Aspects of Mathematical Logic (C.I.M.E. Summer Schools, 48)

E. Casari ( E d.)

Aspects of Mathematical Logic Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Varenna (Como), Italy, September 9-17, 1968

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-11078-8 e-ISBN: 978-3-642-11080-1 DOI:10.1007/978-3-642-11080-1 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 Reprint of the 1st ed. C.I.M.E., Ed. Cremonese, Roma 1969 With kind permission of C.I.M.E.

Printed on acid-free paper

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CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E. 3'

Ciclo - Varenna dal 9 a1 17 Settembre 1968

t

l

OF MATHEMATICAL ~ ~

LOGIC^'~

~

~

Coordinatore : Prof. E. C a s a r i

H. HERMES

:

Basic notions and applications

of the

theory of decidability. D. KUREPA

Pag. 1

:

On s e v e r a l continuum hypotheses.

'I

55

A. MOSTOWSKI

:

Models of set theory

"

65

A. ROBINSON

:

Problems and methods of model theory

A. SOCHOR - B. BALCAR :

181

The general theory of semisets. Syntaktic models of the s e t theory.

"

267

C E N T R O INTERNAZIONALE MATEMATICO ESTIVO (C. I. M. E . )

H. H E R M E S

BASIC NOTIONS AND A P P L I C A T I O N S O F T H E THEORY O F DECIDABILITY

Corso tenuto a

Varenna dal 19 a1 17 Settembre 1968

BASIC NOTIONS AND APPLICATIONS O F THE THEORY O F DECIDABILITY by

H. Hermes P r e l i m i n a r y Remarks.

( F r e i b u r g r Germany)

The f i r s t t h r e e l e c t u r e s contain an exposition of the

fundamental concepts of some main t h e o r e m s of the theory of r e c u r s i v e functions. One of the m o r e difficult theorems of the theory of r e c u r s i v e functions i s FI-iedberg-Muxniks theorem which a s s e r t s the existence of non-trivial enumerable d e g r e e s . In L e c t u r e s

4 and 5 we prove this theorem, following the t r e a t -

ment given by Sacks, but s t r e s s i n g somewhat m o r e the combinatorial p a r t of the proof ( L e c t u r e 4). Lecture 6

deals with problems in the theory

of

primitive r e c u r s i v e functions. As a typical example of the application of the theory of recursitivy we give in L e c t u r e 7

in detail a proof for the unsolva-

bility of the domino problem in the simplest c a s e of the origin-restricted problem and ehow the

in Lecture

8

how the domino problem i s connected with

c a s e of the Entscheidungsproblem. Lecture

has

6

been given before L e c t u r e s

4 and 5. The inter-

change is due to systematical reasons. The interdependece of the l e c t u r e s may be indicated a s follows: 1

2

3

4

5 , 2

6 , 2

7

8.

Contents L e c t u r e 1: Computability, Enumerability, Decidability L e c t u r e 2:

- Recursiveness, Turing Machines, Degrees of Unsolvability

Lecture 3: Kleenesl Normal F o r m Theorem; the Jump Operator Lecture 4: Theorem of ~ r i e d b e r ~ - M u g n i kP,a r t I

v .

Lecture 5: Theorem of Friedberg- Mucnlk, P a r t I1 Lecture 6: Primitive Recursive Functions Lecture 7: The Domino P r o b l e m Lecture 8: AvA- Case of the Decision Problem of Predicate Calculus. Bibliography

H; Hermes L e c t u r e 1: C o m p u t a b i l i t y

Enuii~erability,becidability

1. Algorithmic procedures and calculi always have been a n essential part of mathematics. In the l a s t thirty o r forty y e a r s a theory has been developed in o r d e r t o study the fundamental notions which a r e connected with this part of mathematics. Everybody knows algorithmic procedures for computing the sum of two decimals. The existence of such procedllres shows that the sum-function is a computable function.

If a

mathematical theory T is given based

on a finite number of axioms and on the r u l e s of first-order logic calculus, we may generate

one by one the theorems of T. Hence the s e t of theorems of

T is' a generable set. Using lexicographical principles it is possible co get nd theorems in a sequence, s o that we may speak of the (Ith, lst, 2 ,

...

theorem of T. In this way we get an (effective) enumeration of T, and we call T an enumerable set. The notions of generability and enumerability may be identified.

F o r any natural number it is decidable whether it is a prime o r

not. Hence the set of p r i m e s is called a decidable set. The concepts of computability, enumerability and decidability a r e narrowly related (cf.no.4).

In order to be able to develop a mathematical

theory concerned with these notions it is necessary to replace intuitive concepts by p r e c i s e mathematically defined concepts.

F o r each of these concepts

different definitions have been proposed and proved to be equivalent t o each other. Practically everybody is convinced that the precise notions correspond llexactlyllt o the intuitive concepts. This fact, the so-called

Church's Thesis

(1936), may be compared with the statement that there exists no perpetuum motiile. In the following (cf. no. 5 , 6 , 8 and Lecture 2 ) we give several p r e c i s e concepts which lead to definitions of enumerability and computability. Referring to such definitions we have notions like Turing-computability, recursir veness,

/u,-recursiveness

etc. But since these concepts can be proved to be

extensionally equivalent, we l a t e r on may interchange them arbitrarily.

H. Hermes 2. In o r d e r t o compute (calculate) it is necessary t o manipulate objects, i.e.

to t r e a t objects by manual means. Not every s e t

property that every element of ch1 s e t of r e a l numbers).

S can be used in this way

S has the

(e.g. the classi-

A s e t of objects which can be used for computa-

tion may be called a s e t of manipulable objects. Typical example f o r manipulable objects a r e the words composed of l e t t e r s from a given finite alphabet A. If A h a s only one element, these words may be identified with the natural numbers. An infinite s e t

S

of manipulable objects is denumerable.

If

S1 and S a r e two (infinite) s e t s of manipulable objects t h e r e exists 2 a 1 1 mapping f from S onto S which is effective in both ways, 1 2 i.e. : if any x S is given it is possible t o compute f(x), and if any 1 1 f (y) Such a mapping is often y S2 is given it is possible to compute

-

-

.

S coincides with the s e t of natural num2 (in this c a s e f(x) is called the Gtidel number of x). In principle it

called a ~'ddelization,especially if bers

is irrelevant on which

(infinite) s e t of manipulable objects the theory is

based. Very often (following Gb'del) we

choose for this purpose the

set

of natural numbers. But many applications may be much e a s i e r if other s e t s a r e chosen. fixed s e t

- We

speak of an enumerable o r of a decidable s e t S

St of manipulable objects is given and if

only if a

=St.

3. F o r most questions concerning computability if is irreleveant whether we consider

1-place o r n-place functions ( o r similarly 1-place o r n-

place predicates). L e t us consider e.g. n=2 t h e r e exist computable functions CS 2, (1.1)

d 2 ( 6)21(~)s %(x))

=

x

(1.2)

d2(x9~))

=

x

=

Y

(1.3)

GZ2(g 2 ( % Y))

. It

may be easily shown that

d21s &22,

S.t.

for each natural number x for each p a i r x, y of natural numbers

Using these functions we may associate with e v e r y 2-place functionj f a 1- place function g,

defined by

H. Hermes

(1.4)

Now we get (1.5)

f(x,y) As

place

g( 6'2(x, Y))

=

.

f a r a s questions of computability

a r e concerned we may

re-

f by g. 4. The following statements hold intuitively:

(1.6)

A s e t i s enumerable i f f it i s void o r the range of a computable function.

(1.7)

A

1-place function is computable, iff

the

2-place relation

R is

enumerable, where R holds for y and x iff y=f(x). (1.8)

A set

(1.9)

A set

S

is decidable iff S and i t s complement a r e enumerable.

S is decidable iff i t s characteristic function f i s computable.

f (x) has the value

0 or

1 according a s

x ES

or

x 4 S.

5. Here and in no.

6 we give two definitions of the notion of enu-

merability. Here we a r e concerned with s e t s whose elements a r e words over a finite

alphabet. Let be given four mutually disjoint alphabets A, B, C, D. The ele-

ments of

A a r e called constants, the elements of

,*.

B variables, the elements

of C predicates. With each predicate i s associated a natural number a s i t s place number. the words over

AvB

a r e t e r m s , then

Ptl;,

D =

);

The words over

t e r m s . If P is an n-place predicate and t

. .;t n

by substituting a proper word for a variable, system Rule relation

and a formula p+F

in a formul:

system

@

Rule 2

p 1' 2'"" etc. a r e called

F to a formula G

the transition from an

t o the Formula

(Smulyan) is given by a finite s e t

derivable

p14p2w3

Rule 1 p e r m i t s the transition from a formula

atomic formula p

.,tn

is called an atomic formula. If p

a r e atomic formulas, then the words pl, pl+p 2, formulae.

nl 1 A called proper words,

F. A formal

4 of formulae. A formula is

, if it can be obtained by applications of

1 and/ o r Rule 2, starting with the elements of

R between words over a finite

@.

An n-place

alphabet A. is enumerabld (in the sen-

H. Hermes

s e of Smullyan) iff there i s a formal system

A, B, C, D, where A n-tuple w ble in

4

0

C

@,

belonging to the alphabets

A, and a n n-place predicate P, s. t. for each A the formula Pw o w n

w of words over 1'"" n iff R holds for w1,

.;w

.

...,

is deriva-

n

6. Another way to define enumerable relations is given by F i t c h f s

minimal logic. We s t a r t with

4 ( , ),*) . A

the 3-letter alphabet

word over

this alphabet is called a n expression if it coincides with i , o r if it may obtained, starting with words a and b

;W.

, by the rule which permits to go over from

to the word

sion. Take the s e t of all

(a, b).

(r (*+))

is an example for an expres-

expressions a s the underlying s e t of manipulable

objects. We choose certain expressions and call them = 12, 13,.

.. .

,/\,V,V,

11,

With these expressions a r e connected certain rules. We confi-

ne ourselves h e r e to indicate the r u l e s connected with (1.10)

,4

v,

=,

V, and

11:

F o r each expression a we may write down the expression = a a ( t h i s is an abbreviation for ((=a)a) (parentheses to the left, also in the

following). (1.11) F o r a l l expressions

a, b

(1.12) F o r all expressions

a, b

(1.13) F o r all expressions a b

we may go over from a t o we may go over from

sult of substituting c llabc

for

Va. the r e -

a in b, we may go over from d to

.

expression is called (lerivable (w

to

a is variable, and d

(These rules a r e s i m i l a r to r u l e s of logic, hence derivation

.

b to v a b .

we may go over from ab

(1.14) F o r a l l expressions a, b, c, d, where

Vab

if it can be obtained by the rules. E.g. the

(d), (Vjt),V\/) dhows, that

T

between expressions

r

s. t.

f o r each n-tuple

r a l . . . a n i s derivable iff

is

"minimal logicm.) An

(vV)

is derivable

.A

relation

(Fitch-) enumerable iff there i s an expression al,

R

...,a n holds

of expressions, for

a

.. , a n .

the expression

H. Hermes

7.

The l a s t example shows that the enumerable s e t s (of expressions)

a r e manipulable themselves, because they may be given by expressions, and each expression determines such an enumerable set. Unfortunately we do not have this pleasant fact for the computable functions. In o r d e r to show this l e t us assume that we have an enumerable set s . t . (a) each element of ctinn

and that

the elemets of

determines effectively a unary

of words computable fun-

(b) each such function may be given in this way (think of S

get a contradiction n

S

S

being descriptions of the computing processes). Then we a s follows : We get in a effective way for each

a prescription how t o compute a certain unary function

.

We introfn duce a new function f by postulating that f(n) = f (n)+l. According to n our assumptions t h e r e is a n m s.t. f=f This leads to a contradiction m' for the argument m (A diagonal argument of this kind is often used in

.

the theory of recursive functions). It is possible t o remedy this defect by enlarging the s e t of functions hitherto

considered. Until now we only have admitted total functions.

The domain of an

n-ary total function consists of all

n-typles of objects

in question. We now consider partial functions. The domain of an

n-ary

total function consists of an n-ary partial Tunction does not necessarily have all

n-tuples a s elements, it may even be void. Intuitively a partial

function is called computable, if there is a procedure which terminates for a given argument iff the function has a value for this argument which determines in that c a s e that value. With partial function we do not get the contradiction of no. is only possible to conclude that

f is not defined f o r the argument

If we admit a l s o partial functions, the statement

7 . It h.

