear
say,
topology
ordering
2 ~ is c o n s i d e r e d
discrete i.e.
space
the b a s i c
sets
i E ~] is a d e n s e
2 ~. The f o l l o w i n g
- s2) U (s2
as the p r o d u c t
2 = {0,1}, open
of
{ai;
relation
subset < is
of 2~:
sl < s2 ~ Min((sl Since
that A ~
we may endow are of the
- sl))
E s2
of ~0 copies 2m with
of the t w o - p o i n t
the p r o d u c t
topology,
form
b r = {f E 2~; f D r} where
r is a f i n i t e
set of f u n c t i o n s The space since
partial
(=real
2~ endowed
function
numbers)
with
2 ~ is h o m e o m o r p h i c
this
from ~ into
topology
of the real
line w i t h
see e.g.
Ph. Dwinger:
Introduction
Verlag
W~rzburg
Springer
Verlag
1961, Berlin
2. Hence b r is the
2 which
is c a l l e d
p.49-50, 1964,
the usual
r.
(considered
as
interval-topology)
to B o o l e a n
or R . S i k o r s k i :
p.43,
extend
the C a n t o r - s p a c e ,
to the C a n t o r - d i s c o n t i n u u m
a subspace -
from ~ into
Algebras, Boolean
and t e x t b o o k s
Physica-
algebras,
on T o p o l o g y ,
133
e.g.Kelley.
The s p a c e 9 ~ w i t h
the topology
disconnected
compact
also closed,
and hence regular
Lemma
Hausdorff-space
3. Every basic or better function Hence
Proof.
stated:
~"b
a generic
~ A # ~ for every
r
real ai,
finite
partial
r : m ÷ 2". in ~
there
that A is a dense
is a finite
such that b r N A = ~ holds
in ~
is forced by some condition
defined ~
and the basic open sets b r are
b r N A ~ ~ does not hold
set b r. Hence, ~,
~
is a totally
open sets.
open set b r of 2~ contains
it holds
suppose
as given above
)~ there
in ~
subset of 2 ~.
for every basic open
partial
function
. Since
everything
in
the complete
r
: m ÷ 2 in which holds
sequence
in
~ (which
is p E ~ such that
p i~ b r ~ A = Since
p is finite,
n e ~, (n,i0)
there
~ Dom(p).
is a natural Define
subsets
i0 such that for every
the following
q = p U {((m,io) ,1) ;m e ~ ^ r(m) Identify
number
extension
q of p:
: 1} U {((m,i0) ,0) ;m e ~^r(m):0}.
a of ~ with their corresponding
characteristic
function ~a(m)
: {i
• then q I@ E C ai0 model
~0
and second
and hence
defined
an element
'~"ai0 q I~
by any complete
element~
to the assumption, every basic
if m ~ E a
it holds
open set
variables
sequence
(i.e.
@~) forces
every absolute a dense
or the constant members
there
a sequence
exists
is in contradiction
interval7
of 2 ~ contains
with no free
that ~ contains
holds:
constants
(br~ .... ,brn)
if h = (hl ,...~hn>
of pairwise
in ~
x
,... ,hn).
, then
disjoint
basic
(for 1 ~ ~ ~< n) and the is any sequence
of A such that h~ 6 br~ for'l ~< ~ ~ n~ then
7[
none
~. Let g ----(gl ~...,gn > be a sequence
of A. If %(gi ,...,gn ) holds
open sets of 2 ~, such that g~ E br~ following
first
b r N A = ~ . Thus
ai (for i E ~), but % may contain
of different
in the
subset of 2 ~, q.e.d.
other than x, ,...,x n and suppose
for x e ~
that
which has p as
Let ~(x, ,...,x n) be an i - f o r m u l a
of the symbols
members
~
that b r N A ~ ~. This
that p (and hence
of A and A is in 9 ~
Continuity-Lemm~.
e b r- It follows
of different
134
Proof.
Suppose that for sets gl ,...,gn E A the sentence ~(g, ,...,gn )
holds in ~
. Let tl ,...,t n be constant terms of £, such that g9 =
val(t v) for I <~ v ~ n. Consider the following £-sentence: (0) ~
(~(t, , . . . , t n)
vr -'
^ tI ~ A
''" - - n [
^
...
^ tn
e A)-+
2k) ^ the basic open intervals n brv (for 1 ~ v ~< n) are pairwise disjoint ^ / ~ 1%v g b r ^ n
Ax, ...
AXn(~/AXI(X~
The con%inuity (0) holds in ~
l(rv e
C br v ^ x v e .~.) ÷ ~(x,
lemma is proved as soon as we
in ~
which holds in ~
• ,xn))].
have verified that
. So suppose that the sentence
. Then the negation of (0) holds
,..
in ~
(0) does not hold
, and sinee everything
is forced by some condition of the complete
quence 6] (which defines ~ p I~*-] (0). The statement is equivalent to p I!.~"
~l
se-
), there exists p E 6~, such that (0) has the form ~, ~ ~2. Hence p I~f 7 (0) ^-t~2,
Thus p 11-*~,
and p 11'-*-1 ~2. We shall
obtain a contradiction by showing that there exists an extension P0 of p such that P0 tt.2. ~2. First, p I1-*'~, n (*) p IF*#(t, ,... ,t n) ^ v/~k~_ltV g It follows from the forcing definition,
is:
that there are i,,... ~ln
such that p' IF tl = ai1^'" "^ tn = aln for some extension p' of p.Hence: (**)
P' l~'~(ail ' .... ai )• n Extend p' to a further condition p" such that (1)
If ( m , i )
(2)
If 1 ~ v ~< n, then there is m 6 ~ such that (m,i v) E Dom(p"),
E Dora(p") and m' < m, then ( m ' , i )
(3)
If j, ~ J2 then there is m 6 m such that (m,jl) (m,j 2) E Dom(p") and p"((m,j~))
(4)
6 Dom(p,')3 6 Dom(p"),
~ P"(<m,j~)),
i
p" extends p' .
It is possible to find conditions
p" satisfying
(1), (2), (3) and
(4). We shall not explicitely describe such a condition, but assume that we have obtained such a p". Define interval designators functions
(i.e.
from finite proper initial segments of ~ into 2) for
1 ~ v ~ n, called rv, by rv(m) ----p,,(<m,i v)) for m e {m'; (m',iv) e Dom(p")}.
By condition
(I), r v is a funotion
defined on some initial segment of m. By condition
(~), r v is for
no v with 1 ~ ~ ~ n the empty function and by (3) the basic open sets brv = {f 6 2~; f _D r v} are pairwise disjoint
(in the sense
135
of the mete-language).
Hence this is weakly
thus: n/'~k, ]~ v:l ( F n c ( r ~ ) =
~ 0) ^
rv ^
forced by every condition,
4 A (i ~< v, p ~ n ^ ~ ~ ~ ÷
(~)
brv ~ brp = 9).
Since occ(~(~i i ,...,a i )) = {ii ..... i n } by our assumption on @, the restriction !emma,napplied to (**) with p' replaced by p", yields : (~)
P* '~'*~(ai ,...,ain )
where p, is the restriction
of p" to {ii ,.,.,in},
{((re,i) ,e ) e p"; i 6 {ii ,...,in}}. holds :
id est p, :
Now we claim that the following
n , ,.,
n [
= l(xv£ A
b r v)
@(x~
÷
We have to prove that if t,',...,t~ are constant any extension
of p" such that q ]~*v~_ l(t~~" e ~
term of £ and q is ^
-
exists a further extension
._''''Xn)]"
t v!
E
b r ), then there
q' of q such that q']~* ~(t~,?~..,tn).
Hence suppose terms t i' ,..., t'n and a condition q are given, where q extends p" and q (weakly) forces the conjunction of the statements
t~ e A ^ t~ e b r . Then there exists an extension q' of
q and numbers
j, ,...,3 n e ~, such that
q C q'l~ t: = Ajl --
Hence q'l~- aj,
c br!
^ t' : " . ^" " " n ajn ^ . .. ^ a]n" e brn. Thus,
of ~ with their corresponding q'l~ r, C aj,
^
. . ,
characteristic
o(jg)
functions
subsets
:
^ r n _C aj n" This implies q, _C q' , where
q, : {((m,j~)) ,e) ; 1 ~< 9 ~< n ^ rv(m) Define the following
if we identify
permutation
= e}.
~ of ~: o(i v) = Jv for I ~ v ~< n,
= i , q(i) = i for all other natural numbers
i. It follows
that q, : c(pl ) Hence p, C p,' C q C q' and c(pl ) : q, C q,, and it follows that Pl
U g(p, ) is contained in the condition
condition.
o(p,)
: q,l~ ~(Aj,
Hence, by the extension forces forces
q'. Thus, Pl U ~(Pl ) is a
Applying c to (8) we get by the symmetry-lemma .....
Ajn)
lemma q' I~*#(Aj,
:- ). Now, since q'
t~ : aj~ ^ ... ^ t'n : ajn" we infer that q' also (weakly) ¢(t~ ..... tn). This proves
Both, non-empty,
(7).
(e) and (7), show that p" (weakly) pairwise disjoint
absolute
forces that there are
intervals
brv such that
136
azg" e brv
, and if
(t~,...,t~)
#(ail ,...,ain),
with t v~ e A ^ t~ e brv
p _C p' _C p,' , we have obtained
then ~(t;...t~)for (1 ~ ~ ~ n ) .
the desired
every
sequence
Hence p" I ~ 2 .
contradiction.
Since
This proves
the continuity-lemma. Lemma
4. Let r ~ T f u n e t i o n br~
A is infinite.
infinite Proof.
from some finite
By lemma
Further,
in the usual
ordinal
A is Dedekind-finite
3 every absolut
b r : {f e 2m; f D r}
interval
[f E 2~; r0 ~ f ~ rl} contains
a generic
v (m ~ Dom(r)
Since
^ e = r(m))
every absolut
absolute
interval
intervals,
infinite. Notice, lemma
^ e = r(m))
(m 6 Dom(r)
includes
it follows
Remgrk.
has an
and occ(~)
3~-definable
Mathias'
theorem
=
^ e = 1)}. many pairwise
was proved
in section
also follows
disjoint
Hence A is C, page
102,
from the continuity-
th.10).
such that there
u : val(E e ~ ~(x))
where r0 :
countabiy
Since we want to be able to prove
sets u of 0 ~
:
^ e = 0)} and rl
that b r N A is infinite.
That A is Dedekind-finite
[~],
real,
v (m ~ Dom(r)
that the Dedekind-finiteness
(see e.g.
(though
sense).
{(m,e);(m 6 Dom(r) {(m,e>;
into 2. Then
exists
C d, where
wellordering,
(see p . 1 2 0 - 1 2 1 ) a n d
that every class
a constant
d is a finite
we proceed define
V d of
term E ~ ~ @(x) subset
with of ~,
as in the proof of
in ~
a ramified
language
£*. The alphabet of £* contains besides the usual ZF-symbols and the useful
limited
sion operators
existential
E e (for ordinals
a name A* for A and names tion ~* for the constant ~*(A')
: A, ~*(x*)
sion terms
situation
ble wellorderings
Le~ma
terms
~* is ~ - d e f i n a b l e on page
121).
an interpreta-
~*(a*)
to all limited
(for more details to obtain ~
V d = {x e ~
; supp*(x)
see -defina-
C d}
will depend
of the continuity-lemma. with no free variables
than x~,...,x n and let ¢* = {ai; ai occurs
If gl,...,~ n is a sequence and if 9(gl,...,g n) holds sequence
= a, comprehen-
In order
5. Let ~(xl,...,x n) be any £-formula other
Define
The proof of the support-lemma
generalization
comprehen-
), a name a* for each a E A,
of £* by induction:
of the ~ - c l a s s e s
we need a support-lemma. on the following
e of ~
x* for each x 6 ~ .
: x and then extending
of £*. Hence
the analoguous
q u a n t i f i e r s V ~ and limited
of different in ~
br, ,. ..,brn of absolute
members
, then there intervals
in ~}. of A-c*
exists
a
of A, pairwise
137 disjoint
and d i s j o i n t
f r o m c',
and ~(g~' .... 'gn') holds
( 1 ~ V ~ n)
g~'''''gn' of d i f f e r e n t for 1 ~ v <~ n. Proof.
such that
members
gv 6 b
for every
of A such that gv' 6
Let ~(xl ,...,x n, Y, ,...~Ym ) be the formula
replacing
each o c c u r r e n c e
variables
are used
variables
y~ ,...,ym
..,gn ) holds
in ~
ail ,...,aim)
holds
ty lemma
there
constants
do not o c c u r
in ¢).
for d i f f e r e n t in 9 ~
obtained
f r o m ~ by different
(it is assumed,
of A-o*,
members
absolute
Hence
~(gl ,.-.
of A. By the c o n t i n u i -
intervals
~(g;,...,g')n
that the
that
then ~(gl ~...,gn,
such that~/(g' '" " " 'gn+m' ) holds,
for 1 ~ v ~< n+m.
for $ < v < n, since
disjoint
brv
Yv' w h e r e
If we suppose
members
for d i f f e r e n t
are p a i r w i s e
brn+l,... ,brn+m of A, gv' e brv
of air by the v a r i a b l e
for d i f f e r e n t
r~ sequence
holds
brl ,...,brn,
whenever
whenever
~(g,', .... gn,ail .... ,a i ) holds
in ~
gv' 6 br v . Since
m a{
~ b~
1 ~< v ~ m and
for
"9 ~n+v disjoint, it follows and d i i j o i n t
that
9rom c*
Support-lemma.
q.e.d
Let tt
prehension
all b ~ (1 < v <~ n+m) are p a i r w i s e ~v the b r for 1 ~< v ~< n are p a i r w i s e d i s j o i n t
= E a x ¢, (x) and t2
terms,
an Z - f o r m u l a and for t3
and suppose
~ (x) such that
= E ~ x %,(x)
p U-*t,
Proof.
Since
lemma,
that (re,i) E Dom(p)
and assume only
p into
(without
~
~ t,
only
loss
holds:
: oec(~,)
of g e n e r a l i t y ) ,
(the general ~0
= po(c)
case
= p/c,
that d~ follows Pl
: p/dl
permutation
of w h i c h
leaves
the
syn~metry-lemma
applied
po(c) where
P2(Jo)
Hence
both
relations p'
Obtain
: po(c)
an e x t e n s i o n
such
that
page
[$~),
where
Define
- c = {i0}
fixed
Split
and
and let and maps
to Po (c) U p, (d,) ~ P2 (ie)
U P2(jo) and
%
Ib't~(c~d,)
• be a io onto =p
Jo.
:
the r e s t r i c t i o n
U P2(jo)
l~'t2(c,io)
p" of n' as in the proof conditions
condition
= t~(c,jo)
: t, ( c , d , ) , . % ~ ( c , j o )
together~using U p2(io)
p" satisfies
= t3.
yields
U p~(d,)
= T(p2(io))
(~2).
= occ(~2)
= p1(dl)
c U d~ p o i n t w i s e
~ tl
by induction).
Let j0 e ~, j0 ~ c U d~ U {i~}
Then
exists
- c,
: p /d2.
p [b't~(e,d, ) = t2(c,io)
~
if i 6 occ(%l ) U oce d~
There
by the r e s t r i c t i o n
: P2(io)
P2
= t2.
com-
occ(%3 ) = occ(~, ) N oco(~2 ) and
the f o l l o w i n g
(~ occ(~2),
one element
subconditions
that
= t2 for p 6 ~, we may assume
c : oec(~,) has
= E e x }2 (x) be limited
(1),
(2) reads
(2)~
of
= E a x T(#(x)). lem~na entail:
= t2(c,jo). ~he c o n t i n u i t y
(3) and
in the present
(4)
(listed
context
lemma, on
as follows:
138 (2) If i e c U {io,j0}, then there is m e m such that (m,i) Define r i by: ri(m)) :p"((m,i))
e Dom(p").
and bri = {f e 2~; r i _C f}. Then
the br. are pairwise disjoint and a i e br. for i E c U {io,j0}. 1
In
1
particular b r i o N brj0=
~ and br.
as well as b
lo
are disjoint from
~]0
{ai; i 6 c}, see lemma 5. It was shown in the proof of the continuity-lemma,
(t)
p"l~" Ay[y
where t~(c,i0)
~ i
^ y , brj ~ ~ t2(c,io)
that
: t2(c,y)]
= t2 (c,x) is to be taken as ~(x). Now define ~'2(y,x)
to be the £-formula obtained from ~2 (x) by replacing ai0 by the variable y at all places of oceurrenee
(it is assumed,
that y does
not occur in ~2 (x)). Define t3 : E ~ x ( V y [ Y
e A ^ y e br.
^ ~(x,y)]).
i0
By (*): p" I~*t2(c,io) = t3, where oce(t3) to be included in
= c. Since p" can be chosen
~J6~, hence in some q 6 ~, it follows that q [~ t, =
t3, and the lemma is proved. Lemma 6. The ordering principle Proof.
For u 6 ~
(OP) holds in ~
define supp'(u)
minimal cardinality
as the finite subset of A of
such that there is a term t* of £" mentioning
only names a" for a E d with ~ ( t * ) supp*(u)
then the
is always defined.
= u. By the support-lemma,
Put
o = {u 6 ~ ; supp'(u) = d} Vd ~ -class V d has an ~ - d e f i n a b l e
with the lexicographic one concludes,
that in ~
Notice, that every ~ - c l a s s
every set has a total ordering
V d = {u 6 ~
well-ordering in~
(for all
; supp*(u) C d} has also
namely the one inducec (via ~') by of the constant terms t of 2" with
occ(t) C d. Notice further, t h a t t h e that (0P) holds
Together
proof in section E, page 120-121).
wellordering~
the ~ - d e f i n a b l e
wellordering.
ordering of the set of finite subsets of A
details see the corresponding
an ~ - d e f i n a b l e
.
proof of lemma 6 shows that more
,namely the Kinna-Wagner ordering principle
(KW-0) holds i n ~ .
Next we want to show that also
Prime Ideal theorem
(BPI) holds
in ~
the Boolean
. First we need the following:
Lemma 7. Let ¢(x0,xl,...,x n) be a formula of £ with no free variables other than xo,...,x n and suppose that occ(~) C d. If t, , .... t n are constant terms of Z such that
Vx
~(x°'t*'''''tn) 0
holds
139
in ~
and if val(t
) 6 V d for 1 < 9 ~ n and some finite
subset d of A, then there is a constant that ~(t0~t,,...,t n) holds in ~ Proof.
Since
Vx
#(x0,t, .... ,tn) holds
~rm
tQ of £ such
and val(t0) 6 V d.
in ~
, there is a constant
0
term
f of £ such that ¢(t' ,tl,...,t n) holds in are valuations
~
(since sets of
of terms of £). Suppose val(t') ~ V d. Let c =
{il,...,i m} be the set of numbers such that ai~ occurs in t' and 1 ~ 9 ~ m and air ~ d. Hence c ~ 9. Let t'= E e x ~(x) and let ~'(x,y,,...~y m) be the formula obtained from ~(x) by replacing the constants ai9 by y9 for 1 ~ 9 < m, different variables
for different
constants.
Hence t' = E e x ~*(x,~ii,...,~im). By lemma 5 we find • 6 br~ (1 ~ ~ ~ m) and by intervals br,,..,brm such that al~
absolut
ai~ (I ~ ~ ~ m)
lemma 3 generic sets aj~ such that aj~ E br~ and aj~
and further the br9 are pairwise disjoint and all are disjoint from d. Hence
~
~ E ~ x ~*(x~ai,,..,aim)
= E e x ~*(x,aj, ~...,ajm) and by
the support lemma there is a term to such that 9 ~
t'=t0
and to m e n t i o n s
only generic reals in d, hence val(t0) 6 Vd, and lemma 7 is proved. Lemma 8. If ~
is a Boolean algebra in ~
J of ~
in ~
, then there is a prime ideal
such that
supp*(J) C supp'(B). Hence the (BPI) holds in ~ Proof.
Let B
meetoperation
.
= < B , N ,- ) be a boolean algebra in ~
~ where ~ is the
(i.e. product, or greatest lower bound) and - is the
complementation
operation
( u
is definable by means of n
Let d = supp*(~), hence ~ 6 Vd. Since the operations: the i st (2 nd, 3rd resp.) lemma 7, that B, ~
coordinate,
projection to
are single valued it follows from
and - are sets of V d. Further
gest and the smallest, resp.)
and -).
1B and 0 B (the lar-
are in Vd, since they are unique.
Consider the set Z of all pro~er ideals of ~ , Since {0 B} 6 Z, Z ~ ~. Since V d has an ~
which are in V d.
--definable wellordering
and Z C V d, it follows that Z has maximal elements.
Let I be a
maximal proper ideal of ~ with I 6 V d (id est: I E Z). We want to prove that I is prime Suppose I is not prime. Then for some b E B, b ~ I and (-b) ~ I. Since b E ~
, there is a constant term t b of £, such that b = val(tb).
We shall derive a contradiction by showing that 1 B E I. It holds that supp*(b)
- d ~ 9, since otherwise the ideal genera-
ted by I and {b} would be in Z, contradicting
the maximality of I.
140
Hence
let t b a constant
t b = E e x T(x) cardinality.
term of £ such that b : val(t b) and,
then oct(T)
: {i e ~; ai occurs
in T} is of minimal
Write
oct(T)
- d = {i e occ(T);
The case k = 1 is especially fop the proof-procedure, [~}.
We, however,
case,
but recommend example
as an illuminating
in detail
shall
k : I in [~S]
2). Before we formulate
example
in Halpern-L~vy
start directly
our reader
We shall need a combinatorial
we have to introduce
and,
is discussed
and Halpern
of the illuminating
i ~ d] = {ii ~ .... i k}
simple,
the general
theorem
if
[~]
with handling
to look at the discussion an~ L ~ ]
theorem
of Halpern-L~uchli
a particular
([33] ,
case of that theorem~
some notation.
A tree T = ( T , ~ } is a partially
ordered
set such that for
each x 6 T, {y E T; y < x} is totally
ordered
by ~. The cardinality
of this
set is called
the order of x, or the level at which x occurs.
A fan (F~cher) is a non-emDtv tree such that all elements of it bare finite order and each level is a finite set. Hence, if ord(x) is the order of x in the tree < T , ~ )~ then (T~<)
is a fan ~
Define
T/n
/~x6T(Ord(x)
(restriction
e ~ ^ Ane~{x
e T; ord(x)
: n} is finite).
of T to n):
T/n = {x e T; ord(x) ~ n}. Definition.
Let D, and D2 be subsets D, dominates D, supports
D2 ~ A X 6 D ~
a (m,1)-dense cartesian Theorem
subset
product
of T i (with respect
i~kDi
is called
(Halpern-L~uehli).
if there
is an
= m + 1 ^ x < y}
n such that
for every
id est i~k(Ti/n) includes
elements.
in lemma
is a positive
(Correct
Q, or Q~
for some m < n.
in [33] ,p.364~ quantifier
quantifier,
on the left side of
many
integer
Q = (Q, ~ Q2 )of i~k(Ti/n),
= Q, u Q2 , Q, N Q2 = ~, either
1 the second
d is an existential
There
2-partition
an (m,l)-matrix
For a proof see [39|.
to ~ i ) for i < k. Then the
an (m,1)-matrix.
Let ( T i , ~ i) , for i < k, be finitely
maximal
quantifier
(m,1)-dense
= m and {y 6 T; ord(y)
by D. Let ( T i , ~ i) be fans for i < k 6 ~ and let D i be
fans without
misprints:
.
~y6D, (y ~< x).
is called
x of T~ such that ord(x)
is dominated
(T,~)
D2 ~ A x e D 2 ~ Y y 6 D ' (x ~ y).