(1.7) remains

true. (1.6) may be simplified : (1.15)

A s e t is enumerable iff it is the range (or the domain) of a computable partial. function.

H. Hermes

8. There a r e different important p r e c i s e

ty

definitions for computabili-

for partial functions. F o r Turing-computability and

Lecture 2.

/1L. -recursivity

cf.

Here we mention only the concept of Markovfs algorithm.

Let be given a finite alphabet A A Markovfs algorithm

and words

A 1., B.1 ( i = l ,

...,p) over A.

i s given by sequence

where "(. )I1 indicates that t h e r e niay be a dot behind the arrow o r not. (1.16) determines a unary partial function. f

. The domain

and range of f

tained in the s e t of all words over A. F o r any word mine

... of

uniquely a sequence

w=w(O), ~ ( l ) w , (~),

iff the sequence terminates, and in that case

W

over

words.

a r e con-

A we deter-

f is defined

f(W) is the l a s t element of

the sequence.. If w ( ~ + ' ) is defined we will have a uniquely determined number ( l < ~ , + ~ < p )which , describes in the sequence (1.16) the rule which responsible for the transition from We call a word

K

a

w ( ~ to )

w ( ~ + ' ).

is

part of L iff t h e r e a r e words K1, K2 S. t.

L=K KK Given K, there may be different decompositions of L of this 1 2' kind. If K has minimal length, the decomposition of L is uniquely deter1 mined and called the normal decomposition.We now procede to define w ( ~ + ' ) and p

-

n+l ' w(n+l) and pn+l a r e only defined if there i s

an i s. t.

A . is a part 1

.

n=O o r (n>O and the pth t e r m of (1.16) has no dot ) In n this case let be p the smallest i, 5.t. A. i s a part of w ( ~ ) Let be n+ 1 w ( ~ ) = KA K the normal decomposition of W(') relative to Ai. Now W(n+ll l i 2 = K1 BiKZ. of w ( ~ )and if

.

A unary partial function &whose domain and range i s contained in

the s e t of dl words over a finite slphabet) is called computable by a Markovls algorithm over an alphabet A, iff

A CA and if for each word 0

W

- 10 H. Hermes over A (a) if f (the function determined by this algorithm) is defined f o r W 0

if

f(W) is a word over A g is defined for

(b) if

0'

W

then g is is defined for W and g(W)=f(W), and then also f is defined f o r W and again f(W)=

=g(W). R E F E R E N C E S: ------------

Davis. [l], of Fitch). Kleene lyan

L e ct u r e

-------we--

2:

111 ,

117

[I,

Hermes [23

.

.

(also f o r the minimal logic

Markov [I],

+Recursiveness, Enumerability, ......................

Rogers

[I) , Smul-

Decidability.

1.

In no. 1 we use natural numbers a s manipulable objects. Let be 1 the 0-place function with value 0, S the 1-place successor-function and n U. the n-ary function whose value coincide with the i-th argument 1 O l n (i = 1, ,n). The functions Co, S , U. a r e called initial functions. The

...

initial functions a r e computable total functions. The process of substitution leads from function a function f = g(h (2.1)

1'""

f(xl,

h ),

r

....x,)

g, hl,

..., hr t o

where = g(hl(xl,

...,xr ), ..., hr(xl. ...,xn)) .

Substitution p r e s e r v e s totality and computability. The process of primitive recursion leads from functions g, h t o a function f, where (2.2)

f(xl..

..,xn ' 0) = g(xl.. . ., xn),

H. Hermes

Primitive recursion preserves totality and computability.

r-

The process of application of the tion

g

to

a function f,

( p y = the l e a s t s.t.

for all

z

y. p y


..., xn,y) = 0; in this

/IC

g(xl,.

where

g(xl,.

..,xn,y) = 0

..,xn,z)

case y =

operator leads from a

is defined iff

#

is defined and

y g(xl,.

t h e r e is a

0 and if g(x

..,xn,y)=O.

func-

y

la***

Application of the

operator p r e s e r v e s computability but in general not totality. Computabi-

lity would not generally be preserved if we would not postulate that g(xl,.

..,xn, z)

be defined for z < y.

~ h e p - o p e r a t o rmay a l s o be applied to a relation R. We define

py

Rxl.. .xny

by p y g(xl,.

..,xny ) = 0,

where

g

i s the characteristic

function of R. The functions which may be obtained starting with the initial functions and using substitution and primitive recursion a r e called primitive r e c u r s i v e functions. If we admit in addition the application of the r a t o r we get the

/L

ction is total and computable. The the computable

, h - r e c u r s i v e functions coincide with

(partial) functions. There a r e total

which a r e not primitive recursive. 2. Turing machines.

qO.

...,qp

A TM (Turing machine)

-recursive functions

M

is given by (a)

ao,...,a

of quadruples i s often called the table of M.) q a b ql, where

r

, (b) a finite (ordered) s e t of N and a ffnite (ordered) s e t of quadruples. (The s e t

a finite (ordered) alphabet states

pope-

-recursive functions. Every primitive recursive fun-

q, q1 a r e states,

a is a letter

either a l e t t e r o r one of the symbols

A quadruple is of the form (of the alphabet), and b

R ("right1' o r L (llleftll). (We assu-

H. H e r m e s

R ,L

m e , t h a t t h e s e t of l e t t e r s , t h e s e t of s t a t e s a n d t h e s e t

a r e mutually

d i s j o i n t . ) We r e q u i r e t h a t t h e r e i s a t m o s t one q u a d r u p l e i n t h e s e t of q u a d r u p l e s which b e g i n s with

qa,

where

q

is a fixed s t a t e a n d a is a f i x e d

letter. A t a p e is a two-way infinite s e q u e n c e of s q u a r e s . ( S o m e t i m e s a l s o one-way infinite t a p e s a r e c o n s i d e r e d . ) An i n s c r i p t i o n the

is a m a p p i n g of

s e t of s q u a r e s into t h e a l p h a b e t . We a s s u m e t h a t i n I

s q u a r e is b l a n k ( e m p t y ) , i . e . m a p p e d o n t h e

-c-o-n-f i g u r a t i o n

C = (I, s, q ) , w h e r e

C i s a triple

s c anned -

square

of

C, a n d

almost every

loid letter a

A (complete) 0' is a n i n s c r i p t i o n , s is a

I

s q u a r e a n d q a s t a t e . We w r i t e C = (I(C), s ( C ) , q ( C ) )

re -

I

.

s ( C ) is c a l l e d t h e

a ( C ) = I(C) ( s ( C ) ) t h e l e t t e r o n t h e

of C. C is c a l l e d t e r m i n a l , i f t h e r e is no q u a d r u p l e i n M

with

q(C), a(C)

figuration

C

. O t h e r w i s e we a s s o c i a t e with , c a l l e d t h e --s u c c e s s o r of C :

q u a d r u p l e of M which b e g i n s with C1 = ( I ' , s ,q l ) .

We put q 1 = q

q(C) a(C)

.

I' a n d s

C l e t be

. We

scanned squawhich b e g i n s

i n a unique way a c o n q ( C ) a ( C ) b q*

the

want t o define

depend o n b.

We d i s t i n g u i s h

two c a s e s : (1) If

exception

b e is a l e t t e r ,

that I ' ( s l ) = b.

( 2 ) If b = R(L), s t is t h e r i g h t

If w e s t a r t with a n a r b i t r a r y quence

C, C1, C v = (C1)I , C1I1,.

s t a r t with

C" m a y b e e x p r e s s e d

q ( C ) = q O . I'To p r o c e e d f r o m we h a v e c a s e

(I), a n d a s

h a l t s , s t a r t i n g with

C, a t

(left) n e i g h b o r of s C,

. ., w h i c h

m a y b e f i n i t e o r infinite. " T o

C t o C1 Itmay b e e x p r e s s e d a s p r i n t i n g , i f

going t o r i g h t

c*"

n-ary partial

(left), if we h a v e c a s e (2). I1M

means, that the sequence

function

C,

C:.

..

.

C"

3 . With e a c h T u r i n g m a c h i n e an

a n d It = I.

w e o b t a i n in a unique way a s e -

a s t 1 t o a p p l y M o n I(C) i n s ( C ) " , if

h a s a l a s t t e r m i n a l t e r m which is

associate

s t = s a n d I1 = I w i t h t h e G o s s i b l e )

we p u t s

M

and each natural numer

n

we

f b . (We a s s u m e in t h e following

H. H e r m e s

M

t h a t t h e a l p h a b e t of

h a s a t l e a s t t h e l e ~ ; e r - s a (blank) a n d a l . ) 0 M Let be x x natural n u m b e r s and a c o n f i g u r a t i o n with 1'"" n Cxl....,rn M M l(Cxl.. x ) a n inscription where the a r g u m e n t s q ( C x l . . . x n ) = q,, n

.

.

a r e r e p r e s e n t e d by s e q u e n c e s of (x + l ) , . . , ( x + l ) c o l ~ s c c u n 1 a and a r e s e p a r a t e d f r o m each other t i v e s q u a r e s which b e a r t h e l e t t e r 1 M ) i s the f i r s t s q u a r e on the tape by one blank s q u a r e , a n d s(Cxl. . x n which b e a r s t h e l e t t e r a F o r any configuration C l e t b e v (C) 1' ( t h e v a l u e of C) t h e n u m b e r s q u a r e s which b e a r t h e l e t t e r a 1 in I ( C ) x1,...,X

.

f,

Now we i n t r o d u c e a

s.t.

C*

case

.

x-

x ) = v(C )

n if t h e r e i s

computable

a s follows:

cM

M, s t a r t i n g with

.. .

f&(xl,

fi(xl,

. An

x ~ 9 * . *n1 x n - a r y function

s. t .

C

t h e q u e s t i o n w h e t h e r a p a r t i a l function s o m e o t h e r p a r t i a l function

i s called

f

of

g

a r e given.

t h e v a l u e s of

g

f f

that g

is c o m p u t a b l e ,

If

Turing

is c o m p u t a b l e , but a l s o

f

f

we c a l l

is c o m p u t a b l e

g-computable g-recursive) and w r i t e 2 g ,< g f o r e v e r y function

g

(where iff

may

there

under the assumption that the t h e v a l u e s of

is c o m p u t a b l e u n d e r f

,< g. A s

g

is

values

are

given

the assumption

c o m p u t a b l e r e l a t i v-e f

in

is c o m p u t a b l e if w e s u p p o s e

It i s not a s s u m e d t h a t

by a n y e f f e c t i v e p r o c e d u r e .

this

n f = f M'

b e c o m p u t a b l e o r not). T h i s q u e s t i o n h a s a p o s i t i v e a n s w e r a procedure t o get

*; i n

M a t h e m a t i c i a n s not only a r e i n t e r e s t e d

i n t h e q u e s t i o n w h e t h e r a p a r t i a l function

that

) i s defined iff t h e r e i s

, halts at

a Turingmachine

4. R e l a t i v e c o m p u t a b i l i t y .

.

. . ., x

to g

or

a n example we

have

f: .

A p r e c i s e definition f o r t h e r e l a t i o n We have

f

4g

fo'r e x a c t l y t h o s e f u n c t i o n s

f

f

4

g

r u n s a s follows :

which may b e obtained

H. H e r m e s

s t a r t i n g with the initial primitive r e c u r s i o n

and

functions (cf. no. 1)

and the application

of

g , using

the

-& is reflexive and t r a n s i t i v e

It is obvious that

fined by (f & g and g&f)

substitution,

p - operator. .

divides the c l a s s of all

Hence the relation de( p a r t i a l ) functions

in mu-

tually disjoint s u b c l a s s e s . T h e s e c l a s s e s a r e called d e g r e e s (of unsolvabi-

Ilty). L e t be -f the c l a s s t o which

<

g

set.

D

f

by

f ( g.

Hence the s e t

h a s a l e a s t element

ctions.

0

f belongs. It i s possible D

of d e g r e e s

which c o n s i s t s of

t o define

i s a partially o r d e r e d

p-)r e c u r s i v e fun-

all (

Not in e v e r y d e g r e e we leave a total function (Medvedev

cf. R o g e r s [I]

.