A subset D of the fan ( T , ~ ) element
of the fan
~d
cJ~n e
the following
and two~below is a
two
on the left side of in 1.1.
universal
the third
quantifier).
141
Notice that in [3~]
and [3~]
the terminology
used. We do not like this philosophically
"finitistic tree" is
sounding word "finitistic"
and use the word "fan" which is also used in Intuitionism. Now we return to the proof of lemma 8. Let tB, t n terms of ~ such that val(t B) : B, val(t n ) : ~ of the boolean algebra B) and val(t
) = - (the complement operation
of B), such that occ(t B) C d, o c c ( t ~ Further,
and t_ be
(the meet operation
) C d and occ(t ) C d.
let t b : E ~ x ~ (x) be the term obtained above, and let
t I be the ~ - t e r m with oce(t I) C d such that val(t !) : I. Define: g(xl,...,x k) to be the g-formula
obtained
from (t b e t B ^ D tb.e t I) ^ (t (t b) ~ t B ^
^ ~ t (t b) e t I) by replacing
each occurrence
E ~ x ~(x)) by the variable x
for 1 < 9 < k. In the formula used
above t_(t b) denotes the "complement" Since
~(ai, .... ,aik) holds
in ~
of al~. in t b (i.e.
of tb, id est vai(t
(tb)) = (-b).
, there exists k absolute
intervals brl,...,brk which are pairwise disjoint and di{sjoint from d, such that (1)
c A ^ x~ ~ br v +
~
&(xl ..... Xk))
holds in
~ (this follows by lemma 5). We put S o = {b r }, for 9 1 ~ 9 ~ k, and S o = (S~ .... ~S~) . We continue and define k-termed
sequences of absolute intervals, we prove that the sequences
S n, for every n E ~. Simultanously
S n have the following properties
(S n v
denotes the 9-the coordinate of the sequence sn): (PI) S~ is a finite set of absolute subintervals S~ -1 for m > 1 and 1 ~ 9 ~ k. Therefore,
of the members of
by the definition of
S o and by induction on m, the members of S~ are subintervals
of
br~ for 1 ~ ~ ~ k. (P2) Every member of S~ -1 has at least two subintervals
in S~ for
m > i and 1 < 9 ~ k. (P3) The members of S~ are pairwise disjoint, (P4) If
ZE
for m > O, 1 ~ 9 < k.
E S m-1 and G is a finite set of elements of A which
contains exactly one member out of each member of ~ R a n g e ( s m ) , then
~ { v a l ( t b ( u l ~ . . , U k ) ) ; ( u ~ , . . , u k) e ~ ( Z N G)} e I, and 9:1 ~ { - v a l ( t b ( U l , . . , U k ) ) ; (ul .... u k > e ~ ( ~ N G)} e I, where v:l tb(U,,..,u k) results from t b by replacing ai~ by u9 for 1 < v < k.
The only one of (P1),...,(P4) which applies
in the case m : O, is
(P3); but this holds since the br9 's are pairwise disjoint.
142 Induction.
Let us assume now that for m ~ n, Sm is defined and
(pl),...,(P4)
hold. We shall define S n+l and prove that (PI),...
...,(P4) hold for m = n+l.
By (P1) the members of S~n are subinter-
vals of br~ (for 1 ~< ~ ~ k). Therefore, where S* = (br, ,...~brk).
are understood to be 1-1-functions with
domain some element of ~). Hence, Ze
Z E K S n, then ~ Z _C HS*,
(The symbol ~ is used to denote the car-
tesian product and sequences
(2)
if
ES n ^ (u,, .... u k) e ~( ~
if
E6
~S n then by (1):
~ A) "~
~(u, ..... u k)
Consider the ideal J generated by I £) {va!(tb(ul ,...,Uk)); (u,,...,u k} e ~( ~
~ A)}.
Since J e V d and I is a maximal proper
ideal in V d and I _C J we must have 1 B 6 J. Hence~ by lemma 1, there is a finite subset G, ( D
of n(E9 A A) such that ~ {-val(tb$u , ~..,Uk))
(u,,...,u k) e G,(E)} e I. By (2) G,(E) has at least two members. By considering,
in the same ~ay, the ideal generated by
I U {-val(tb(u , ~... ,Uk)) ~(u, , .... u k) e ~(E~ N A)}
one obtains
a finite subset G2(E) of [(Et ~ A) which has at leat two members such that ~ {val(tb(ul ,...,Uk));(u I ,...,u k) e G2(Z)} 6 I. Define G~(E) to be the set of those elements which occUr on some place in an k-tuple of GB(7),
! ~ = 1,2. GU(E) is a finite subset of
A and it has at least two members
in common with each Z9 for
I ~ v ~ k. Put Then G* is a finite subset of A which has at least two members common with each member Of O R a n g e ( s n ) .
Since G (E) C ~ ! ( 7 =
in
N G*)
for 1 ~< ~ ( 2, we get (3) E 6 ~S n ~
~
h
k ~ { - v a l ( t b ( u i .... Uk)){(u I ,..,Uk)¢ ~ (E ~ G*)} E I ~:I k ~ and ~ {val(tb(u I ,..,Uk));(ul ,..,u k) ¢ ~i(7 ~ m G')} 6 I,
since Y, C y2 C B ~ ~ Y ~
~< ~ Y , ,
and I is an ideal. By lemma 5 we
obtain from (3) that there are absolut intervals b, ,...,bk, pairwise disjoint
(and disjoint from d) such that
then aj
e b~ for 1 ~< ~ ~< X and
,
if G*
(4) E e ~S n ^ ~ _ _ [ (x9 e A ^ x 9 s b ) +
=
{aj, ,. ..,ajl},
k
~ { - v a l ( t b ( U , ..... Uk));(u , ..... u k) e
~ (E9 r~ {x, ,. .. ,xl})}e I ~=1 /k ~{vai(tb(U, ,.-.,Uk));(u, .... ,Uk> 6 ~ ( 7 ~ {x~ ,...,xl})} 6 I. ~=i We can assume that each b ~) (for 1 ~ 9 ~< l) is a subinterval of some member of ~ R a n g e ( S n) since this can be attained always by taking appropriate
intersections.
Moreover it follows from our construction
143
of G*, that if s 6 U R a n g e ( s n ) ,
then s includes at least two sub-
intervals out of the sequence (b, ,...,bl) . Let us define now the k-sequence
S n+l by (for 1 ~ ~ ~ k):
sn+Iv = {by; I < y ~ I ^ by is a subset of a member of S~}. What we just said concerning the sequence (bl ,...,b~) S n+l satisfied the requirements Notice, by induction
(PI), P2),
shows that
(P3) and by (4) also (P4).
that we defined the infinite sequence S°,S* ,...,sn,... in ~
where at each step we made arbitrary choices,
namely by selecting G~(Z) for ~ = 1,2. But at each step we made only finitely many of them and each set S n is in V d which has a definable wellordering previously.
(in terms of members of d) as we have shown
Hence the construction of So ,S* ,.,,,sn...
above can be performed
inside of ~
To apply the Halpern-Lauchll
as given
.
theorem we define the following trees.
T~ = n ~ S ~ is the converse of the inclusion relation C. (for I ~ 9 < k). It follows from (P1),...(P4) of ( T ~ , ~
is exactly S~ and < T v , ~ )
tree-tops.
Hence all the requirements
that the n-th level
is a fan and by (P2) has no of the Halpern-L~uchli
theorem
hold in the present case. Let n be a natural number as in the consequence of that theorem. Let H be a choice function on the finite set W = ~ { S ~
m ~ n ^
1 ~ 9 ~ k} such that Range(H) C A. Let y be the k-sequence given by y9 : {H(s); s e ~ { S ~ ;
m ~ n}} for i ~ 9 < k. We shall show that
for every z C Ny either (5)
~{val(tb(U, , .... u k));(ul ~...,u~
e z} e I,
or ~ { - v a l ( t b ( u , .... ,Uk));(u, , .... u k) e z} e I. Once (5) is proved then we shall obtain the desired contradiction 1 B e I. Let us prove(5)
k
We define the following 2-partition of
~ (T~/n): ~=i
QI = {g 6 9=l(T9/n) ~ ;( H(g,),... ,H(gk)) 6 z} k Q2 = ~ (Tv/n) - QI, ~=1 where g is the sequence (gl,...,gk). By our choice of n the HalpernL~uchli theorem asserts that there exists a positive integer m0 < n such that either QI or Q2 includes an (mo,1)-matrix M. Suppose M C Ql. By definition there are (mo,1)-dense M =
subsets A9 of T v such that
~IA
Since all the sets A
are (m0,1)-dense,
we may choose a k-sequence
144
of intervals
and for all s E S~ m°+l,
if s C_ T~, then r _C s for some r 6 A~. With this choice of (T,, .... T ~
define a function f on U R a n g e ( S m°+l)
(using lemma 3) by:
if for all v with 1 ~ ~ ~ k, s ~ Tg, then let f(s) be an arbitrary member of s A A. f(s) =
if ( ~ ) ( 1
~ ~ ~ k ^ s _C T~), then take an arbitrary member
r of A~ for whieh r _C s holds and let f(s) be H(r). Thus for every s 6 U R a n g e ( s m o + l ) ,
f(s) 6 s. Define
G = {f(s); s e U Range(sm°+l)}. By requirement (6)
(P4) for m0+l we get
~{val(tb(U, ,.. ,Uk));(u , ..... u ~
6
~ ( T q G)} 6 i. ~=I v We shall now prove that ~ (T9 n G) C z. Once this is proved, then ~=1 (5) obviously follows, since x, ~ x2 ~ 11 ~ x, 6 1 is a property of ideals. Hence let us prove that the cartesian product of T v n G is k included in z, and let h = (h, ,...,h k) £ ~ 1 ( T 9 N G). Since h~ ~ T N G, there are s~ E U Range(Smo +1) for 1 ~ v ~ k such that f(sg)
= hg. Since f is a choice function, h9 6 s~.
Since also h9 6 T 6 S m0 v , it follows from (P1) and (P3) that s9 C Tv . But in this case h~ = f(sg) : H(rg) where r~ 6 Ag. Thus h =-(H(r ), .... H(rk))
where (r, ..... r k) e
of Q, this implies h = (H(r,),...,H(rk))
~ A. C Q,. But by definition ~:I v 6 z and hence ~(T9 N G) C z
holds. To deal with the other case, namely M C Q2 , let us write z ~ = (~y) - z
and proceed exactly as in the case M C QI ~ replacing z
by z* and Q, by Q2. Where
(P4) was used to obtain
(6), we use (P4)
now to obtain ~ { - v a l ( t b ( u , ,...,Uk)) ; .... } e I and get {-val(tb(ul ,...,Uk));( u~ ,...,u k) 6 z*) e I, so that (5) holds. Thus we have shown that for every z C ~y~ (5) holds. This will be used to obtain by means of lemma 2 the desired contradiction that 1B would be in I. Let P be the set of all functions
~ defined on
X = {val(tb(u,,...,Uk));(u~,...,u k) 6 ~y) such that for x e X, ~(x) e {x,-x). Consider z = {(u, ,...,u~ e ~y; ~(val(tb(u , ,...,Uk) ) = val(tb(u , ,...~Uk))} . Then z C ~y and by (5): ~ { v a i ( t b ( U , .... ,Uk))~(u, ,...,u k) e z} e I or ~ { - v a l ( t b ( u I ,...,Uk));(u, ,...,u k) e z} e I. Since one of these elements
is in I, their intersection
~{#(x);
x e X} e I for every ~. Since P is finite, the union of
these intersections
is in any case in i. Thus
is again in I. It was shown in lenuna 2, that this
145
union equals leads,
1 B. Hence
hence,
1 B 6 I. The assumption,
to a contradiction,
id est:
that
I is not prime
I is prime.
This proves
lamina 8. This proves (AC). (BPI)
the theorem,
The axiom of choice
that
(AC)
~[a0
,a, ,...,A]
is therefore
~ ZF + (BPI)
not provable
+
from
in ZF.
We may use the model constructed further
independence
result.
above
We consider
in order to obtain
the following
a
two defini-
tions of continuity: Definition
(L.Cauchy): nuous that
Definition
The function
f from reals
at x0 iff for every Ix - xol < ~
(Heine
- Borel):
to reals
is conti-
e > 0 there is a 6 > 0 such
implies
If(x) - f(xo)I
< e.
f is continuous
f(x0) kevery sequence
at x0 iff lim f(x n) = n-M~ {Xn}n6 ~ convergent to X0.
{o~ In elementary
analysis
one proves,
that both definition
lent, but the proof uses the axiom of choice. is no longer
true if we drop
L~vy in [~5]
and independently
[$~]
M.JAEGERMANN: nuity.
Theorem
(AC) has been discovered
by Halpern-
by
The axiom of choice
and two definitions
of Conti-
Bull .Acad.Polon. Sci .vol. 13 ( 1965)p. 699-704 •
(Halpern-L~vy;
Jaegermann):
every
from reals
x0)
are equiva-
That the equivalence
function
It is not provable to reals
in the sense of Heine-Borel
which
in ZF, that
is continuous
is also continuous
(at
(at x0)
in the sense of Cauchy. Notice, nuous
that
it is obviously
function
Proof.
By lemma
and by !emma
in ZF that every Cauchy-conti-
continuous.
4 (see page 136) A is dedekind-finite,
3 (see p.133)
of the product
provable
is also Heine-Borel
topology).
A is a dense
subset
while
infinite
of 2 m (in the sense
The function 1
: x ~ n~0 2n+~ for x 6 A, is a one-one
mapping
of A into the interval
closed,
left open).
subsets
of ~ are all in the g r o u n d m o d e l ~ ) .
of A are non-empty,
¢ is one-to-one the image ~(A)
since
]0,1]
no x 6 A is finite
is a dense
Since absolut subset
(right (finite intervals
of [0.1],
and
146
¢(A) is a dedekind-finite~ val [0,1].
while infinite,
subset of the closed inter-
Define the following function f: f(x) = ~ 0
for x E [0,1].
if x E ~(A) otherwise
L Then
f(0) = 1, since 0 ~ #(A). Since ~(A) is dedekind-
finite, every sequence
{Xn}nE ~ convergent to 0, for 0 < x n < 1, can
contain at most finitely elements from $(A). Hence lim(f(Xn)) f(0) and f is continuous
in the sense of Heine-Bore!.
is dense in [0,1], f is obviously not continuous
= I =
Since %(A)
in the sense of
Cauchy. q.e.d. H) THE AXIOM OF DEPENDENT CHOICES In his paper "Axiomatic and algebraic on sums of cardinals" p.96) A.Tarski
aspects of two theorems
(Fund;Math.35(1948)p.7g-104,
considered the following axiom,
la~te~ b y P . B ~ r n ~ y s
w h i c h was first formu-
(J.S,L. 7 (19~2)~p.86):
(DC) AXIOM OF DEPENDENT CHOICES: set x such that ( V y a countable
in particular
Let R be a binary relation on the
E x ) ( ~ z E x)((y,z)
sequence y0,yl ,...,yn,...
such that (yn,Yn+l)
E R), then there exists
(n E m) of elements of x
E R for all n E ~.
The name "dependent choices"
is used,
since (DC) asserts that there
exists a choice sequence where Yn is chosen in dependence of the choice of
Yn-l"
Betrays mentions that (DC) follows from (AC) and that
(DC) implies the countable axiom of choice
(AC ~) obviously
(see p.100
of these notes for the formulation of (AC~)). Both axioms,
(AC ~) and (DC), are powerful weakened forms of
(AC); e.g. in analysis
(AC ~) is sufficient to prove most of the
"positive" results such as the first fundamental theorem of Lebes~uemeasure.
In addition,
(DC) is sufficient to prove such results as
the Baire category theorem.
Further, we mention~
that A.L~vy has
shown in his paper "A Hierarchy of formulas in set t h e o r y " that (DC) is equivalent Sko!em theorem
~8],
(in ZF) with some forms of the LSwenheim-
(see [%8]p.72-74).
In the formulation of (DC) the choice of Yn is made in dependence of the choice of Yn-l"
In this formulation,
(DC) can not be generalized
to yield the existence of sequences of length larger than m (if certain hypothesis a "predecessor".
are fullfilled)~
since e.g. y~ can not depend on
Bu~ we get the idea to let depend y~ on the set
147
{Yn; n E ~}. More (i.e.
finite
generally
Choices:
and elements card(y)
for c a r d i n a l
Let R be a b i n a r y
of a set x, such that
< ~ there
a function
f'8
formulate
numbers
or an aleph):
(DC ~) D e p e n d e n t
Here
we
= f(S)
is an z 6 x w i t h
f : ~ ÷ x such
and f"8
relation
(y,z)
subsets
• R, then there
that (f"8,f'8)
= {f(Y);
between
for every y C_ x w i t h 6 R for every
is
ordinal
Y < 8}. The f o r m u l a t i o n
of
(DC e) is
of c e r t a i n
of the
due to A.L~vy: [SZ]
A.LEVY:
The
Interdependence
axiom Lemma
I: Lgvy's
of choice;
axiom
of d e p e n d e n t Proof.
(I) Suppose
ned b e t w e e n (card(y) subset
(DC)
function
there
f : ~ ÷ x by:
f satisfies
L4vy's
(2) Now for every
suppose
S n : {(zo~...,z~. and S : ~J{Sn;
to
n 6:m};
T ~nd ~ e m e n t s
to
• R).
• R* ~
f(n)
g
(3z
*(z> denotes
(DC ~) and
Define
let R be a r e l a t i o n
the
e x)(~to
: w ÷ x by g(O)
• T)(t
on x such
that
~ R. Define
relation
Let x be any set.
i.e. Since
= v0,...,g(k)
ZF °
~
Then
e R}
R* b e t w e e n
to
.: ( v o ~ . . ~ V n , ~
f(O)
f : ~ ÷ S = f'O
: Vk,g(n)
g satisfies
such
= (v0 ,..,v k) e S
= the
Bernays'
if
last (DC),
coordiq.e.d.
A e ( D C ~) ÷ (AC).
Define
R between
: to
is a f u n c t i o n
• R'.
[SZ]):
relation
following
concatenation,
Lemma Proof.
of Un+ 1 - u n. then
t Qf S by:
for n ~ k+l.
a binary
element
a
;( zo ,z,) • R ^ (z, ,z2) • R ^ ... ^ (Zn_l,Zn)
nate of f(n-k), 2 (A.L~vy
(n 6 ~)
all- n 6 m. D e f i n e
is z • x such that (y,z)
for all n • ~: ( f " n , f ' ~
define
e R).
uo ,u, ,...,Un,...
• R * for
is the only
defi-
C x)
R" on S:
= u2 ^ ( u l , ~
a sequence
that (Un,Un+1)
relation (~y
S = {u; u is a finite
relation
U {z}
= (vo ,...,v n) . By (DC m) there
that
Consider
the f o l l o w i n g
exists
axiom
(DC~).
y • x there
(T,t)
and let R be a b i n a r y of a set x such that
• R*.~-~.(~z • x)(ut
of S such
subsets
(DC)
< ~ ÷ ( 3 z • x)((y,z)
By B e r n a y s '
w i t h Be~nays'
(DC).
and elements
of elements
where
choices
of x} and define
(ul,u2)
consequences
54(1964)p.135-157.
(DC ~) is in ZF e q u i v a l e n t
Bernays'
subsets
Fund.Math.
W = {y C x; y is w e l l o r d e r a b ! e }
subsets
of x and e l e m e n t s
of x by:
and
148
(y,z)
6 R ~ (y 6 W ^ z ~ x ^ z @ y).
Let ~ = ~(x)
= sup{l;
is a cardinal satisfies
I is an ordinal
number.
the hypothesis
Suppose
and e m b e d d ~ b % e
x is not well-orderable;
of (DC~),
and hence,
x. Hence
~ 6 ~(x)
well-founded, Corollary
implies~
= ~, a contradiction,
there
exists
6 R for all ordinals
that ~ is embeddable since ordinals
into
are allways
q.e.d.
3. ZF °
~
Next we shall axiom of choice ees between
of R this
then R
by (DC~),
a f u n c t i o n f from e into x such that (f"8,f(8)) 8 < ~. By definition
into x}.
A ( D C ~) ~ (AC). discuss
another
and investigate
family
of weakened
the interdependences,
them and %he family of statements
Let ~ be an ordinal: that card(x)
(AC~):
= e~ then there
forms of the resp.
{(DC~);
if x is a set of non-empty exists
a function
indepen-
~ a cardinal}. sets,
f defined
such
on x such
then f(y) 6 y fom all y 6 x. A.L~vy
has obtained
in [~2]
(among other things)
the following
results: Lemma
4: Let ~ be an aleph, ~2 be alephs (DC~I ) and
For a proof
then
The following
(DC ~) ÷ (AC~).
such that ~i < ~2 , then
(AC ~2 ) ÷
(in ZF)
Let ~I and
(DC ~2 ) ÷
(AC ~I ).
see L~vy [5~J p.138~
(unpublished).
(in ZF):
nice result
We are grateful
p.140
and p.142.
was obtained
by R.B.Jensen
to A.R.D.Mathias
in 1965
for telling
us Jen-
sen's proof. Theorem
(R.B.Jensen):
ZF
~ A~( AC ~ ) ÷
Proof.
Suppose,
this
binary
relation
R on X such that
there
is no sequence
(Xn,Xn+ ~
is not true.
(DC) rather
than
Then there (Vy
~ ,xl ,...,Xn,...
6 R for all n 6 ~. (DC~)).
(DC~).
(Notice,
exists
a set X and a
E X)(~ z 6 X)((y~J (n E ~),
6 R), but
such that
that we consider Betrays'
Consider
W = {Y; Y C X ^ Y is wellorderable}. Then for all Y 6 W we get that ( Y , R - I ~ Y) can introduce relation
a rank-notion
R -1 (restricted
is w e l l - f o u n d e d
Oy on Y by induction
to Y).
Since
and we
on the well-founded
U W = X (since
all singletons
149
are in W) we can define for every z 6 X: l(z) = sup{Py(Z); Y e W}. Step I: If z e X, then ~ ~ ~(z). Proof. l(z)
Suppose not; then l(z) = n < m for some z E X, and hence
= Py(Z) for some Y e W. Thus there are elements xl,...,x n
in Y, such that (x, ,x2) 6 R -i, ( ~
,x3) 6 R - l , . . . , ( X n _ l , x ~
E R -~
where x n = z. But for given xl E X~ there exists xo 6 X such that ( x l , x o ) E R, and hence <xo,xl) get py.(Z)
e R -I. Hence for Y* = Y U {x0} we
= n+1~ a contradiction.
Step 2: For every z 6 X there exists Y e W sUch that l(z) = @y(Z). Proof. For ordinals y ~ l(z) consider the following sets: Ty = {(Y,M) ; Y 6 W ^ M wellorders
Y ^ py(Z) = y}
and define K(z) = {Ty; y < l(z)~. The set K(z) is we!lordered type < %(z) + I. Hence by (AC l(z)+l)
which selects from each Ty one element. define Y* = U { Y y ;
of
there exists a function f Write f(Ty) = (Yy,My) ~ and
Y ~ %(z)}. We claim that Y* is wellorderable.
In fact~ put Y0 = Ae, AI
= Y, - Y0 .... , A
for y < l(z) and N o = Mo,...,Ny Ay~ and Y* = 0 { ~ ;
= Yy - ( ~ { Y s ;
= My N ( ~
y < l(z)} and the ~
× ~);
8 < Y}),...
then Ny wellorders
ame pairwise disjoint.