)

1955,

D e g r e e s which have total functions a s e l e m e n t s a r e cal-

l e d total d e g r e e s .

A total d e g r e e may be identified with t h e s e t of a l l to-

t a l functions belonging t o it. The s e t

T

of total d e g r e e s i s not

only

partially o r d e r e d but in addition a semi-upper-lattice:

Let

unary functions and

h(x, y) = d 2 ( f ( x ) , g(y)). Then

i s the l e a s t upper

bound of

with

f and g 1.f

5.

g

r e s p e c t to

quadruples

q a

---

C is not t e r m i n a l guration

b qt

where

(no. 2) we

total

< .

i s a unary total function the relation

fined using a n extended concept

be f, g

of T u r i n g machines.

b is a state a s s o c i a t e with

(cf. no. 2)

f < g may be de-

H e r e we admit a l s c .

. If

a configuration

C a s i t s s u c c e s s o r a confi-

C1 g begins with

depending on g. C r i s determined by the quadruple which g q(C) , a (C) (cf. no. 2). L e t be q a b q' this quaduple.

If b

a s t a t e we define

is

not

C1 = (I(C), s(C), qf ) w h e r e q9 = q n ask

otherwise.

C' = C1 (no.2). If b = q" we define g qa = q t if g( G2l ( v ( ~ ) ) ) )= G' 2Z(v(C),, and

Hence in o r d e r t o get

Cf one h a s in g e n e r a l t o g a n "oracle" (Turing) about the value of g f o r a c e r t a i n argument. With

g and

each

Turing machine

e a c h natural n u m b e r

n

(of the extendet kind), e a c h function

we a s s o c i a t e an n - a r y function f'" M, g' n The definition r u n s l i k e the definition of f M in no. 2, 3, but with

in place of

C'

. Turing computable relative t o n (off the extended kind) s.t. f=f M, g

An n-ary (partial) function i s called iff there

6.

is

It

a Turing machine

.

is often convenient to identify a predicate (set, relation) with

its characteristic function

(which i s a total function). Hence we may speak

of the degree of a relation and extend

REFERENCES : -------------

See

Lecture

<

to relations.

1.

Lecture 3: Kleene's Normal F o r m Theorem; the Jump ............................................... 1, degree is not

dl

With each total degree (the j u m p of d)

an upper neighbor

troduce Kleene's

2.

d

it

Operator,

i s possible to associate a

which has the property that

d<

of d\. To prepare the definition

normal form for

d'

of

(but d'

d' we in-

g-recursive total functions.

In the following we a s s u m e that we have mutually disjoint infi-

nite seouences

ao, a l , a 2 , .

..

Bnd

qO, q l , q 2 , .

. .,

and that

the alphabets

and s t a t e s of

any Turing machine a r e initial segments of these sequences

(which i s not

a s e r i o u s restriction). A Turing machine

M (of the extended

H. Hermes

kind, cf. Lecture,

no. 5) may be described by its Godel number

We a s s u m e in addition that a l s o finite figurations

a r e described by

sequences

Giddel numbers

Co, C1,

. . .,Cm

.. . ,Cm ).

G(CO,

G(M). of con-

The follo-

wing constructions depend on the (fixed) ~ b ' d e l i z a t i o n G. We want to introduce a unary total function

and for each n an

U

T~ (which depends on a total function g) in o r d e r to g F (cf. also Lecture 2, nos. 2, 3 , 5 ) Let be M, g

(ni-2)-place predicate

.

describe the function

the value (3.1)

v(C ) i f t h e r e is a sequence of configurations m cO, 'm s.t . g = G ( c ~ , , c,),

...,

u(g) = 0

(3.2)

...

otherwise.

n T zx . x y iff there a r e a Turing machine M and configurations g l M , Cm), C = C CO, , Cm S. t. z = G(M), y $ G(CO, 0 Xl...X ' n C. = (C.)' depending on M) and Cm is terminal. 3+1 J g

..

...

.. .

Now we repeat the definition of

fn (Lecture 2, no. 5) by writing M, 2

This i s Kleenets Normal F o r m T h e o r - It shows that each n-ary sive function may be represented

.

by U ( ) ~ ~ T ~ .xny) X ~ .with

It is e a s y (but somewhat tedious) to show that

g-recur-

suitable z.

U is a recursive function

T~ a g - r e c u r s i v e predicate (i. e. it-s characteristic function is g n g-recursive (g-computable)). F r o m this we infer that U ( p yTg zxl..

and

is an n-ary g-recursive (partial) function for every z. Hence,

the number

z = 0,1,2,

.. .

of

varying

we get e v e r y n-place g-recursive function.

In addition it can be shown that in o r d e r t o obtain r a c t e r i s t i c function

.xny)

U and the cha-

Tn starting with the initial functions Bn'd t h e fung

H. Hermes ctions g, (cfr. Lecture 2, no. 4)

it is not necessary to apply the

.

F

-Ope-

r a t o r ( U and T~ areItg-primitive r e c u r s i v e f f ) Hence (3.3) shows that g we get every g-recursive function by applying the -operator at most

P

once.

F o r l a t e r application it i s convenient to notice that M, starting with

Finally we r e m a r k G(Co,

. . ., C m )

, halts

C:

iff

that for the usual choice of

g

and of

21

we have

1

This shows that in o r d e r to check whether for

1 VYT G(M)xy g

T zxy holds o r not, the oracle g is asked' only for arguments which a r e l e s s than y

.

3.

F o r each total unary function

g

we define

1 g' = characteristic function of the unary predicate VyT xxy. g We want to show that (3.5)

(3.6)

g < gf

(3.7)

-g # -g1 (hence

(3.8)

f

Using (3: 6), (3.7), (3.8)

with (3.6) g < g1),

,< g 4 f t < g ' . we may extend the operator '(jump) to elements

of T(tota1 degrees) by defining ():I

4. We obtain

>

=

(a. It follows that

(3.7) by proving (Church) :

d < df

.

H. H e r m e s

Otherwise let be g t

5

We introduce a total function h by

g.

{

h(x) =

(3.10)

U ( p T g x x y ) + l , if t h i s i s defined f o r x , 0

Under o u r a s s u m p t i o n

gt


otherwise. we find

.

h

i s g-computable Henee a c c o r d i n g 1 g f o r e v e r y x. We t o no. 2 t h e r e is a n u m b e r s. t . h(x) = u ( r y ~ zxy) 1 h ( z ) = U(/U y T zpy), which c o n t r a d i c t s ( 3 . 10). get a s a special case g 5. In o r d e r t o obtain (3.6) and ( 3 . 8 ) we p r o v e t h e following T h e o r e m (Kleene). L e t be g

a u n a r y and t o t a l function, R a 2-place r e l a t i o n a n d

R < g. Then t h e r e i s a computable total u n a r y function (3.11)

1

VYTg

V y ~ x iff y

Proof:

C, M(x)

s t a r t i n g with

L

n a t u r a l n u m b e r ) we get

s.t.

.

r(x) r(x) Y

x i t i s p o s s i b l e t o c o n s t r u c t effectively a n e x -

F o r each number

tended T M M(x) s . t .

r

a sequence

co

=

(where M(x)

ct

t

is an

c,

,

arbitrary

=(Co);:,

C2=

=(C )I of configurations s. t. t h e following s t a t e m e n t s hold : T h e r e 1g"" CM(x) i s a k s . t . I(C ) is void. F o r a n u m b e r k > k we have k 0 x, 0'

-

Using t h e a s s u m p t i o n not, t h e r e will be a whether halt.

Rxl

R k

M now

> k o s.t.

o r not, e t c .

But if t h e r e a y

"checkstt whether Ckl

be

s. t.

r ( x ) = G(Nl(x))

-

M(x) - cx, 1

.

Now

If t h e r e e x i s t s no y s. t. Rxy, M(x) will halt.

M(x) , s t a r t i n g with

(3.12) Now l e t

1

< g,

0

.

RxO

o r not. If

M ttcheckstf

R x y , M(x) d o e s not

Hence we have

c ~ ( ,~ h a) l t s iff t

Vy Rxy.

r i s a computable total function. F r o m (3.4)

we infer: (3.13)

~ ( x ) s, t a r t i n g with c ~ ' ~ ) h, a l t s iff t

VY T

1

G(M(x))) t y

.

H. Hermes Comparing (3.12) and (3.13) we get (introducing r(x)) (3. 14)

1 V y T r(x)ty , g

V y 'Rxy ; iff

which gives (3. 11) for

.

t = r(x)

6. We now apply Kleenels Theorem in o r d e r t o prove (3.6) and (3.8). Proof of (3.6):

We introduce R by postulating RxY

iff

. Hence

It is obvious that

R< g

ble total function

r

(3.15)

vy R x y

according to Kleene we have a computa-

s. t.

The left side i s equivalent t o obtain

g ( G z l ( x ) ) = Gz2(x) A Y = Y.

g < g l ( r ) , and trivially

V ~ T ' r(x)

iff

g

g(

r(x) y

cg 2 1(x)) = 6/ 22 (x) .

gl(r)

which shows that

7.

f1 = g f ( r ) <

The upper-semi-lattice

g1

Hence from (3.15) we

6 g1 .

Proof of (3.8) : L e t be f 6 g. We define Rxy by 1 Tf \< f < g. Using Kleenels theorem we have

. ~ ~y: x x y

.

1

Tf xxy

1 iff V ~ Tr ( x ) r ( x ) y g

.

R \< g, since

.

. T ' (lecture

2, no. 4) with the additional

jump-operator i s a very cornplrex s t r u c t u r e which h a s been intensively studied.

I want to mention only two results:

(1) Every countable partially ordered s e t i s imbeddable in

T.

H. Hermes

(2)

The complete degrees (e.g. de with the degrees

> -

the degrees of the form dl) coinci-

0'.

8. Of special interest a r e the degrees of enumerable sets. These de-

g r e e s a r e called enumerable degrees. About the enumerable degrees we --have the following elementary facts: (a) 0 i s an enumerable degree, since every decidable s e t belongs to 0 and every decidable s e t (b)

i s enumerable.

Going back to the intruitive notion of enumerability it is easy

t o s e e that every enumerable s e t may be expressed in the form Vy Rxy, where

R

is decidable. Conversely each s e t of this form with decidable

1 is enumerable. If is a computable total function (e.g. f = S ) then 1 1 Tf xxy i s decidable. Hence YyT xxy is enumerable. This shows that Y O 1 is an enumerable degree (cf. (3.5))

R

.

(c)

As we have s e e n

expressed in the f o r m

in

(b) , each enumerable s e t

S may be

V y Rxy with decidable R. Using Kleenets Theorem

(3.11) for a computable total unary function g, we obtain the result that degree of We have shown that

S = g t ( r ) < g1 = 01.

---

-

0 and 0' a r e enumerable degrees and that for

every enumerable degree we have 0 < d < O t . Post (1944) has asked whether

- -

t h e r e a r e enumerable degrees other swered (positively) not before

R E F E R E N CES: ------------

then O , O 1 . This question has been an-

196617. Cf. Lecture 4 and 5.

See Lecture

1.

- Kleene

and P o s t

[I]

, Post

ll]

H. H e r m e s

L e c t u r e 4: T h e o r e m of F r i e d b e r g - ~ u & n i k , P a r t ....................................

I

1. T h e F r i e d b e r g - M u c n i k T h e o r e m a n s w e r s question

in t h e a f f i r m a t i v e t h e

whether t h e r e a r e enumerable degrees besides

.

L e c t u r e 3, no. 8)

0 and

0 ' (cf.

We follow t h e t r e a t m e n t of S a c k s who t r i e s t o s e p n r a -

t e a combinatorial

part

of t h e proof

(which h e c a l l s " p r o r i t y method")

f r o m t h e r e s t which u s e s r e c u r s i v e concepts. T h i s l e c t u r e is devoted t o the combinatorial part.

T h e p r o o f is finished i n t h e next l e c t u r e

. For

o t h e r p r o o f s cf. t h e r e f e r e n c e s . 2. E, F, F'

T h e individuals c o n s i d e r e d h e r e a r e n a t u r a l n u m b e r s . u n a r y a n d H, D

binary predicates, and g

Let be

a total unary

ction (whose a r g u m e n t s and v a l u e s a r e n a t u r a l n u m b e r s ).