Define M" = {(a~b) ; V y V 6 ( a
6 ~
^ b 6 A~ ^
[y < 6 v ( y then M* wellorders
= 6 ^
Y*° Hence Y* 6 W~ and py.(Z)
follows that @y.(Z)
It
= l(z), q.e.d.
Step 3. The final step: Take an element minimal in {l(z);
e Ny)])} is defined.
zo e X such that l(zo)
z e X}. By the Hypothesis
is
on R, there exists
z, 6 X such that < z0 ~z,) 6 R. For any Y 6 W such that zt 6 y define Y+ = y U {z0 }. Then always py+(Z0 ) = py+(Z! ) + I. By the result of step 2, there exists Y 6 W, such that l(z~ ) = py(z, ). Hence for this Y: X(z,)
: Oy+(Z,) < Oy+(Zs)
A contradiction~
+ I : Py+(Zo) ~ ~(zo).
k(Zo) would not be minimal~
This proves Jensen's
theorem. Independence Results Mostowski has shown in 1948 by means of models containing urelements ZF°~(DC
that (DC) does not imply (AC) in Z ~ , or even:
~) ÷ (AC~*):
150
[68]
A.MOSTOWSKI:
On the principle
Math.35
(1948)p.127-130.
We shall describe in whioh there reflexive
briefly
of dependent
Mostowski's
are Mt many reflexive
model. sets.
sets x = {x}~of cardinality
Ry = ~ J { P ( R s ) ; of reflexive
8 < Y} and V = ~JyRy. sets
Every ordinal
(called
"atoms"
y can be written
a limit ordinal
choices;
Consider
Fund.
a set theory
Let A be the set, of
M,. Define
Re = A,
Take an enumeration
in the sequel):
(in a unique way)
of the set
Re = A = {ay;y < m,}. as 8 + n, where
8 is
and n 6 e. Write y ~ 0 iff y = 8 + n with n even,
and
write y ~ 1 Iff y = 8 + n with n odd. A permutation
f of Re
one-one-mapping
iff f preserves
pairs, Def.
of R0 onto Re)
~ A y<~l[f(ay)
admissible
pointwise
permutations
fixed.
a countable fixed}.
~ a T ÷ [ (y ~ 0 ^ f(ay)
on R0 leave
subset
H of G a countable
support
support
as in chapter
subgroups
III,
since e.g.
section
B E~,
(AC e* ) is false
in ~
Mostowski
that
shows
Choose
that (Xn,Xn+l)
6 R
(DC),
fixed} in~
in~
(DCe), holds
) a countable
of type ~,.
in~
. In fact,
if
the hypothesis
sequence
x0,x, ,... such support
H = {f E G; f ].eaves
the sequence
xo ,xt ~...~Xn,..-
the following:
(A.Mostowski containing
violates
. Moreover
in~
the subgroup
is in F and fixes
e pointwise
= ~[G,F]
set i n ~
, such that R satisfies
of ~
on is
~ q.e.d.
Thus we have proved
Thus
id est
~
obviously
for all n E e. Each x n has a countable
~Js n pointwise
Theorem
Define
54. Then ~
is again countable,
which
is hence
as filter basis. B~ page
but B has no choice
(outside
S n, Since n~mSn
if there
of G which has the set of
, since B is w e l l o r d e r a b l e
R is a binary relation of (DC).
permutations
subgroup
e of R0 such that H = {f 6 G; f leaves
Let F be the filter of subgroups
countable
= ay+ 1) v
(y m I ^ f(ay) = at_l)]]. B = {{ay,ay+l};Y e e,^ y m 0}
Let G be the group of all admissible
R0. Call a subgroup
(AC),
admissible
id est:
f admissible
Hence
is called
(i.e.
[~8]):
atoms
There
in which
exists
a permutation
(DC ~) holds,
(DC) ~ (AC), or better
(DC) ~
but
(AC~*),
model
of ZF °
(AC e* ) is violated. is not a theorem
of ZF ° . A.L~vy's L4vy's
paper [5~]
paper
contains
is "pre-Cohen",
only to ZF ° and the method
further
independence
as Mostowski's
[68],
is by construction
results.
Since
the results
apply
of Fraenkel-Mostowski-
151
Specker models. Lgvy asks is provable.
in [ ~ ] p . 1 3 7
R.B.Jensen
We are grateful proof.Jensen
his result
in order to illuminate model.
for sending
the basic
the method
We follow Jensen
during
the sequel we make,
of course,
in 1965
(unpublished). of Jensen's
the L o g i c - C o l l o q u i u m model containing
idea in his
independence
to the construction
and present
(AC e) ÷ (DC e)
us his abstract
He first gave a permutation
then translated
whether
has solved this problem
to F.R.Drake
presented
in Leicester.
and in [~9]p.224,
proof,
and
of a Cohen generic
first his permutation
the tacit
1965
urelements
assumption,
model.
In
that ZF ° is
consistent. Theorem
(R.B.Jensen): containing true
Proof.
There
atoms
in~.
exists
in which
Hence
a permutation (AC e ) holds,
reflexive
sets
(called
Consider e~
first
by:
; x,
(DC e) is not in ZF °.
is a set R0 of
I ~ m < m}. There
<
...
< xm e
is a natural
one defines
M~. We
on Ro.
set e, and consider
< x~
sl < < s= ~ the sequence preceeds
of ZF °
such that Ro has cardinality
the w e l l - o r d e r e d
of s~. More precisely immediately
in which there
tree-ordering
: {<x,,...,x~
and H = U { e ~ ; defined
atoms),
a certain
while
(AC e ) ÷ (DC m) is not provable
Take a set theory with choice
want to define
model ~
e,}
partial
ordering
s, is an initial
first
< < on ! segment
sl <. s2 to express
that s,
s=:
sl ~ s2 ~ (s, : (x, ,...,x 2 ^ s2 : <x, ,...,Xm,Y) ) s, ~ s2 ~ ( ~ y E m,)(s2 = s,*
Hence
sequence.
of
Now define
s, < < s2 ~
there
are finitely
h,,...,hn, Since H has eardinality
many elements
such that sl = hl ~
M, , there exists
R0 onto H. Via f one carries
of H, say
h2 ~
a one-to-one
the tree-structure
.. ~ h n : s2. mapping
f from
< < over to R0 by:
a < b ~ f(a) < < f(b). Hence < is a strict
partial
Notice
has no tree-tops,
that ( R 0 , ~ )
of R0 is finite successors
ordering
on R0 and < R0,~ ) is a tree. that the order of each element
and that for every a 6 Re the set of immediate
{b 6 Ro ; a < b ^ ord(b)
= ord(a)
+ 1}h~S
Let G b e the group of all orderpreserving from R0 onto R0. subgroups
In order to define
we define
the notion
one-to-one
an interesting
of a "small
~ar~mg~{{7 ~ mappings
filter F cf
subtree".
First:
.
152
B is a c o m p l e t e and
branch
in ( R o , ~ )
iff ( B , ~ )
if a < b 6 B, then a E B and t h e r e
element Definition.
d E R0
such that
If T is a s u b s e t
of R 0 , t h e n
a < b 6 T implies
Definition.
branch
A subgroup exists
in ( R 0 , ( )
subtree
F to be the set of t h o s e
subgroup.
M[y]
= {x} U x U
as in c h a p t e r
e F)}.
Speeker's
ZF ° . We shall
Ux
III,
The a x i o m of d e p e n d e n t
Proof.
Since
is a small
subtree
is a b i n a r y (DC m),
and h e n c e
countable
subtree
pointwise K|T]
fixed}
But
Hence
permutation b,
(by d e f . s e e
exists
set of i m m e d i a t e
ord(e0)
successors
£x e R0 ; x >I b0}
T e K[T]
and since
of bo,
If
be a
of Ro
in~
is a c o m p l e t e
is a T
branch
(or even p o s s i b l y the f o l l o w i n g
- T such that~
if
then bl < co and ord(co) and co and b0 are
= eQ,
,
then there
bo ~ T. D e f i n e
and T(bo)
onto
y).
= S, id est
of b~ ). Let T be the
K[T]
Ro~
Obviously
= {o 6 G; o leaves
co 6 Ro
of
{b 6 R; b ~ a}
e R0)(x<
n(S)
~
in ~ .
E F and h e n c e
of e l e m e n t s
T N S" is f i n i t e
= ord(b0)
{x 6 Ro ; ~ (x > bo v x > co)} the s u b s e t
and h e n c e
6 S)(b ~< a)}
bo 6 S w i t h
predecessor
not h o l d
then t h e r e w o u l d
entails
of x.
6 TC(x)
.
If S is in ~ ,
(~a
= x}
is a m o d e l
<~ is in ~ .
if ~ 6 KIT]
z of Ro : pick any e l e m e n t
ord(b, ) + 1 (hence
Hence
in ~ ,
p.57)
T is small,
there
is an i m m e d i a t e
order,
a nice
closure
Ay(Y
in~
e R0)(~y
....
such that
S* = {b E Ro;
in (R0 ,~ ) • Since empty).
... < a n <
T of Ro
include
that ~
(DC m) does
S = {ao,a, .... ,an,...}
such that a0 < al < small
shows
that K[a] --- H[a]
hold
iff t h e r e
= {~ e G; w(x)
= {x;
(DC ~) h o l d s
(~x
is a
fixed}mK[T]
H[x]
~[G,F]
finite
satisfying
(DC), w o u l d
sequence
subgroup
... be the t ~ a n s i t i v e
and the t r e e - o r d e r i n g
relation
(id est:
in ( T , ~ )
of G w h i c h
Define
ehoiees
it f o l l o w s
of R0
of
such that
T pointwise
~ :
^ 7
for a 6 Ro, a has
element
a nice
(T,()
(see p.54)
(AC ~)
Lemma:
every
U
subtree
( T , ( } is a s u b t r e e
subgroups
p,54:
theorem
show that
an
.
F is a f i l t e r of s u b g r o u p s .
and let TC(x) Define
a small
a 6 T) and no b r a n c h
H = {7 e G; ~ leaves Define
not exists
(T,()is
H of G is c a l l e d
a small
does
ordered
a < d for all a 6 B.
complete
is t o t a l l y
in the same
identity
T(co)
=
= bo
on and • maps
{x 6 Ro ; x > co} and m e v e r s e l y .
~ H[ S] , it follows
that
T(S)
= S.
153
But b0 6 S i m p l i e s
e0
proves
the
Lemma.
The c o u n t a b l e
Proof.
Let z = {xi;
i ~ j ~ x i N xj
z,
K[T]
holds {Yi;
= S, a c o n t r a d i c t i o n •
a x i o m of c h o i c e
Z is in ~ .
~ K[Z]
(AC ~) holds
i 6 ~} be a c o u n t a b l e
= ~, x i ~ ~ for all
=
~ H[Z]
p.57-58)•
see
6 ~(S)
Write
Hence
~
Z
set theory,
(not in G) w h i c h
is in ~
.
Since Yi 6 ~
transform
, there
there
exists
of ~
subtrees
subtree.
K[T']
~ M[C]
obtained Case
~ M[C]
2. T"
C for
is not a small
permutations subtree.
and C is in ~
a choice-set
~'I
K[Tz]
6
We c o n s t r u c t
II)
for
III)
Suppose
that
Construction
set C =
for
set C* w h i c h
Then o b v i o u s l y
and we are done
since we have
z in ~ .
subtree.
these
identity
of
(AC)
of (R0 ,< ) such
We shall
construct
such that T o = ~ { ~ i ( T i ) ;
I) Let ~0 be the
a choice
choice
Ti
Case
is a small
Since
. We are l o o k i n g
by cases.
i 6 m}
T such
i 6 ~. The set C need n o £ b e
C into some
are small
subtree
of ~ •
that K[T i] < H[Yi ] for i e ~. P r o c e e d 1. T" = U { T i ;
such that
Z is a w e l l o r d e r i n g
outside
by the t r a n s i t i v i t y
mappings
in ~
is a small
(since
. Proceed
i 6 m} such that Yi 6 xi for all but C C ~
set
in ~ .
i e ~ and the s e q u e n c e there
H[z]
T = T
in the s u r r o u n d i n g
in ~ ,
This
lemma.
{(xi,i) ; i 6 m} that
: T(b0)
permutations
ni by
a sequence
i e ~} induction
of
is a small on i,
on R 0 .
0 < i < n permutations
~i 6 K[Tz]
are defined.
of n
Of course,
the c o n s t r u c t i o n
that the tree immediate
structure
successors
cardinality H e n c e we may
M, w h i l e shift
of ~n takes
~ on R o has
of any small
element ~ubtrees
(displace)
place
outside
the p r o p e r t y , of R 0 has always
(outside
have
the tree T n into
For a E R 0 let [a]
be
{x 6 Ro;a < x}.
ord(a)
is for a E R 0 always L(~)
For y 6 L(k)
= {x e R0;
let I(y)
I(y)
finite.
ord(x)
be the
of ~ )
cardinality
Further
from
~ ~0"
L(k)
~{~i(Ti);
call the e a r d i n a l i t y
of a, in symbols Define
. Remember set of
a tree nn(T n ) so that
T z N Tn = Tz N ~n(Tn) but ~ n ( T n ) - Tz is d i s j o i n t 0 ~ i < n}. In details:
of {y 6 R0 ; y ~ a ^ y ~ a} the o r d e r
of ~
that the
ord(a).
to be the k-th
: k}
set of i m m e d i a t e
= {x 6 Ro ; y < x ^ ord(x)
successors
= ord(y)
+ i}
of y:
Hence level:
154
Thus yl ~ Y2, Yl, Y2 e L(k) implies I(y I ) • I(y 2) cardinality M, and further L(k+l)
= U{I(Y);
=
~lalways has
y E L(k)}.
When we consider ([ a] ,~< ) we mean of course the ordered pair consisting of [a]
and ~ restricted to [a] . If a, , a2 6 L(k), then
([a,] ,( ) and <[a2] ,~< ) are isomorphic
(and isomorphic with
( R 0 , ~ ) ). Let ~a2 a, be such an isomorphism which maps [al] onto [a2]
in an orderpreser~ing
way. If ord(a, ) 4= ord(a2)
one-to-one then
o al a~ is undefined. For m < ord(x) define k(m,x) to be the element z 6 R0 with ord(z)
= m and z < x. Since ~ is a tree-ordering of height m,
l(m,x)
is unique.
With this amount of notation we are able to define by induction (on k E w) a sequence of permutations
gnk 6 G which approximate
n"
0 is the identical mapping on R0. Suppose gnm is defined for gn 0 ~ m ~ k. Define gnk by means of gnk-l,o and the following function which we are going to characterize: Since Tz, T n and U { ~ i ( T i ) ; each set I(y) for y 6 L(k-1)
0 ~< i < n}
are all countable and
is uncountable we may find a function
from L(k) onto L(k) which preserves the partition id est ~"I(y)
{I(y); y e L(k-1)}
= {~(x); x E l(y)} -- I(y) for all y 6 L(k-1), and
leaves L(k) N T z pointwise
fixed and
n-1
{g(x); x 6 L(k) N T n ^ x ~ T z} N {x 6 L(k); x 6 T z U ~ Now define g< :
i(Ti)}
=
0
.
let x be in R0:
I gkn-l(x)
if ord(x) < k,
gk(x) : ~*(gnk-1(x)) ~(k,x) |O~. k ....
if ord(x) = k, (X)
> If ord(x) k.
Thus the sequence gnk for k 6 w is defined by induction for all k 6 and we define ~
n
: let x 6 Ro:
ord(x) (x) ~n (x) : gn It follows that ~n E KIT z ] and ~n (T n) is "disjoint from i=o~i(Ti ) modulo T z'' . Since we . have defined . the . sequence
~o
,~I,
'~n'"""
by induction it follows that every branch B which is included in To : ~ { ~ i ( T i ) ;
i e ~}
is either included in T z or in one of the small subtrees wi(Ti). Hence T o itself is small. Hence KIT o ] E F. Define C o : {~n(Yn ); n 6 ~}; then K[T °] < K[~n(Tn)] ~ H[*n(Yn )] since KIT n] ~ H[Yn ] . Thus KIT °] ~ K[C °] ~ HI C °] and it follows that C ~ is a set of ~ .
155
Since ~n 6 K[Tz ] ~ KIz]
it follows from Yn e Xn 6 z that
~n(Yn) 6 ~n(Xn ) = Xn 6 z. Hence C o is a choice set. This proves the lemma and Jensen's theorem is established. Discussion.
The function ord can be given in ~
is a countable set in ~
. By (AC e ) in ~
and {L(k);k e e}
, ~.~j~,contains a choice set
C for this set. It follows that {x 6 R0; (~ y e C)(x < y)} is a small subtree~ Another point is, that the method of proof given above can be used to yield the following generalization: Theorem:
If ~ is an aleph, then (AC ~) + (DC e) is not provable in ZF ° (provided ZF ° is consistent).
The proof is analoguous to be U { e ~ ;
to the one given above.
m e e} one takes U { ( ~ + ) m ;
Instead of defining
m 6 e} as ~, where ~+ is
the successor aleph of ~. Small subtrees of (R0, ~ ) are subtrees without complete branches of cardina!ity < e. Notice, that here R0 has cardinality ~+. The proof can be carried over to the present ease~ since e + is a regular cardinal. This generalization
shows, that the other result of Jensen~
namely ZF e ~ A ~ ( A C ~) ÷ (DC e) is the "best possible" result. Translation to a Cohen-~ener!c model Theorem (R.B.Jensen 1965): Every countable standard model ~
of
ZF + V = L can be extended to a countable standard model of ZF such that: (a) The ordinals of ~
are exactly the ordinals o f ~
(b) the alephs of ~
are exactly the alephs of ~
(c) (AC e ) holds in ~
,
, ,
(d) (DC m) does not hold in Proof. ~
Let ~
be a countable standard model of ZF + V = L. We extend
by adding to ~
a generic copy of H (the tree defined on p.1[~)
and generically all small subtrees T of H together with wellorderings on each T. Hence take the first uncountable in ~
ordinal e,9 ~
and the tree-ordering < on ~ as before.
language £ in ~
of ~
and define
Next define a ramified
which contains besides the usual ZF-symbols,
the
limited quantifiers V ~, the limited comprehension terms E e (for
156
ordinals e of ~
), constants x for each set x o f ~
, constants
ah for each h E ~, a binary predicate < and binary predicates
ST
for each small subtree T of H. A condition p is a finite partial function from ~ x H into 2. Define a strong forcing relation Ib containing the following keyclauses: p !b t ¢ ah ~ (3 n 6 ~)(p I~ t = n & p((n,h} ) = 1), p tb ST(t, ,t2) ~ ( ~ h
E ~)(h e T & p ~ t, = ah & p I~ t2 = h).
where t, t, , t~ are constant terms of Z. Obtain a complete sequence of conditions
and thereby a valuation val(t)
terms t. By the Hauptsatz,
~
model of ZF which contains ~ and ~ hence
of the constant
= {val(t); t constant term of £} is a as a complete submode!.
have precisely the same ordinals
Further
(by lemma T, see page 90);
(a) holds. Notice that 1) a h = val(~h ) _C ~ 2) ~ = val ( ~ ) is a tree-ordering
the field of which is
just the set {ah; h 6 ~}, 3) S T = val (~T) = {(ah,h ) ; h e T C i} more precisely ~ is the valuation of Em+lx( V v V z ( x
= (y,z)
^ y ~ z))
J
and similar for S T . Since we took symbols ST into £ only for small subtrees T which are in ~ a n d there is in ~
~
satisfies V = L, hence
a we!lordering W T for T. Since S T is a one-one-function
from {ah; h 6 T} onto T~ and val (W T) = W T is in ~ in ~
(AC),
the sets {ah; h e T}
it follows that
(T a s-mall subtree of ~ in ~
) are
wellorderable. Let G be the group (in ~ )
ef all orderpreserving
one-to-one-
mappings ~ from H onto H. For a condition p define ~(p)
= {((n,w(h)} ,e) ; ((n,h) ,e) 6 p}
and define the action of G on £ in the following way: h 6 !~ then z(a h) = a~(h) ~ ~(ST ) : Sn"T w"T = {w(h); h e T}
if s E G and
~ where
and for an g-formula ¢ let w(¢) be result/from ¢ by replacing any occurence of ah by w(ah ) and of ST by W(ST ) remain unchanged). Symmetry-Lemma:
(the other symbols of
Then the following holds:
If w E G, p is a condition and ¢ an £-sentence~ p II- ¢ ~ ~(p) if- ~(¢).
then
157
Proof by induction
(see page 99).
This lemma enables us to prove: Lemma: The countable axiom of dependent in ~
choices
(DC m) does not hold
.
Proof. One shows that no condition p can force that {ah; h 6 H}, partially ordered by ~, has a complete branch.
See the proof of
(DC ~) in the preceeding FraenkelMostowski-Specker In order to show that (AC ~) holds in ~ which says, that any set of ~ table in ~ thus proving
are preserved
we need a lemma
which is ~ - c o u n t a b l e
) is also ~ - c o u n t a b l e .
that cardinals
model. (i.e. coun-
More generally we shall prove
in the transition from ~
to ~
,
(b).
Combinatorial
Lemma.
exists i n ~
Let B be i n ~
a set of conditions.
There
a subset B' of B such that B' is in ~
coun-
table and for every p 6 B there is a p' E B' such that p and p' are compatible. For a proof see e.g. Jensen's
lecture notes [ 3 9 ]
(Springer-Berlin)
page 147, or these notes page 106-107. The lemma implies obviously that if B is in ~ incompatible, (these notes),
a set of conditions whose elements are pairwise
then B is countable.
We have shown on p.106-108
that this implies that cardinals
finite ordinals)
(i.e. alephs or
are absolut in the extension from ~
to ~
;
thus we have shown: Lemma. For every ordinal y of ~ of y in the sense of ~ in the sense of ~
hence ~
and
9~
(and hence of ~
is equal to the cardinality of T
:
have precisely the same ~lephs.
In order to show that (AC ~) holds in ~ notation:
is a complete
p.96-97), £ is not in 0 ~
we introduce the following
for a small subtree T of (H ~ )
of sets which are explicit definable and ~
) the eardinality
submodel of ~
~-definable.
let V(T) be the ~
-class
from T. Since • is ~ - d e f i n a b l e (see the necessary remark on
Since the correspondence h ~ ah is
(for all h e H), we cannot interpret £ inside of ~
but what we can do is to interpret certain sublanguages £(T) of £
158
in ~
. Namely
let for a small
which
contains
besides
further ~ for x E ~ symbols
is a constant
-definable
term t s of 2(T)
we can define
in ~
~(~ ) = <
and ~(x)
and then extending
= {~(t);%is
V(T)
is an
~-definable
~-definable
C(T)
sequences
of
weliordering
of £, but only for
in the present
case~ q.e.d.
is a small suhtree T such that that p I~t ~ ~ ~ t N C(T) ~ ~. term of 2 so that V(T)
B : {q 6 Cond;
of conditions.
a ~-countable
subset
is t' of £(T)
=
q I~" V x x g t},
Now apply the combinatorial B' of B so that for every
is q' 6 B' with q U q, 6 Cond,
Then for p' E B' there
. Hence the class
as finite
for all terms
is the unlimited
Cond is the ~ - s e t
and
it is clear,
has a (lexicographic)
p it holds
For given t consider
lemma and obtain
}
in
via ~.
but this is sufficient
Sketch of Proof.
e e on9~ in ~
(considered
of £(T))
over to V(T)
are wellorderable
wel!orderings,
For every term t of £ there
q 6 B there
subtree}
}, { V e ;
that Q can not be defined
vaI(C(T)).
then V(T) has an
is wellorderable
t of £(T)
for every condition
where
of (H ~ )
a set of ~
from the alphabet
is carried
Thus
term of £(T)}
class.
subtree
of /(T)
terms
~ for constant
to all t's of £(T).
have ~ - d e f i n a b l e
of all constant
terms of £(T);
and the
wellordering.