We i n t r o d u c e

t h e following a b b r e v i a t i o n s :

-

(4.1)

Lrs

for

O
(4.2)

Ps

for

Vr(gr)
(4.3)

Qs

for

V r ( g ( r ) = g(s) A L r s ) ,

(4.4)

$) s k

for

(4.5)

sk

for

(4.6)

@ (k)

for

[s:

(4 7)

p(k)

for

p:Y/sk]

T h e derivations

in

O<s A

s A Fs

TFI

Vr(r<s A

4rkA ~

4 rk)

A2: A3:

g(s)

-

.

t h i s l e c t u r e a r e b a s e d on t h e following

S

A

FsC)~ES

~ H r . s- 1 A H r s --, D r s A y E s Hss +Es

1,

= k,

H r s 1 4 Hrs),

,

Axioms: Al:

n

fun-

H. Hermes

A7:

-

Hrs

1

Hrs

4

We f i r s t prove s e v e r a l lemmata. The most important a r e Lemma Lemma 5 which relate the predicates used to derive sets

Lemmata 8

(k) and

Axioms

6

y(k)

and 9

C$

v . These lemmata

and

which show that for each

k

4 and are the

a r e finite. This immediately leads to Lemma 10.

and 7, not used hitherto (and no

in o r d e r to derive Lemma

other axiom) will be used

11. In the next lecture we apply only Lemmata

10 and 11. 3. Lemmata

Lemma 1.

Proof: Lemma 3.

1.

Axiom

ysk--) 7 E s . Axiom

4 sk-

2.

+ss.

Lemma 1, Axiom

Proof: 7-

r <s

Lemma 4: Proof:

( p s k we have

have

0 < r < s, 7F1r

@ r k gives

Lemma -

5:

4 rk

@ s k ->Vu(r
A

g ( r ) = g(s). Hence and F r (from u s.t.

yuk.

,Qs

gives

4 rk), we get

r < u < s,

4

/\ Y u k )

and

1Hru-1

i(i
-

.

1,Axiom 5). Using O r k

(Lemma 3 ) . Comparing -->Hrr

that t h e r e i s a number with

A

3.

F r o m @ s k we get -Qs(Lemma

and

w e obtain ,Hrr

5.

4s k 47Es.

Proof: Lemma 2:

1 to

.

L r s . Since we

Hrs-1. F r o m

rk

Hrs-1 we fina Hru.

This together

H. H e r m e s P r o o f : U s i n g t h e definition of qrk,

r s. t .

we h a v e a n u m b e r

'?+/$K

r<s,

H r s - 1 , H r s . L e t b e i = g ( s ) . R e m e m b e r t h a t k = g ( r ) ( f r o m @-k).

7

T h e n we h a v e g ( r ) = g ( s ) a n d

If we a s s u m e k~ we get a c o n t r a d i c t i o n :

L r s ( u s i n g 4 r k a n d 7 H r s - 1 ) . H e n c e we get Q s a n d , with Axiom 5, E s . But w e a l s o h a v e i E s by L e m m a 2. If we a s s u m e k
--

-

< g(s),

L r s ( a s a b o v e ) a n d D r s ( u s i n g A x i o m 2). Hence w e g e t Ps and (with A x i o m 4) E s . But h a v e i E s

b y L e m m a 2.

1

Hence w e h a v e i

< k. F r o m y s k we g e t 7 E s ( L e m m a 2), t h e n -,Ft s a n d Fs (Axiom 1). S i n c e 0 < s a n d g ( s ) = i we h a v e $ s i 7 4. L e m m a t a

6 t o 9.

L e m m a 6. Proof:

Lemma card

L e m m a 7.

r < s,

5.

($ (k)

Proof;

According to L e m m a

($r k

and @sk,

is infinite t h e n @(k) =

1

also

So,...,S

'j

s

with

0

< s <.

y(0)< 2'

1

n

c a r d y ( k ) < 2k by induction:

. (b)

Let

(a)

Lemma

s. t . p u k .

If $ ( k )

. .< S

n

t

.

:-

< < -

we h a v e n u m b e r s

Yulk

,..., y'unk.

(0) is void (cf. L e m m a 5 )

iCk ick

2 ick

be

i

. Hence

( c a r d 2fl/(i)

card

< k. T h e n we h a v e

c a r d @(d)(~emma 6)

2i

Hen-

.

8 be true for all

card fik) <

@ (k)

H e n c e if

is f i n i t e l e t

. Then

s

.

t w o n u m b e r s r, s with

between

v ( k ) is infinite.

a. Proof

4

card y ( k ) + l

t h e r e is a n u m b e r u

s.t. s c u <S n + l = c a r d $(k). ce card Lemma

5

+

1) (Lemma 7)

(induction h y p o t h e s i s )

H. H e r m e s

zk .

<

c a r d @ ( k ) & Zk

L e m m a 9. Proof:

L e m m a t a 7 , 8.

A r /I

i (u
L e m m a 10:

Ak v u

o f :

T h e g i s t of L e m m a t a 8.9

finite f o r e a c h

is t h e fact that $(k)

k. F r o m t h i s we i n f e r i m m e d i a t e l y

L e m m a 11:

and y ( k ) a r e

L e m m a 10.

L e t be

(1)

g(s) = k ,

(2)

-F'

s, 7 H s s - l ,

(3)

u < s,

(4)

/\ r /\ i(u < r A i < k 4 l @ r i A

(5)

Ar(r

,<

7y

ri),

.

u+qDrs)

Then we have

V r ( q r k 4 ,dm Proof: F r o m a t t h e definition

(4) we get f o r

-I

Hrm)

r = s and i = k t h a t - j @ s k . Looking

(4.4) we find that TFS. Hence

Ps V Q s

V

H s s - 1 (by

Axiom 6). H s s - 1 is excluded by (2). Ps

c a n b e excluded a s follows: A s s u m e Ps. T h e n t h e r e is a num-

b e r r , s. t . i = g(r).

r

\< u.

0 < r < s,

T h e n we h a v e Hence -,Dm

F r

4ri. Now

0 < r < s,

have

@ rk.

H i

-

1

= g ( r ) < g ( s ) = k.

Using (4) we g e t

Q s . By definition of Q w e have a n u m b e r r,

F

F

lHrs-l,

g(r) = g(s) = k

. For

this

r we

W e want t o show that ~ H r m f o r e v e r y rn.

We have

-

~ H r s 1. Hence by Axiom

If now H r m f o r s o m e m , we have a n u m b e r and H r m .

g(r) < g(s), Drs. Let b e

by (5), c o n t r a d i c t i n g D r s .

T h e r e f o r e w e have s.t.

F

7 we h a v e - t H r m f o r m < s - 1.

m - 2 ~ 9 .s.t ~ H r m - 1

T h e r e f o r e w e would have q m k (by definition (4.5)). Now (4) s h o w s

t h a t 7 H r m which c o n t r a d i c t s H r m . R e f e r e n c e s : F r i e d b e r g r17, Mucnik

rll , Lachlan Ll]

, S a c k s [I],

Shoenfield

1

H. H e r m e s

Lecture 5: T h e o r e m of F r i e d b e r g ------------ ~ u F n i k , P a r t I1 ----------------------1. We want t o s n o w t h a t t h e r e a r e e n u m e r a b l e d e g r e e s

do$' d l a n d dl&do. 0

<

d

Since for e v e r y enumerable d e g r e e

d

do, d l , s . t .

we h a v e

,< O f (cf. L e c t u r e 3, no. 8) n e i t h e r d n o r d

can be 0 o r Of. 1 In t h i s l e c t u r e e v e r y s e t ( r e l a t i o n , function) h a s n a t u r a l n u m b e r s a s 0

elements (arguments, values). Small l e t t e r s r e f e r t o natural numbers. F o r 0 1 0 e a c h s we w i l l d e f i n e s e t s A.s, A s t . t h e b i n a r y r e l a t i o n s x E . A and s S 1 a r e r e c u r s i v e ( i . e . d e c i d a b l e ) . Now we i n t r o d u c e t h e s e t s AO, A x&A1 S

by

x PA'

F r o m the equivalence b l e (cf

. Lecture

3, no. 8)

define s e t s

Ts

s.t.

(5.2)

a function

S

x eTs

n 2. T h e s e t s

. The A'

T h e d e f i n i t i o n s of

iff v s x e ~ : w e i n f e r s a m e holds f o r

a n d A'

A0 is e n u m e r a -

.

w i l l b e i n t e r r e l a t e d . We l a t e r o n

is a b i n a r y r e c u r s i v e r e l a t i o n a n d i n t r o d u c e

1 n € A n iff 2 n t l e T s-1'

s iff 2n E T s- I '

CAO

Ts

A

that 1

will b e d e f i n e d t o g e t h e r with s e t s

FS, HS a n d

g ( s ) by s i m u l t a n e o u s r e c u r s i o n . We i n t r o d u c e t h e following a b b r e -

viations:

FIS

for

c

F'

Hrs

for

B' n

Es

for

T

S

F f o r F' c T

T ~ -,

=

s

T~ j

O.

Ts-l

-

D r s f o r H~

S

n F'

,

j 0,

H.. H e r m s s

T h e d e f i n i t i o n s of

s

F' , H',

= 0. F o r t h e c a s e

n i t i o n s (5.9), (5. 1 2 ) f o r (5. 14)

T

S

a n d g ( s ) a r e g i v e n by (5.6) f o r t h e c a s e

s > O we d i s t i n g u i s h C a s e 1 w h e r e we h a v e t h e defi-

(5. l o ) , (5.11) a n d C a s e 2 w h e r e we h a v e t h e definition

FS, H',

and g(s)

. In

both c a s e s

T

. 0

F

(5.6)

= H

0

s is g i v e n by ( 5 . 1 3 ) a n d

= T o = 0 (void s e t ) ,

g(0) = 0.

s- 1 In o r d e r t o define FS , H' and g ( s ) f o r s > O we s u p p o s e t h a t F , 1 A : a n d A' a r e g i v e n by a n d g(s-1) a r e defined. H e n c e a l s o 1 (5.2). Let b e f , f t h e c h a r a c t e r i s t i c f u n c t i o n s of A,: A:. L e t b e

-

~ ( s )= 0

(5.7)

if

s is e v e n ,

e ( s ) = 1 if s i s odd,

e ( s ) = t h e n u m b e r of p r i m e f a c t o r s

(5.8)

3 occuring in the pri-

m e n u m b e r r e p r e s e n t a t i o n of s. Consider

t h e following condition ( w h e r e p is t h e

kth p r i m e n u m b e r ) :

We h a v e

C a s e 1 if (*)

C a s e 2.

be

r(s)

the greatest

is s a t i s f i e d , o t h e r w i s e

m

f o r which

V y (...)

,

In C a s e 1 l e t

T h e n we define :

H. H e r m e s

In C a s e 2 w e put

s S F = H =0,

(5.12)

g(s)=O.

In both c a s e s l e t b e (5.13)

T s = Ts- l , of Ps V Q s V H s s - 1 ,

(5.14)

Ts = Ts,l

Here

P

and Q

a r e introduced

F~ o t h e r w i s e .

U

b y (4. I), ( 4 . 2 ) , ( 4 . 3 ) , w h e r e F', F, H, D

a n d E a r e defined i n (5. 3), ( 5 . 4 ) , ( 5 . 5 ) . Using (5.13), (5.14) a n d (5.2) we g e t i m m e t i a t e l y

In t h e definition quantifiers i n (*))

.

( e . g. V n ( n

of

5

s

F',

HS, T

s

a n d g ( s ) w e h a v e only r e s t r i c t e d

~. .) i n (5.10) and V m V y ( m 5 s A y

In t h e t h e o r y of r e c u r s i v e functions i t is p r o v e d

5 s ...

)

that this fact

g u a r a n t e e s t h a t e v e r y t h i n g is r e c u r s i v e .

By i n s p e c t i n g (5.9), ( 5 . 1 0 ) a n d (5.12), (5. 13), (5.14) a n d (5. 2) we s e e that (5. 16)

S

S

0

F h a s a t m o s t o n e e l e m e n t . H , T s, A S a r e finite. 3. Connection with L e c t u r e 4. F i r s t we find t h a t

S

T h i s is t r i v i a l e x c e p t f o r C a s e 1. H e r e a c o m m o n e l e m e n t of F a n d m u s t b e both of t h e f o r m

HS

2 p + l - & ( s ) and 2n+ & ( s ) which i s i m p o s s i b l e .