{~; x
}
that the alphabet
Notice,
9~-definable,
an interpretation
Since T and {D C T; D a small
and further
Lemma.
hence
for h E T, D C T, D small subtree,
a constant
If T is a small
symbols
and has an
= x,
by induction
V(T)
{Ee; e 6 On ~
which are explicit
-definable
by setting
~(ah ) = a h, ~(SD ) = S D
which
is ~
a h ~ h for h 6 T and S D ~ D (since D = Range(SD))
t of £(T)
Proof.
from
such that s = val(ts).
of all sets of ~
is ~ - d e f i n a b l e ,
for D C T are in ~
Lemma.
definable
wellordering.
since Z(T)
correspondences
and
is explicit
from T. We claim that V(T)
In fact,
ah for h E T and
D for D _C T.
Now let V(T) be the collection definable
the language
the symbols V e and E ~,
, only constants
we say that the set s of ~
T if there
terms
T be ~(T)
the ZF-symbois~ and ~
~D for small subtrees
Notation:
subtree
and construct
the tree T.
so that p' i~*~ ~ ~ ~ t' ¢ t.
159
Lemma.
(AC m) holds i n 3 ~
Sketch of proof.
.
Let z be a countable set i n 3 ~
, z : {zi; i e ~}
so that each z i E z is not empty. Let t z, t i (i E ~) be constant terms of £ so that z = val(t z) and z i = val(ti).
By the preceeding
lemma there is a sequence of small subtrees T i such that l~ti ~ ~ ~ ti N C(Ti ). Obtain a sequence of permutations leaving
t z invariant but such that T* : ~ J { ~ i ( T i ) ;
small subtree
(use the construction presented
Mostowski-Specker
version of the model).
i 6 ~} is a
in the Fraenkel-
Hence in ~
for all i E ~. Since V(T*) has an ~ - d e f i n a b l e obtain a choice sequence in ~
~.z
: z i ~ V(T ~) ~
wellordering we
. This proves Jensen's theorem.
Again, as in the case of the permutation model, the theorem can be strengthened to Theorem
(Jensen):
Let ~ be an infinite cardinal in a countable
standard model ~ to a countable
of ZF + V = L; then
standard model ~
~
true but (DC m) is false and furthermore ~L
and ~
can be extended
of ZF in which 0n 9Tt
(AC ~) is
= On 9~
and
have the same alephs.
The !ndependence of the Axiom of Choice from the Principle o f Dependent choices Mostowski
showed in 1948 that (AC) is independent
ZF ° . W.Marek in Warsaw translated Mostowski's model ~ o f ~
from (DC) in
construction of a
ZF ° + (DC) + ] (AC ml) to yield a Cohen-generic model
of ZF plus
(DC) + 7 (AC). Thus
(DC ~) ÷ (AC) is not provable in
full ZF. [ST] W.MAREK: A ~emark on independence
proofs; Bull. Acad. Polon.
Sci. vol.14(1966)p.543-545. Marek ~ust immitates Mostowskits standard m o d e l ~
model by adding to a given countable
of ZF + V : L generically a set X of ~I many
unordered pairs Bp (p E ~i) where each Bp contains two Cohen-generic reals ~p and U~. The extension ~ closure
(using ordinals of ~ ) ,
is obtained as the constructible but besides G6del's eight fundamental
operations Marek takes a nineth one which will serve to add countable sequences.
Unfortunately the proof in [57 | is only briefly sketched.
We shall consider in the sequel Feferman's model ~ [ a 0 , a 1 , . . a n . . ] . Feferman showed that in this model the (BPI) fails and Dana Scott pointed
160
out that even the axiom of choice for unordered pairs violated - see Feferman [46]
(AC2) is
(we cited this paper on p.98).
Several
people observed that (DC) holds in Feferman's model.
Probably
R.M.Solovay was the first who made this observation.
A proof is
published in: [~]
G.E.SACKS:
Measure - Theoretic Uniformity
in Recursion Theory
and Set Theory; Transactions Amer.Math.Soc.vol.142(1969)p.38142O The proof is presented in [}3] in the language of measure-theoretic uniformity rather than in the forcing approach. [~3] are announced
in: G.E.Sacks:
Bull. Amer. Math. Soc. vol.73(1967)p.169-174, title)
in the G~del-Festschrift
FEFERMAN'S MODEL Let ~
The results of Sacks
Measure Theoretic Uniformity; and (under the same
(Springer-Verlag,
Berlin 1969)p.51-57.
~[a0,at,...].
be a countable standard model of ZF + V = L. Define in ~
a
ramified language 2, which contains besides the usual ZF-symbols ( 7, v , V ,
:, e a n d variables),
limited existential q u a n t i f i e r s V ~
and limited comprehension operators E ~ (for ordinals s of ~ constants ~ for eagh set x of ~
)~
and unary (generic) predicates
ai
for i 6 m. We suppose that this is done in such a way that the correspondences
x ~ !~ i ~ at' ~ ~ v e and e ~ E e are a l l ~ - d e f i n a b l e .
This can be attained e.g. by the standard-method Define conditions
p to be finite partial functions
2 = {0,1}. The (strong) ~-sentences
- see page 79. from ~ × m into
forcing relation I~ between conditions
is defined as usual.
p and
The definition contains the following
key-clause: p I~ at(t) ~ (~ n e ~)(p I~ t = n & p((n,i))
= I)
where t is a constant term of £. Obtain a complete sequence ~ and thereby a valuation val(t)
of the constant terms t of £. Let
= {val(t); t a constant term of £}, then ~
is a countable
standard model of ZF. It holds that a m• = val(Ai ) C ~. We use the following notation ~ ~"~.,"[ a0 ,a, . . . . ] i.e. ~
results from 9 ~
reals a i (i E ~) to ~ .
by adding countably many Cohen-generic Notice, that the correspondence
a i ~ i~
id est {(at,i) ; i 6 m}, is not added and that we did not_add a set A which just collects these generic reals ai?hThis rence to the model of Halpern-L~vy:
~[
is the main diffe-
a0,a, ~...,A]
(see pages I01-
103 ~nd p.131). Halpern-L4vy's model satisfies the (BPI) and since
161
(BPI) ~ (0P) ~ (AC2) where
(BPI)
principle ordered
is the Boolean
and
We shall
~[a0,al,...].
is no set in an,...
ideal theorem,
~[a0,a,
,...]
mely different
which
~ust collects
of Feferman
(in~)
(AC2) does
the reals
ao,a,,..,
and of Halpern-L~vy different
seem
and have extre-
~
and define
model of all one-to-one
Let ~(9) be the result ~(p)
of substituting
p. Then the "classical"
p ~P ~ considers
mappings
a~(i)
= {((n,~(i)),e);
"If ~ 6 G, ~ is an Z-sentence
formation
that
set of un(AC2) holds
features.
for conditions
Feferman
that
it will be shown that there
they are considerably
Let G be the group onto ~.
says that every
show in the sequel
Symmetry-proper ties of Feferman's
the Z - f o r m u l a
(0P) the ordering
it follows,
Moreover
Though the models
to be very similar~
which
choice-function,
in ~ [ a o , a l , . . . , A ] . in
prime
(AC2) the statement
pairs has a
not hold
(in ZF),
for ai in
((n,i),e)
Symmetry-lemma
~ of 6
p}
says:
and p a condition,
then
iff ~(p) Jb ~(~)" in [16]
(we use L@vy's
If r is a set of ~
p.330-331, notation
a different
in [51]
and r C w × ~
kind of trans-
p.147-149).
, then r defines
a trans-
formation: Definition:
Let Q be a function particular [r,Q]
on a subset of m × ~ into 2 (in
Q may be a condition).
= {((n,i),e)
We define
; (((n~i)~e) e Q ^ (n,i) 6 r) v v(((n,i) ,l-e) e Q ^ (n,i)
Definition.
Let ~ be an Z-formula. replacing
or a constant
of ai(~)
term of Z)
[r~c]
s ~ ^ (Ai([r,cl)
where
[r~]
stands
We write [r,~]
each occurPence
~ r)}.
for the result
(where
of
~ is a variable
in % by
"(It,C],!>
~ r),
~- ~ if ~ is a variable
and ([r,%] ,_i) ~ r
for:
Vx tX s r ^ A
(Y
x-
(A:(z
(the Kuratowski-definition ted s e n t e n c e
of
/?)
and
y
[r,~]) v
<'> Z
of an ordered
z = [r,~l
for
pair as a limi-
x(V
e
z ,e, v e [ r , ~ l
).
162
It follows, ~,
that
then [r,p]
£-formula,
if p is a c o n d i t i o n is a c o n d i t i o n
then [r,#]
(in ~
is a limited
so is [r,%] . If t is a limited then [r,t]
~ E~x[r,#(x)]
used
for [r,~]
Lemma
above
If QI c Q2,
and ~'(x) variable
(see p.79 Lemma
then [r,Q,] = Qi
(for
with
6[r,t]
= 6(t)
for d e f i n i t i o n
c [r,Q2] • Further
then
every
strengthened
it holds
= ¢'([r,u])
if u is a
t e r m of 2. Further
for any c o n s t a n t
that
is an 2 - f o r m u l a it holds
term t of £.
sequence
of c o n d i t i o n s
, then
term t: val~(t)
proof
iff
~[r,~l
see L~vy~s
= val[r,~l]([r,t])
iemma.
for
For every £ - s e n t e n e e
~[r,¢].
paper
[51]p.148.
Lemma
III can be
to :
~)
[16] ): Let ¢ be an £ - s e n t e n c e ~
and r C ~ × m, r a set of ~ p J~ ~
Next we shall
Lemma
in ~ .
sequence.
constant
IV.(Feferman
not hold
C 2 and
of 6).
~¢
Lemma
If %(x)
[r~¢(u)]
r and 6~ as in the p r e e e e d i n g
Fo~ a d e t a i l e d
is
~--Def ([r'Po] ,[r,p~] , .... [r,Pn] .... )
is a c o m p l e t e LemmallI.For
(this
Q,, Q= and r are
if r C m × m is a set i n ~ [r,~]
term
Dora(Q1 ) C m × m,
!l.lf ~ -~ (p0 ,pl ,. .. ,pn,. ..) is a c o m p l e t e and
say Eex#(x)~
C_ 2, Range(Q2)
that
of ~ or a c o n s t a n t
then
term).
i = 1,2).
= [r,¢(x)],
term,
comprehension
if ~ is a c o n s t a n t
suppose
if ¢ is a limited
if ~ is u n l i m i t e d ,
comprehension
C m × m and Range(Q,)
[r'[r'Qi]]
that
). Further
is a limited
let r C m × m and
and r C m × ~, r in
£-formula;
I. Let QI and Q2 be functions Dom(Q=)
(in ~ )
in ~
present
iff [r,p]
V. Let P(m)
some
of ~.
lemma,
. Then
which
says,
that
(BPi)
does
lemmata.
be the p o w e r s e t
If I is not p r i n c i p a l ,
(in
IP It,C] .
Feferman's
. We need
p a condition
of ~ and I a prime
then
I contains
ideal
all finite
in P(m). subsets
163
Proof.
Suppose,
there
is a f i n i t e
that S ~ I. T h e n a - S e I since - { b 2 } , . . . , B n = a - {bn}. is p r i m e
there
a proper
ideal,
exists
Then
i with
subset
S = { b t , . . . , b n } _C ~ such
I is prime. ~{Bi;
Write
B,
= a - {bt},B2
1 ~ i ~ n} e I and since
I
1 ~ i ~ n such that B. 6 I. S i n c e I is l H e n c e I is a p r i n c i p a l ideal,
I = {x C a; x C B,}.
q.e.d. Lemma
(Feferman
[~6]p.343):
sense of ~
Let P ~ ( a )
. Every prime-ideal C } is a p r i n c i p a l
Suppose
there
(m), _C ) . Thus
exists
of ~ in the
in the B o o l e a n
algebra
ideal.
in ~
I contains
be the p o w e r s e t I 6 ~
a non-principal
all f i n i t e
prime
subsets
i d e a l I of
of m. Let t I be
a constant
t e r m of £ such that
I -~ v a l ( t I) and t I -~ ESx¢(x).
Let occ(¢)
= {i e ~; ~. o c c u r s
in ¢} and n e ~ such that i 6 occ(¢)
l
implies
i < n. We shall
(a n generic). Case
Proceed
1. S u p p o s e
show that n e i t h e r
by cases:
a n 6 I holds
in ~
is f o r c e d by some c o n d i t i o n s define&
~
,
there
exists
. Since
in ~
in 6{, the c o m p l e t e
that [r,p]
p I~ [r,¢(&n)] , h e n c e b [r,¢(&n)] ~
that holds
sequence
in
which
~ ( k , n ) @£ Dom(p).
:
r = {(k~m); It follows
everything
p 6 ~ such that
p t~- ¢ ( ~ n ) . Let ko be c h o s e n so that for all k ~ k0 Define
a n nor m - a n is in I
m ~ n v (m = n ^ k < k0)} C ~ x ~. = p. Let ~'(x)
= [r,¢(x)] . L e m m a
~
since [r~p]
~ [r,¢(an)]
~ % ' ( I t , & n] ). By c o n s t r u c t i o n
¢'([r,ln])
~ ¢([r~n]).
IV implies
= p e 6{. But
of r:
Hence
~[ ~ ¢([ r,&n] ) and t h e r e f o r e v a l ( [ r , ~ n] ) 6 I. But val([r,~n] ) -- [ (a n N ko) U (~ - an) ] N (~ - k0). Since
prime
follows
ideals
J satisfy:
that e i t h e r
But I c o n t a i n s
all f i n i t e
H e n c e a - k0 c a n n o t Thus we get that by our a s s u m p t i o n Case
subsets
(a n N k0) U an 6
This
proves
it
is in I.
in p a r t i c u l a r
k0 U (~ - k0)
k0 .
= ~ E I.
(~ - an) 6 I. H e n c e ~ - a n 6 I. But
I, a c o n t r a d i c t i o n , in ~
(~ - k~)
of ~, h e n c e
be in I since o t h e r w i s e
2. a - a n e I holds
a contradiction.
a rl b 6 J then a e J or b 6 J,
(a n ~ k0 ) U (~ - an) or
. Proceed the
lemma.
~ would
be in I.
in a s i m i l a r w a y and o b t a i n
=
164
Corgllary: Proof.
The Boolean Prime Ideal Theorem does not hold in 0 ~ .
It is well-known that (in ZF) the (BPI) is equivalent to the
statement: ideal"
"Every infinite Boolean algebra has a non-principal prime
(see e.g. Tarski's abstract in the Bull. AMS 60(1954)p.390-
391). It follows,hence~from
the previous
lemma, that (BPI) does not
hold i n ~ Remark:
Dana Scott showed that in Feferman's model ~
even there
does not exist a choice set selecting reals from the cosets of the rationals
in the reals - see Feferman's
paper [96]p.343-344.
of all constant terms Let C(~,Ai1,~i ,... ,~.ik ) be th~ ~ - s e t t of £ such that ~(t) ~ e and symbols ~j occur in t only for j e {ii ..... ik}. Let ~ [ ~ , a i ~ sets (of ~
..... aik] be the
~-set
of those
) which are denoted by members of C(e,~il , .... ~ik ).
It is clear that £ has a constant term ~(~,~il,...,~in)
which denotes
~[~,ail,..,aik] . Define C(Ail , .... ~ik ) : U { c ( ~ , ~ i l ~...,~ik); ~ e 0n 91Z } and let ~(Ai
,...,&ik ) be (an unlimited)
~{~T~[~,ai, Lemma.
.... ,aik] ; ~ e On yro }
=
0 ~ [ a i , ..... aik]
For each finite subset {i~ , .... i k} of ~ has an ~ - d e f i n a b l e
~[ai
~ .... aik]
well-ordering.
For the proof use the techniques Lemma.
constant term of £ denoting
presented on pages 97-98~
Let ~(x) be an k-formula whose only free variable that oec(~(x))
= {i e m; ai occurs
138 and 15B
is x such
in ~(x)} _C k e m. Then for
every condition p: p IF
Vx ¢ ( X )
~ (
¢(x)
^ x s ~(~,&o~1,...
The idea for the proof is the following: £~ say E6x~(x)~ with 8 < ~
if u is a constant term of
such that ¢(u) and u mentions
generic reals ~j at most for j E {0,1~...~m}~ a term u* = ESx~*(x)
achieved by replacing ak,ak+l,...,am of a k. More precisely one defines
r(E xai+k(X))
if
(in~)
0 ~
(names for)
: k + 1. This can be
in u by pairwise disjoint
from constant terms to constant terms : Emx~l.(x)
(names for)
then transform u into
such that ~(u*) and u* mention
generic reals ~j at most for j E {0~l~...~k}
. r(E m xai(x))
).
= E~X~k((m-k+l).x+!)
the following function r
(assume k ~ m):
i <
subsets
k,
if 0 < i ~ m-k,
165
r(E xAi+m+l(X))
= Emx~i+k+l(X)
if 0 ~ i.
Extend r to act on all constant terms of 2 in the following way. If EYxF(x)
is any constant
of ~j(x) by x e E zaj(z) and call the resulting Then replace every occurrence
by r(E zaj(z)) and call the resulting define r(EYxF(x)) be EBxr(~(x)), u* mentions
first in F(x)
.
every occurence formula F'(x).
term of 2, then replace
to be EYxr(F(x)).
id est r(u).
.
of E zaj(z)
formula r(F(x)).
in F'(x)
Finally
With these definitions
It follows
let u*
from the construction~
that
(names of) generic reals ~j at most for j 6 k + 1.
Hence val(u*) 6 ~ [ a o , a 1 , . . . ~ a k ] .
A symmetry argument
shows that
~(u*) holds. Lemma (R.Solovay): Feferman's
The axiom of dependent
model ~
Outline of proof.
~ ~[ao,a,
choices
(DC m) holds
in
,...,an~...] .
Let ~(x,y) be an C-formula whose only free variables
are x and y~ such that if ~. occurs in ~, then j < m. Let us assume 3 for simplicity that m = I. Suppose Ee(x~y)~(x~y) defines in ~ a binary relation R on a set s such that for all x E s there exists y e x with (x~y) 6 R i n ~
. We intend to find in ~
from ~ into s such that for all n 6 m, (f(n),f(n+l))
a function f 6 R and
f e ~(a0,al). By the previous
lemma it holds
in ~
that the following
two
formulae are e q u i v a l e n ~
(1)
A~Vytx ~
~(~o,~,,
(2)
~x
~(~o ,~ ,...,~n~ ~ (~(x,y) ^ y E ~(~0 ,i, ,... ,~n+~))].
~j[ x ~
Consider the following rn+!(E~x~o(x))
....
~n ~ ~ ~ ( x , y ) l ,
transformation
rn+l:
= E~x~0(x),
rn+l(E xai+l(X))
= E~x~l(2 i+1 "x + 2 i - 1)
rn+!(E~XAn+2(x))
= E~xg~ (2n+l.x + 2n+l - I),
rn+!(E~Xan+j(x))
: E~X~l+j(x)
This transformation of constant proof.
terms
for 0 < i < n,
for 3 < j 6 ~.
r can be extended to act on the whole ~ - e l a s s
in the same way as it was done in the preceeding
A symmetry argument yields that A x ~y[X £ rn+l(~(~0, .... An)) ~ ¢(x~y)] , and
(4)
A x Vy[X e rn+l(~(~0, .... ~n )) ~ ~(x,y)
^ y ¢ rn+l(~(Ao .... An+l))]
166
are equivalent
(namely,
apply rn+ 1 to (1) o (2)). Notice that
rn+l.(~(~0,...,~n)) and rn+l(~(Ao .... ,~n+l )) mention (names of) generic reals =j only for j = 0 or j : i. Hence val(rn+l(%(~o,..~ak)) ~|a0 ~al] for k e {n,n+l}. But ~ [ a 0 , a , ] has an ~ - d e f i n a b l e wellordering.
Hence there is in
~
a function
C
f : ~ ~ s such that
f(0) • ~ [ a 0 , a l ] , and A
f(n + 1) • va!(rn+l(%(~0 .... ,~n+l))) C ~ [ a 0 , a l ] , (f(n),f(n + 1))6 R holds in ~ . We can insist that
' "nT~
]
f 6 ~[a0,a,, rings needed members Theorem
, since the constant-terms in the definition
that denote the we!l-orde-
of the term tf (for f = val(tf))
of C(~e,~,).
This proves the lemma.
(R.Solovay):
If ZF is consistent,
then so is ZF +
are
A ~ ( A C ~) +
+ (DC ~) + ~ (DC~'). The model used by Solovay is the Cohen-generic ~[a0
..... ay~...]y<~17tL
ZF + V = L which results ~)
of a countable
from ~ T ~ b y
generic reals ay (y < m,7~) to ~ Another result
[~8] T.JECH:
standard model
~o~
adding ~, many (in the sense of but no set collecting
these reals.
in this area is due to Tom~s Jech from Prague:
Interdependence
of weakened
forms of the axiom of choice;
Comment.Math.Univ. Carolinae, 371, Corrections, Theorem
extension
(J.Jech [38]):
Let ~
voi.8(1967)
Prague,
page 567.
be a countable
ZF + (AC) and ~ a regular
vol.7(1966)p.359-
standard model of
infinite cardinal
in~
. Then
there is an extension ~ of ~ with the same ordinals such that ~ is a gF-model satisfying (AC 8) and (DC 8) for every 8 < ~ but neither H.C.Doets
(AC ~) nor (DC ~) hold in ~
asked, whether there is any interdependence
and the (BPI). The answer follows chapter. satisfies
between
from results presented
In fact, Halpern and L~vy showed that ~ [ a 0 , a ,
(DC ~)
in this ~...,A]
(BPI) + ] (AC ~) and hence ~ (DC m) -- see pages I00-103 and
section G, p.131. model ~
.
On the other hand
[a O,a I ,...] .
(DC ~) + q (BPI) hold in Feferman~s
167
I) A FINAL WORD The reader will have observed that in numerous proofs of this chapter we referred mostly to the weak-forcing relation l~'and in some few places only to s~r0ng forcing I~. For this reason it is best to forget about strong forcing and introduce beginning only weak forcing relation).
(and then using l~ for this weak forcing
If one wants to do so, then one defines
countable standard m o d e l ~ universal quantifiers
(*)
in the given
a ramified language 2" having limited
A e rather than existential ones. As an
example we consider the model ~ [ a 0 , a l Then e.g.
from the
,...,AI of Halpern-L~vy.
(if i~ denotes weak forcing[)
p I~ t e i ~ ( V p '
Since complete
D p)(3q
sequences
strongly, the extensions not differ.
D p')(~i
e ~)(p I~ t ~ ~i).
force weakly £*-sentences
iff they do so
obtained by weak or strong forcing do
Though the weak-forcing
approach is much smoother
going, we presented strong forcing as the basic relation,
since from
the p~dagogieal point of view a definition like ('')
p I~ t e A ~ (~ i e m)(p l~ t = ~i )
is much easier to understand than ('). We intended to present in these lectures a selection of The most important techniques and results of set theory.
from the.field of model theory
Measured by the number of p a g e s ~ h e s e
time of one semester,
notes and the
I hope the reader will agree that the selection
is not so small. But if we measure the selection by the number of results obtained so far we are (strongly) could present only some few results.
forced to say that we
It remains to refer to the
litterature and to the following survey article: [59] A.R.D.MATHIAS:
A survey of recent results
mimeographed,
Bonn-Stanford
in Set Theory;
1968. To appear in the UCLA--
Set Theory Symposium-Proceedings.
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