-

H. H e r m e s

Using

(5.4)

H'

a n d t h e d e f i n ~ t i o nof

it is e a s y t o v e r i f y t h a t

-

(5.18)

~ H S S 1.

Now w e m a y c h e c k ( u s i n g 5. 17) t h a t H e n c e a l s o l e m m a t a 10 a n d

Axioms

1 to

7 ( L e c t u r e 4) a r e t r u e .

1 1 of L e c t u r e 4 hold.

~'1

.

4. We want t o s h o w t h a t A' (In a s i m i l a r way it c a n b e shown 1 1 ' t h a t A' A ) -We a s s u m e t h a t A < A a n d look f o r a c o n t r a d i c t i o n . 1 1 L e t b e fO, f t h e c h a r a c t e r i s t i c f u n c t i o n s of A' r e s p . A According to

j .

.

K l e e n e ( L e c t u r e 3 ) t h e r e is ( u n d e r o u r a s s u m p t i o n ) a n u m b e r e , s . t . 1

(5: 19)

r

f ( x ) = U(

yT

1

exy)

P

.

We now f o r m u l a t e a condition (5.20). In no. 5 we s h o w t h a t we g e t a c o n t r a d i c t i o n a s s u m i n g t h i s condition, a n d i n no. 6 the negation is a s

(5.20). This

finishes t h e proof

we show t h e s a m e a s s u m i n g . 1 0 for A < A . T h e condition

follows:

(5.20)

F o r each finite s e t

5 . We n e r e a s s u m e tc L e m m a

L t h e r e is a n u m b e r s, s . t .

(5.19) a n d ( 5 . 2 0 )

10 ( L e c t u r e 4) t h e r e is a n u m b e r

. Let

b e k = 2e+l. According

u s.t.

L e t be

L is f i n i t e

. By

(5.20) we h a v e a n u m b e r

s

s. t .

(*+j

.

We now c h e c k t h e

- 29

H. H e r m e s

a s s u m p t i o n s of L e m m a 1 1 ( L e c t u r e 4). ( 1 ) g ( s ) = 2 e + l = k. ( 2 ) Hss

-1 /

H e n c e F' re

u

by (5. 18). (3) S i n c e g ( s ) = 0 we h a v e s S

If s < u then

0.

L fI F 3 F

< s. (4) is i d e n t i c a l w i t h

D r s , i. e .

H

r

fI F

s

(++). H e n c e 7 D r s . q r k and that

#

0, we would h a v e

- NOW L e m m a

s)

> 0 and --C a s e 1.

= 0 corltradicting

(5.21). ( 5 )

Let

L fI F

S

be

#

w). T h e r e f o -

r 5 u. 1:' we a s s u m e

0 in c o n t r a d i c t i o n t o

11 shows that t h e r e i s a number

r s. t.

~ H r m for each m.

F o r e a s i e r reference we rename former

S

F's by (4X),

r by

s (and f o r g e t about

the

.

H e n c e w e have : (5. 23)

a n d ~ H s mfor each

@sk

B y (5.23) we h a v e c e by (5. 11)

g(s) = k

g(s) = 2e(s)

= 2e+l

+ &

rn.

{ 0. We t h e r e f o r e h a v e

(s)+l. That gives

E ( s ) = 0,

C a s e 1. Hene ( s ) = e and

by ( 5 . 9 )

where ber y

r ( s ) i s determin2d a s

in

( 5 . 9 ) . A c c o r d i n g t o (*)

we h a v e a num-

s. t.

(5.25)

r(s)

5

1 s , T f0

s , y L

e per(S)y, U(y) = 1

S

H e r e we m a y r e p l a c e -------

f0

s

by -

that in o r d e r to check wh?ther

fO: In L e c t u r e 3 , no. 2 we h a v e m e n t i o n e d

T

1

zxy holds o r n ~ t ,we h a v e l o a s k t h e g o r a c l e f o r g only f o r a r g u m e n t s which a r e l e s s t h a n y . In o u r c a s e we

y < s . H e n c e i t i s sufficient t o show t h a t m e values for each x <s. ------have

(a)

0

If f ( x ) = 0 t h e n we h a v e x

e AUs'

f0

S

and

-

f0

x e A O , cO(x) = 0

h a-v e -

.

the sa-

H. H e r m e s (b)

If

fO(x) = 0

n, 2x B T n wise

let

by be

we would h a v e

0

then

xe A , x

A0 n ( 5 . 2 ) . We want t o show t h a t

0

fs(x) = 1. H e n c e

H' nTn-

4 0,

X ~ A ; . Then

i. e . Hsn-

for a certain number 0

fs(x) 2x

H

= 0, if x S

-< s.

Other-

b y ( 5 . 1 0 ) . But t h e n

c o n t r a d i c t i n g (5. 23).

-

Hence we

have: (5.26)

r(s)

5

s.

Now i n (a) w e d e r i v e fl(p:(s))

y

5 s,

T

1

f0

e p:")

y, ~ ( y =) 1

= 0, a n d i n (b) fl(pr(s))

( a ) We h a v e Fs ( f r o m $sk.

(5.23)), i. r . F'

.

= 1.

C Ts,

hence

2 ~ ' ( ~ ) +1C T s by (5.24). pe e '(') A's+l (5.21, p:(S) A', (b) F r o m ( 5 . 2 6 ) we get u(J.LyT1 e p;(s) y ) = 1 , h e n c e f0 f ( p S ) = 1 b y (5. 19)

fl(~~~(~))=0.

.

6. We no!-assume

L

a finite s e t (*it+)

Let be

m

( 5 . 1 9 andetle_-n_etaJ&n_ of ( 5 . 2 0 ) . s. t . s(s

> o ~ g ( s )= 2 e + l 4 ~ 1 s+ L ~ F ' # 0)

.

m 2p e + l

is g r e a t e r t h e n e v e r y e l e m e n t of L. m ( c ) t h a t t h e a s s u m p t i o n p e~ A' l e a d s t o a c o n t r a d i c t i o n a n d

a number s . t .

We d e r i v e i n

In t h i s c a s e we h a v e

m pe E A 1 a l s o l e a d s t o a c o n t r a d i c t i o n . m ( c ) We a s s u m e t h a t p e e n 1 . H e n c e t h e r e i s a n u m b e r s 1 s . t .

i n (d) t h a t t h e a s s u m p t i o n

H. Hermes

p

m

1

.

6 As

s > 0, s . t .

number

Since T = 0. t h e r e is a 0 1 S m ~ E T ~ - T ~ - THence ~ - ~ . F ={zpe + l ? This

F r o m this we get

1

2p

m

shows that we have Case Now we have

s

>

+

2pe +i,sTs

.

l(no. 2),

and that

.€

( s ) = 0, e ( s ) = 0, g(s) = 2e+l. s 0, g(s) = 2e+l, 7 F ' s (because-F C Ts,,). Applying

w)

m

2 p . +1, the only element we get L Il FS f 0. But this cannot be since e s of F , i s not an element of L according. to the choice of m. d)

We a s s u m e that

pyl.

Using (5.19) we get

e

s. t.

1

e pmy and U(y) = 1. There i s a sefO quence of configurations Co, , Cn) Going through , Cn, s.f. y = G ( Co, this sequence, the T M with Gbdel number e has only a finite number Hence we have a number y

T

.. .

of question

about the value of

a r e also given by

0

.

f

0

. For

.

sufficiently l a r g e

Hence in (5.27) we may replace f s ficiently large s. We may a s s u m e in addition that (5.28)

f

. ..

6(s)=0, e(s)=e,

Now we get from

m

5

0

-s these values 0

by fs

with suf-

s , y L s.

(5.27) :

This shows together with (5.28) that we nave Case 1 (no.2). Therefore thes + 1 = {2pr's)+1j, g(s)= r e exists a number r ( s ) 2 m, s. t . F e = 12pe(s) = 2e+l. We will show in

a moment that - I F ' S . Now applying (ff*Y) we get

2pr(s)+l, only element of F', cannot be an element of e F~ , cannot be a n element of L according to the choice of m , since

L~F'

0.

But

r ( s ) 1 m;

It remains to show that F ' s . Otherwise F

S

C

T

s- 1'

hence

H. H e r m e s

.

'('I + I , g T s - l Hence t h e r e is a n u m b e r q s. t. Z ~ ' ( ~ ) + ~ E - TT '"e e 9 q-1' T h i s Leads t o .rq= )2p:(s)+l] Helice w e h a v e C a s e 1. w h ~ r e F q =

.

-

.

( q ) + ~ &(q)] B y c o m p a r i s o n we g e t 8 (q) = 1. r ( q ) = r ( s ) w h e n c e i 2 p e (q) (i. e . - - t P q c T g(q) = 2 e + l . Now we h a v e q .> 0 , g(q) = 2 e + l:lF1q ). q- 1 Applying (*k4 we get L n F~ # 0. But 2 ~ : ( ~ ) + 1 , f h e o n l y e l e m e n t of F~ , i s not arl

e l e m e n t of

L.

R E F E R -----E N C E-S : S e e L e c t u r e 4. ------

n c-t i-on? -L -e c-t u- r-e- -\'i;--.-P-r i-m-i-t i-v-e -R-e-c-u-r s-i -v e- -F-u1. In L e c t u r e 2 ,

no. I

the c l a s s

-P

of p v i m i t i v e r e c u r s i \ : r t'crn-

c t i o n s h a s been i n t r o d u c e d a s t h e s m a l l e s t c l a s s of ( t o t a l ) functions t o which 1 belong S a n d e v e r y vn and w h i c h is c l o s e d u n d e r s u b s t i t u t i o n a n d C: p r i m i t i v e r e c u r s i o n . T h i s c l a s s i s a p r o p e r s u b c l a s s of t h e c l a s s o f a l l

.

r e c u r s i v e t o t a l f u n c t i o n s , a s h a s b e e n shown by A c k e r m a n n : t h e function d e fined by

H. Hermes

i s total and recursive, but not primitive recursive. The c l a s s of recursive functions i s not an a r b i t r a r y mathematical construction but has an llabsolutenmeaning, because it coincides with the c l a s s of computable functions. It i s not known whether the class of primitive recursive functions i s also *absolute11 in a s i m i l a r sense. The definition of P

depends

on the c l a s s of natural numbers a s the underlying do-

D of manipulable

main

ning of primitive

objects.

We may a s k ourselves: what i s the mea-

recursivity for other c l a s s e s

-P

2. We may consider

C

of manipulable objects?

a s being generated by two different kinds

of operations: (1)

independent of the pecularities of N: To these open n rations belong the substitution, and every U. (a function U . may be considered a s a

Operations

0 - a r y operation )

.

(2)

Operations depending on the pecularities of N: Here we have thf 0 1 0-ary operations C and S and the primitive recursion, which i s a bi0

nary

operation leading from functions

function

f

(cf. Lecture 2, no. 1)

g, h to a a functions g, h

to a

.

N may be considered a s arl absolute f r e e algebra of type < 0, 1 >, 1 i s the 0-ary operation and S the unary operation. This inwhere C: dicates immediately that

0

Co and S'

a r e dependent on

N; the s a m e holds

for the

primitive recursion where in the f i r s t equation occurs 1 in the second equation S

.

C:

. and

3 . How can we transfer: the notion of primitive recursivity of another

domain D

D of manipulable objects ? Let be g a

onto N.

longsn

to

~ s d e l i z a t i o nwhich maps

Then we can associate with each (total) function N a s its

g-transform-

a function

fg

which

f which Itbenbelongstt to D,

H. Hermes

by the following definition :

.

We define

P a s the c l a s s of the g-transforms of the elements of P_ If -g we choose different GBdelizations g l , g2, the c l a s s e s P_gl and P may -g2 differ from each other.

On the other hand we may use the processes (1) and imitate the p r o c e s s e s (2) (cf. no. 2) in a natural way, in order to get a class

P-D

of

functions which henceforth may be considered a s genuine primitive recursive functions. We want to s a y that we have a satisfactor cept

of primitive recursivity for

y

D if it i s possible to introduce

a natural way and if there is a Godelization g s . t .

4.

class

As a f i r s t example for a satisfactory

of words over

solution for the con-

a finite alphabet

ED in

P = P -g -D'

solution we mention the

{ ao, . . . ,a N l

(including the empty

. This < 0,1,. . .,

word E)

c l a s s may be considered a s an absolute free algebra of 1 1 type 1>, where E i s the 0-ary operation, and So ' ' . . ' s N 1 a r e unary operations, defined by S .(w) = wa. (concatenation). In o r d e r to J E, So , , S; and define priimitate (0) we now select the functions

1'

...

mitive recursion by the following N+2 equations:

A s s e r has proved that. we have a satisfactory solution by indicating an appropriate GBdelization.

H. H e r m e s 5. T h e e x p r e s s i o n s of

m a y be considered a s an have t h e

expression

defined by

Fichts

m i n i m a l l o g i c (cf. L e c t u r e 1, no. 6)

absolute f r e e algebra as

of type <0,2>, w h e r e we

t h e 0 - a r y o p e r a t i o n a n d a b i n a r y o p e r a t i o n J,

J(x, y ) = (xy). H e r e w e h a v e t h e following p r i m i t i v e r e c u r s i o n :

In a s i m i l a r way w e m a y p r o c e e d i n the c a s e of a n a r b i t r a r y a b s o l u t e f r e e Mahn h a s 6.

.

n >, w h e r e n a r e n a t u r a l n u m b e r s ( j = 1,. . ,r ) . 1'"" r j shown t h a t we have a s a t i s f a c t o r y solution in a l l t h e s e c a s e s .

a l g e b r a of type


T h e r e a r e d o m a i n s of manipulable o b j e c t s whose e l e m e n t s i n

a

n a t u r a l way m a y b e c o n s i d e r e d a s finitely g e n e r a t e d but g e n e r a l l y not in a unique way.

We c o n s i d e r two e x a m p l e s :

(a) T h e d o m a i n operation

0

of

(empty s e t )

the binary operation

U

0

and t h e u n a r y o p e r a t i o n

{

(unit s e t )

and

of non-emptv w o r d s o v e r a finite alphabet

is g e n e r a t e d by t h e 0 - a r y o p e r a t i o n s

m e n t of t h e alphabet)

5

0-ary

(union).

(b) T h e d o m a i n a ,...,aN

"totally finite s e t s n i s g e n e r a t e d b y t h e

and t h e b i n a r y operation

0

a , 0

. .. , a N

(the e l e -

(concatenation).

In t h e s e e x a m p l e s we have a dificulty with t h e p r i m i t i v e r e c u r s i o n . T h e n a t u r a l way t o d e f i n e t h i s o p e r a t i o n e . g . following t h r e e equations:

in c a s e (a) would b e b y t h e

H. H e r m e s

T h i s o p e r a t i o n p r o d u c e s a function f out of given f u n c t i o n s if --

g, h, k, k u t only

(6.9), (6.10). (6.11) a r e not c o n t r a d i c t o r y . Hence i n g e n e r a l t h e p r o c e s s

of p r i m i t i v e r e c u r s i o n is only defined i n t h e c a s e w h e r e t h e " n a t u r a l e q u a t i o n s t f d o not involve

a contradiction.

Rodding h a s s h o w n that we h a v e a s a t i s f a c t o r y s o l u t i o n in c a s e ( a ) . It is unknown to t h e a u t h o r w h e t h e r we h a v e a s a t i s f a c t o r y s o l u t i o n i n c a s e (b). 7.

Tile next p a r a g r a p h s include t h e d i s c u s s i o n of s o m e h i e r a r c h i e s .

A ------h i c . ~ . a r c h yf o r a c l a s s j the c l a s s

13.. -1

1I.

-J

(Sometimes

H is a s e q u e n c e

i s a proper subclass J

runs over

a s e g m e n t of

of t h e b e s t known h i e r a r c h i e s

;.e f u n c t i o n s o v e r t h e d o m a i n This

F. J

.. . ,

S.

t. f o r e a c h

is t h e union of a l l

l l c o m p u t a b l e l to r d i n a l s ) .

One

E f o r t h e c l a s s P_ of p r i m i t i v e r e c u r s l -n of n a t u r a l n u m b e r s is d u e t o G r z e g o r c z y k .

h i e r a r c h y according to Ritchie

we introdlice f u n c t i o n s

of

El, _HZ, gj+la n d H _Ho,

m a y b e i n t r o d u c e d a s follows:

First

by postulating:

E of t h e G r z e g o r c z y k h i e r a r c h y is t h e s m a l l e s t c l a s s -n a n d which is c l o s e d u n d e r which c o n t a i n s t h e f u n c t i o n s c:. s', u:, F n Now t h e n ' t h

class

substitution and r e s t r i c t e d primitive r e c u r s i o n .

T h e o p e r a t i o n of r e s t r i c t e d

H. H e r m e s

p r i m i t i v e r e c u r s i o n l e a d s f r o m t h r e e functions g, h, j

t o a function

f iff

t h e following t h r e e equations hold:

E c o i n c i d e s with t h e c l a s s of e l e m e n t a r y functions which h a s b e e n -3 introduced by KalmBr-Czillag a s t h e s m a l l e s t c l a s s which c o n t a i n s t h e o n functions U . s u m , difference, product, x (greatest natural Co' 1 number < x ) y ( g r e a t e s t n a t u r a l n u m b e r 5 x y if y { 0, 0 o t h e r w i s e ) ,

1 1 yI

?

se),

and is c l o s e d u n d e r

s u b s t i t u t i o n and t h e o p e r a t i o n s

and

n.

0

8. T h e

Grzgorczyk hierarchy for P -

0

may s e e m somewhat artificial

b e c a u s e i t depends o n t h e p r o c e s s of r e s t r i c t e d p r i m i t i v e r e c u r s i o n . But it is p r a c t i c a l l y i d e n t i c a l with a n o t h e r h i e r a r c h y

introduced by Heinermann and 0 1 which contains Co , S, e v e r y longing t o

R-n- I substitution. If n

A x t. U:

En

R f o r P which h a s b e e n -n is t h e s m a l l e s t c l a s s of functions,

-

e v e r y f obtainable f r o m functions be-

b y one - p r i m i t i v e r e c u r s i o n , a n d which is c l o s e d u n d e r is g r e a t enough we h a v e R

ved f o r

n > 6 by Meyer,

Ritchie

and

for

for

= This has been pro-n %+ 1. n > 5 by K. ROdding, f o r n > 4 b y D. M.

-

n -> 3 by Scwichtenberg.

9. R. W. R i t c h i e h a s p r e s e n t e d a h i e r a r c h y--F f o r the c l a s s of e l e m e n n 0 t a r y functions ( s e e no. 8) _F c a n b e t a k e n a s t h e c l a s s of a l l l i n e a r func-

.

tions

( o r e l s e a s t h e c l a s s of 611 t o t a l functions which a r e I1cornputable by n+ 1 finite a u t o m a t a n ) . A function b e l o n g s t o F iff it i s total and r e c u r s i v e n s. t t h e a m o u n t h of t a p e which is u s e d f o r the a n d if t h e r e i s a function g E F computation of f(xl,

. . . ,xn)

is bounded by

g(xl,

. . . , Xn) .

H. Hermes

Re fe r e nce s: ----------

[I]

ill.

Asser

(primitive r e c u r s i v e functions over different

Grzegorczyk [I) R. W.

Ritchie

, Heinermann

il] , [2]

Lz] , RBdding domains). Axt fi] ,

Hermes 127 , Mahn r l ] , P e t e r

[ly

, Meyer I11 , D.M.

, Schwichtenberg

111

(hierarchies)

Ritchie [l]

,

.

L e c t u r e V I I ; The Domino Problem ........................... 1. The proof of the Ifundecidability of the domino problem" i s a ty-

pical example f o r the application of the theory of r e c u r s i v e functions. The problem (cf. no. 3) h a s the advantage that it does not presuppose the knowledge of a mathematical theory. The fact that a special f o r m of the domino problem i s unsolvable has been related to the Decision P r o b l e m

(~fEntscheidungsproblem") of the f i r s t o r d e r logic calculus. In this way it has been shown that ble (Lecture 8) problem for

.

the

AvA-

c a s e of this Decision P r o b l e m i s

unsolva-

In no. 2 we show a s a preliminary s t e p that the halting

T Ms

i s unsolvable.

2. The halting problem for Turing machines. compare Lecture

2, no. 2.)

( F o r the terminology

With r e g a r d to the intended application it

is convenient to use not extended

T M s with

a one-way infinite tape going

to the right. Hence the tape has an initial square. If we have a configuration

C=(I, s , q) where s is the initial s q u a r e , and if the quadruple q s l q f is an ele ment of

M

we have to a l t e r the general prescription of Lecture 2, no. 2

by postulating that

C

i s terminal.

H. Hermes

M

We say that

Kt!

if M, applied on the void inscription in the

initial square, stops after a finite number of steps. We say that M, applied on the number z, h_aKs_ if Pvl,

starting with

C:

, stops after a finite

number of steps. We want t o show briefly that the halting problem for cidable (i.e. that the s e t of TMs where M

TMs i s unde-

halts, is not decidable).

ad absurdum and assume that the halting problem i s

proceed by reductio

decidable. Then it a l s o would be decidable whether

M, applied on G(M)

(GSdel number of M, under a given ~Edelizatinn), does not halt t h e r e is (7.1)

a TM

We.

.

Hence

Mo, s . t .

F o r every

M: M

0 '

applied on G(M), halts iff

M, applied on G(M),

does not halt. This leads immediately to a contradiction for

M = Mo

.

3. Domino problems. A domino is an oriented square plate with colors

on each edge.

A

d ~ m i n omust not be rotated o r reflected. Two

dominoes a r e of the s a m e --type

if the corresponding edges have the s a m e

colors. (We sometimes confound dominoes with their types). Let be given a finite s e t D

D i s called -good if it i s possible

of domino types.

fill the plane with dominoes who (i.e. whose types) belong to

D , , in such

a way that adjacent edges have the s a m e colors (matching condition)

Fig. 1)

.

i

Fig. 1

to (see

H. Hermes In the origin

- restricted

domino problem

domino types. The p a i r

D,D

0

is given a finite s e t

DO

of

is called good if it is possible to fill

the (first) quadrant with dominoes belonging to c o r n e r domino belongs to

DO

D in such a way that the

and the macthing condition i s satisfied.

If in this definition we replace the c o r n e r by the

(main) diagonal we get

the diagonal-restricted domino problem. In each of the domino problems and the diagonal-restricted) for

(the general, the origin-restricted

it may be questioned whether the property

0

D (resp. D, D ) of being good is decidable. In each c a s e we have a

negative answer, which was given by Wang

(1961) for the origin-restricted

case, by Kahr, Moore and Wang (1962) f o r the diagonal-restricted c a s e by B e r g e r (1966) f o r the general c a s e

. We

and

want to give the proof f o r unde-

cidability in the origin-restricted c a s e in this lecture. In Lecture 8 s o m e r e m a r k s a r e included about the diagonal-restricted case, which

is not s o

easy. Even m o r e complicated i s the general case. 4. We want t o associate with each (one-way) TM M a finite s e t DM of

domino types and a (one-element) subset D

0

M

of DM s . t .

the fol-

lowing holds : 0

(7.2) M does not halt iff DM, DM is good (for the origin-restricted c a s e ) . F r o m (7.2) we get immediately that the property of being good is undecidable. Let be given a o r d e r t o describe sidered 0.1,2

TM M (by i t s l e t t e r s , s t a t e s and its table). In

DM,

DL

we may suppose that the s t a t e s of the con-

TMs a r e given by numbers

O,1,2,.

.. and the

l e t t e r s by numbers

,... . We have the following colors: a, qa, qR, qL, qaR,

qaL,

W, I, V and H ,

H. H e r m e s

w h e r e a r u n s through the l e t t e r s ,

q through t h e s t a t e s and R ( " r i g h t n ) ,

L("leftn), W(nwestn), S(llsouthn), I(tlinitial",

V(nvoidtt) and H(llhorizontaln)

a r e s p e c i a l s y m b o l s . Using t h e t a b l e of M we introduce a function

C

=

qa = C(q, a ) (ltcolor functionl1) whose v a l u e s f o r given a r g u m e n t s q, a a r e c o l o r s

defined a s follows:

V ,

T o the s e t

0

L7

M

if

n o quadruple beginning with q a is i n M

belongs only one type, given by

Fig. -

( H e r e W, I, S. W

a r e t h e c o l o r s and

(K) ( t t K o r n e r t t )i s the n a m e of t h e do-

mino. S i m i l a r l y f o r the o t h e r t y p e s t o b e defined T o the s e t

2

subsequently).

DM Selong ( b e s i d e s (K)) t h e following types ( w h e r e again

a r u n s through t h e l e t t e r s and q

through

the s t a t e s :

H. H e r m e s

Fig. 3 5. Hence

re C .

We a s s u m e t h a t

we h a v e a n infinite s e q u e n c e of c o n f i g u r a t i o n s I

3+1 = 1, 2, 3,

C

P r o o f of (7.2), F i r s t P a r t .

= C

J

. . .)

and

C

C1, C2,.. ., whe-

Co' = (void i n s c r i p t i o n , f i r s t s q u a r e , 0). L e t b e B. ( j =

J

0

the behavior

( P ( " p r i n t ") o r R o r L ) which l e a d s f r o m C.

J- 1

. We s h o w

j with

M d o e s not h a l t .

to

that D is good by defining a filing of t h e q u a d r a n t n, D L (K) in t h e c o r n e r a n d m a t c h i n g c o n d i t i o n s s a t i s f i e d by defining h o r i z o ~ 1 -

tal s t r i p s

of d o m i n o e s

So) SC,Bl~.sC ' ' ' . 2 2

which a r e l a i d down o n e

H. H e r m e s

a b o v e t h e o t h e r a c c o r d i n g t o -Fig. 4 .

Fig. 4 -

Let be

... .

a . .; q a n a r b i t r a r y configuration (where a a is o n o 1 "' t h e i n s c r i p t i o n , n . t h e n u m b e r of t h e s c a n n e d s q u a r e a n d q t h e s t a t e ) . Let be

C = a

Now we define

( ( 7 . 6 ) is d e f i n e d only if

n

> 0.) Now a l l t h e c o n d i t i o n s a r e s a t i -

s f i e d : In t h e c o r n e r we h a v e d o m i n o (K). E v e r y

s t r i p satisfies the hori-

z o n t a l m a t c h i n g condition. T h a t a l s o t h e t h e v e r t i c a l m a t c h i n g condition i s s a t i s f i e d m u s t b e shown f o r S

.

c j + ~Bj+l tion, t h a t S the

S

and S

and for CIBl 'C.B F o r t h e induction s t e p it i s c o n v e n i e n t t o u s e t h i 0

and observa-

and S h a v e t h e s a m e u p p e r c o l o r s . We l e a v e C P ' 'CR CL details to. the r e a d e r .

H. H e r m e s

6.

Proof -

good. H e n c e to

of (7.2Q~cond

--P a r t .

we h a v e a c o r r e c t filling

0

D M' DM

We a s s u m e t h a t

F

is

of t h e q u a d r a n t . We now want

s h o w by induction o n t h e n a t u r a l n u m b e r j : (1) M,

a p p l i e d o n t h e void

at least j (2) T h e

inscription

in the initial square, p e r f o r m s

steps.

j t h ( h o r i z o n t a l ) s t r i p of

F

i s uniquely d e t e r m i n e d

.

C . = a ... a q i s t h e c o n f i g u r a t i o n which w e h a v e a t e r J o n"' t h e j - th s t e p , t h e u p p e r c o l o r s of t h e j- th s t r i p of F a r e

(3) If

We f i r s t ---------(K).

a s s u m e t h a t j = 0. (1) is t r i v i a l . (2): We h a v e t o s t a r t with

T h e r i g h t c o l o r of (K) is I. T h e only d o m i n o which h a s

c o l o r is (I). Hence t h i s way of

(K) m u s t

that the initial s t r i p

h a v e (I) a s i t s r i g h t of

F

I a s i t s left

n e i g h b o r . We find i n

i s So. We v e r i f y

( 3 ) by i n s p e c t i o n

So. We now assume -

t h a t ( I ) , (2), ( 3 L h o l d f o r j

--

. Let be

d e t e r m i n e s t h e next b e h a v i o r of M. In o r d e r t o s h o w n 1, w e h a v e t o exlude t w o c a s e s , n a m e l y t h a t

Now t h e p a i r q, a (1) f o r

j

+

(i) t h e is no

q u a d r u p l e in

M

beginning with

( i i ) n = 0 a n d t h e r e is a q u a d r u p l e in

hn

qa,,

and that

of t h e f o r m

qa Lq' n

(then C . would b e t e r m i n a l a c c o r d i n g t o no. 2). Exclusionof of t h e

jt th

J

(i) : A c c o r d i n g t o t h e induction h y p o t h e s i s (3) t h e r e i s a d o m i n o s t r i p of

t o (7. 3) w e h a v e

C

qan s t r i p with lower color

F

which

= V.

V.

In F

has

a s its upper color. According qan t h e r e m u s t be a d o m i n o i n t h e ( j + l ) s t C

But t h e r e i s not d o m i n o a t a l l with

lower

H. Hermes

color V. Exclusio*

(ii);

Assume (ii). According to the induction hypothesis (3) the

.

second domino 3f the jp!h s t r i p has C

a s i t s upper color According ¶a Hence t f e second domino of the (j+l)st

= q f a L. 0 qao L a s its lower color. Inspection of Fig. 2 and Fig. 3

to (7.3) we have C strip

has q'a

0

shows that this domino i s c e qL of F.

must be the right color of

lqa'

LJ i s

of the jth s t r i p must

be

a* a*

qL

as

of the f i r s t domino in

. Hence

. But

its left color. Henthe (j+l) th s t r i p

>a* L; (where a* i s open). The

We find that this domino must be

lower color is

(qaoL). (qaoL) has

the upper color of the f i r s t domino

by induction hypothesis (3) this color

W. This finishes the proof of (1) for

j+l s t e p s

.

performs a t

least

In o r d e r to show (2) and (3) for h+l we have t o distinguish three

c a s e s whether the quadruple of M printing

j+1. Hence M

which

o r a going t o the right o r a

begins with qa describes a

going t o the left. In each c a s e we

find , starting with the f i r s t domino of the

(j+l; s t s t r i p and going to the

F

i s uniquely determined by the

right, that each domina of this s t r i p of matching conditions. In this way we show to

(2) and (3). The details a r e left

the reader.

R e f e r e n c e--s: --------

Wang

ilf .

111 Buchi

, Kahr-Moore-Wang 117 , Rerger

H. Hermes

Lecture

AVA - C a s e o f t h e D e c i s i o n P r o b l e m of P r e d i -

8:

c a t e -- C a l c u l u s . 1. A formula

y

of predicate calculus i s called satisfiable if it has

a model. It has been proved by Church (1936) that there i s no algorithm to check the

satisfiability for a r b i t r a r y formulae (undecidability of the Deci-

sion Problem).

One may be interested in the s a m e question, where

tJ)

i s restricted to a subclass A of the c l a s s of all formulae. Among these subclasses those a r e of special interest which may be described by a prefix.

E.

2.

t b r e belong to the subclass AAv a l l I\x

ten in the form x, y , z

/\ y /\

I+ , where

z

i s quantor-free

may be writ-

and has a t most

a s f r e e variables. F o r some of these subclasses satisfiability is de-

.

cidable, e. g. for V . .VA/\V. e.g.

I.+

formulae which

for AAV.!.

.. .

F o r others satisfiability is undecidable,

F o r a long time the c a s e AV/7 was

before the y e a r 1962 that Kahr, Moore and

Wang have proved that satisfia-

bility for/\VA is undecidable. They connected this 0

D, D

ty whether

unsettled. It was not

fact with the undecidabili-

is good for the diagonal -restricted domino problem.

E a r l i e r (1961) ~ i i c h iwith a method which he related t o the origin-restricted domino problem has shown that satisfiability for the

qfi A \/A

-cas.e is

undecidable. In this lecture we want to show by reduction to the resul of Lecture 7

that satisfiability f o r the class

t e how the

I\VA P,

which

i s undecidable and to indica-

s a formula

(8. 1)

s: P s

iff

_O of manipulable objects and an undecida-

is defined for those objects. Let us assume that

to each object F o r all

VAv

-case may be settled.

2. Let be given a class ble property

J'A

is associated in an effective way,

(Qs i s satisfiable.

s. t.

H. Hermes

theorem that satisfiability i s unde-

This would be a proof of Church's

cidable. Furthermore it could be that for every s a

subclass

A

of the c l a s s of a l l formulae. Then (8.1)

the theorem that a l s o for A In the following we domino s e t s D, and P

0

D, D

choose 0

where D

O_

has exactly one element and i s included in

a s the property of beidg good for the origin-restricted c a s e , In shown that

is undecidable.

P 0

D, D a formula

YD, DO we a r e interested i n ) , s. t.:

yet the definitive formula 0

D, D is good for the origin-restricted c a s e iff

(8.2)

would lead to

a s the c l a s s of all p a i r s of (finite)

We want to associate with e v e r y not

ys belongs

satisfiability i s undecidable.

the preceding lecture we have

is

the formula

' yD, Do

(which

i s sati-

sfiable D,

DO

is defined a s the conjunction of formulae (8.3),

We a s s u m e that

D = idl,.

the dominoes a r e numerated given by

9.5:

..,dN],

cl,..

., c M

DO

= [dl]

. . .,( 8.9).

, that the r u l o r s of

and that the colors of

dn

are

Fig. 5

The formciIae (8.3),

. . . ,(8.9)

a r e built up using the binary predicate sym-

H. Hermes bols

Dl,.

bols

Z

. .,DN,

C

1

1 . .., 54 ,..., CM ,...,

4 CM , the unary predicate sym-

and a unary function symbol f. These formulae b e a r names whicl:

a r e explained in no. 3. Unicity condition f o r dominoes: .a\xA y v'!n Dn x y (V!nDn x y

is a n abbreviation f o r (D 1 xy y..

.

y DNxy)./\~(Dlxy~DZxy)

A t ( D 1 x y A D 3 ~ y ) . A . ..n - \ ( D N - l ~ y A DNxy). Hence V!nD nxy is quntor-free. vln..

may be read: there is exactly one n, s . t . .

..)

Unicity condition for colors: AX/\ y

1

~ mCm ! x

y

... ~

4 -~,4xAy~!m~~xy.

Domino color relations (cf. Fig. 5) : 1

A X A ~ ( D ~ X Y . xy + CA ~ ...

nc;

xy)

(n=l..

1 4 (Here we have N formulae. )

(8. 6)

Horizontal matching condition:

(8.7)

Vertical matching condition : 1 Ax Ay(Cm

(8.8)

XY

3 . 3 Cm xf(y))

Origin Condition A x Ay(Zx A Z y

(8.9)

Existence of Z e r o

--t D1xy)

.

. ..M

(m=l,

...,M)

H. Hermes 3. Proof

want t o show that (8.9))

D, Do is good.

of (8.2). F i r s t P a r t . We assume that

YD,Do

'

(i.e. the conjunction of the formulae (8. 3),

i s satisfiable. We get a model for

Cnoose a filling F

We

of the quadrant

.. .,

in the following way : D, DO which meets all conditinns for the

'Y

origin-restricted case. Numerate the squares according to

2.6:

Fig. -6

t

Take the s e t of natural numbers a s the individual domain w . Related to w 1 4 N), C m , . Cm (m = 1, M) we now give to the symbols f, D (n = 1 , . n a meaning by the following stipulations:

. .,

. .,

...,

f(x) = s u c c e s s o r of x.

zx -

iff

-Dnxy iff

(8.12)

cP xy

(8. 13)

-m

x i s the zero.

in

F on square (x, y) we have domino

(p=l,.

. ., 4 )

the edges s e e

.. . , (8.13)

.

iff in F on s q u a r e (x, y) on edge p we have color

Under (8. lo),

dn

all formulae (8.3),

(for the numerating oi F A . 5).

. . .,(8.9) a r e

exactly what i s indicated by their names. Hence

7'

f D, Do

valid: they express is satisfiable.

i s satisfia+D. DO D, Do is good. Let be given a (non-empty, pos-

4. Proof of18.2). Second P a r t . We a s s u m e that

ble and want to show that

and a model of YD,DO over w. r etc. the meaning associated with the predicate symbol D n'

sible finite) individual domain w Let be

g,;. . .

' '

by the model. (8.9) guarantees that there i s an element

z r 0

w s. t .

_Z z 0' Take one

H. H e r m e s

of t h o s e a n d c a l l it z

0

o b e -f ( z0 ) = z 0

. Let

' -

f

;+I '

i ( z0 )=f -(f ( z )) (i=O, 1, 2,

We now i n d i c a t e a filling F of t h e q u a d r a n t b y d o m i n o e s d

. . . ). AN

(which d e p e n d s o n t h e c h o s e n m o d e l a n d o n z ): L e t b e (i, j) a n a r i b t r a r y 0

. . ). S i n c e o u r m o d e l s a t i s f i e s (8. 3) t h e (n = 1, . . ., N) s. t . D fi(z ) f J ( z ). We now defi-n-- 0 - 0

s q u a r e of t h e q u a d r a n t (i, j=O, 1 , 2 , . r e is e x a c t l y o n e n u m b e r ne

F

n

b y t h e s t i p u l a t i o n t o put d o m i n o d

It is now e a s y t o (8.8) g u a r a n t e e s

checfi t h a t

(with t h i s n) n F m e e t s e v e r y condition:

o n s q u a r e (i, j). T h e v a l i d i t y of

. From dl w e i n f e r t h a t f o r e a c h i, j, p w e h a v e e x a c t l y o n e m

t h a t w e h a v e on (0, o ) t h e c o r n e r d o m i n o

t h e v a l i d i t y of (8.4) p i . s. t . C f (z )fJ(z ). T h e validity of ( 8 . 5 ) g u a r a n t e e s t h a t c is t h e c o l o r -m00 m of t h e p r t h e d g e of t h e d o m i n o d which i n F is o n s q u a r e (i, j).

n

F i n a l l y t h e validity of

(8. 6)

and ( 8 . 7 ) show t h a t t h e m a t c h i n g condition

is s a t i s f i e d .

5. -.. The formula we s e e

If i n ( 8 . 8 ) we i n t e r c h a n g e Y D , DO i m m e d i a t e l y t h a t t h e r e is a q u a n t o r - f r e e f o r m u l a

exactly three f r e e variables

(where

'Ix) U

x, u, y,

s. t .

is a s u b s t i t u t i o n o p e r a t o r )

TD,DO

. Now

x

and

y

with D, DO is equivalent t o

it is a well-known f a c t

("Skolem f u n c t i o n s f f )t h a t (8. 14) is s a t i s f i a b l e if a n d o n l y if

is s a t i s f i a b l e

. (8. 15)

c u l u s . We identify

(8. 16)

D, Do

(but not (8. 14) is a f o r m u l a of p u r e p r e d i c a t e c a l -

yD,

DO

with (8. 15). Now f r o m ( 8 . 2 )

i s good f o r t h e o r i g i n - r e s t r i c t e d c a s e

T h i s s h o w s t h a t s a t i s f i a b i l i t y f o r \Iv AV

/\

we get

iff s a t i s f i a b l e .

is undecidable.

H. H e r m e s

6. F i n a l r e m a r k s .

VA

i n (8. 15) (e. g.

would

l e a v e out

T h e r e i s no straightforward possibility t o omit V x Z x in A x V u 11 y

by including

3(D,Do).

If we

( 8 . 9 ) we would not b e a b l e t o c a r r y out t h e p r o o f

4, s i n c e we would not b e s u r e t h a t t h e c o r n e r - c o n d i t i o n is m e t .

i n no.

(Wecould

o m i t (8.8) and ( 8 . 9 ) a n d u s e t j r V y ~ l x y i n s t e a d . But t h e n a g a i n we would have existential quantifiers).

We c a n o v e r c o m e t h i s difficulty by r e p l a c i n g i n ( 8 . 2 ) t h e o r i g i n - r e s t r i c t e d c a s e by t h e diagonal r e s t r i c t e d c a s e s a r y condition t h a t f o r r e s t r i c t e d c a s e ) is elements, e.g.

D

0

.

In o r d e r . t o h a v e t h e n e c e s -

D, Do t h e p r o p e r t y of b e i n g good (in t h e d i a g o n a l -

undecidable, w e h a v e t o a d m i t , t h a t =

) dl,.

. ., d p ) . Then

D

0

has several

in p l a c e of ( 8 . 8 )

and (8.9)

.

w e c a n u s e t h e f o r m u l a A x ( D xxV. . VD xx) which d e s c r i b e s t h e d i a g o n a l 1 P condition. T h i s f o r m u l a c a n b e m e r g e d into A x A u A y CXDlD0 ( c f . ( 8 . 1 5 ) ) , w h i c h s h o w s t h a t s a t i s f i a b i l i t y f o r / l v / j i s undecidable. Of c o u r s e t h i s p r o o f p r e s u p p o s e s t h a t b e f o r e h a n d t h e d i a g o n a l - r e s t r i c t e d c a s e is t r e a t e d , w h i c h is, a s i n d i c a t e d i n no. restricted case. T o

conclude

I

1 , not a s e a s y a s t h e o r i g i n -

want t o m e n t i o n t h e e s s e n ! i a l i d e a s

( 2 ) which h a v e b e e n u s e d by K a h r , M o o r e and Wang (K-M-W) ( 1 ) In L e c t u r e 7, no.

(I),

in this case.

the jfth configuration C j C . b y the jlth diagonal

5 we have represented

b y t h e j t t h h o r i z o n t a l s t r i p . K-M-W

represent

( c o n s i s t i n g of t h e s q u a r e s (i, i+j], i = 0, 1, 2 , .

..) .

J

C. J i n t h e j t t h s t r i p . But t h e t o t a l i n f o r m a t i o n of t h i s i n s c r i p t i o n i s included e . g . i n t h e f i r s t 2j s q u a r e s of t h e [ape. K-M-W r e p r e s e n t only t h i s p a r t of t h e (2) In L e c t u r e

7

we h a v e r e p r e s e n t e d t h e t-o t a l i n s c r i p t i o n of

i n s c r i p t i o n in t h e jxth diagonal, References: See L e c t u r e

7.

but t h e y r e p e a t t h i s p a r t p e r i o d i c a l l y .

H. H e r m e s BIBLIOGRAPHY -------------A s s e r , G.

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Axt, P.

I t e r a t i o n of P r i m i t i v e R e c u r s i o n . Z e i t s c h r . f. m a t h . L o g i k und G r u n d l a g e n d e r Math. 11 (1965). 253-255.

B e r g e r , R.

T h e Undecidability of t h e D o m i n o P r o b l e m . M e m o i r s of t h e A m e r . Math. Soc. 66 (1966). 7 2 . p p .

Buchi, J. R.

T u r i n g - M a c h i n e s and t h e E n t s c h e i d u n g s p r o b l e m . Math. Ann. -1 4 8 (1962), 201-213.

D a v i s , M.

Computability a n d Unsolvability. McGraw-Mill, New Y o r k (1958). XXV + 210 pp.

F r i e d b e r g , R. M.

Two R e c u r s i v e l y E n u m e r a b l e S e t s of I n c o m p a r a b l e D e g r e e s of Unsolvability. P r o c . Nat. Acad. S c i . USA 4 3 (1957), 236-238.

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G r z e g o r c z y k , A.

S o m e C l a s s e s of R e c u r s i v e F u n c t i o n s , R o z p r a w y M a t e m a t y c z n e 4 (1953), 1 - 4 5

H e i n e r m a n n , W.

Untersuchungen bber die Rekursionszahlen r e k u r s i v e r Funktionen. D i s s e r t a t i o n M u n s t e r 1961. (Unpublished).

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H e r m e s , H.

1 E n u m e r a b i l i t y , Decidability, Computability. S p r i n g e r ~ e i d e l b e r g t ~ eYwo r k (1965). IX+245 pp.

H e r m e s , H.

2

T h e Concept of C o n s t r u c t i v i t y . In: C o n s t r u c t i v e A s p e c t s of t h e F u n d a m e n t a l T h e o r e m of A l g e b r a . S y m p o s i u m IBM, Z u r i c h . Wiley a n d Sons . ( T o appear).

K a h r , A. S . , M o o r e E . F. 1 a n d H. Wang

Entscheidungsproblem reduced to the V w ~ a s e . P r o c . Nat. A c a d . S c i USA 4 8 (1962), 365-377.

Kleene, S. C.

1

I n t r o d u c t i o n t o M e t a m a t h e m a t i c s . North-Holland P u b l i s h i n g C o . , A m s t e r d a m (41964). X+550 p p .

Kleene, S. C.

2

M a t h e m a t i c a l L o g i c . John Wiley a n d Sons, New Y o r k (1967). XI11398 pp.

K l e e n e , S. C. a n d E . L . P o s t

1 T h e U p p e r S e m i - L a t t i c e of D e g r e e s of R e c u r s i v e Unsolvability. A n n a l s of Math. -52 (1954), 379-407.

L a c h l a n , A. H.

T h e p r i o r i t y Method I. Z e i t s c h r f. m a t h . L o g i k und G r u n d l a g e n d e r Math. 1 3 (1967), 1 - I ? .

1

.

H. H e r m e s M a h n , F. K.

P r i m i t i v - r e k u r s i v e Funktionen auf T e r m m e n g e n T o a p p e a r i n A r c h i v f. m a t h . Logik und Grundlagenforschung 1 2 (1969)

Markov, A.A.

T h e o r y of A l g o r i t h m s . I s r a e l P r o g r a m f o r Scientific T r a n s l a t i o n s , J e r u s a l e m (1962). 444 pp.

M e y e r , A. R .

Depth of N e s t i n g and t h e G r z e g o r c z y k H i e r a r c h y . Notices of t h e A m e r Math. Soc. 12 (1965), 342.

~ u g n i k ,A. A.

Solution of P o s t ' s Reduction P r o b l e m and S o m e O t h e r P r o b l e m s of t h e ' T h e o r y of A l g o r i t h m s I. A m e r . M a t l ~ . Soc. T r a n s l a t i o n s (2) 2 (1963), 197-215.

P e t e r , R.

R e k u r s i v e Funktionen. A k a d e m i e - V e r l a g , B e r l i n (21957). 278 pp.

P e t e r , R.

Uber die Verallgemeinerung d e r Theorie d e r redurs i v e n Funktionen f u r a b s t r a k t e Mengen g e e i g n e t e r S t r u k t u r a l s Definitionsbereiche. Acta Math. Hung. 12. - (1961), 271-314.

Post, E.L.

R e c u r s i v e l y E n u m e r a b l e S e t s of P o s i t i v e I n t e g e r s and t h e i r D e c i s i o n P r o b l e m s . Bull A m e r Math. Soc. 5 0 (1944), 284-316.

.

.

Ritchie, D. M.

Complexity Classification of P r i m i t i v e R e c u r s i v e F u n c t i o n s b y t h e i r Machine P r o g r a m s . Notices of t h e A m e r . Math. Soc. 2 (1965), 343.

Ritchie, R. W.

C l a s s e s of p r e d i c t a b l y C o m p u t a b l e F u n c t i o n s . T r a n s a c t i o n s A m e r . Math. Soc. 1 2 (InrjQ), 139- 173.

Ritchie, R. W.

C l a s s e s of R e c u r s i v e F u n c t i o n s B a s e d o n A c k e r m a n n ' s Function. P a c i f i c J o u r n a l of Math. 1 5 (1965), 1027-1041.

Rodding, D.

P r i m i t i v - r e k u r s i v e Funktionen Gber e i n e m B e r e i c h e n d l i ~ h e rMengen. A r c h i v f. math. Log. und Grundl a g e n f o r s c h u n g 10 (1967), 13-29.

R o g e r s Jr. H.

T h e o r y of R e c u r s i v e F u n c t i o n s a n d Effective Computability. McGraw-Hill, New Y o r k (1967). XIV+482 pp.

Sacks, G.E.

D e g r e e s of Unsolvability. P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , N. J. (1963). IX+174 pp.

H. H e r m e s

Schwichtenberg, H.

1

Rekursionszahlen und die Grzegorczyk-Hierarchie. To a p p e a r in Archiv f. math. Logik und GrundLagenforschung 1 2 (1969).

Shoenfield, J. R.

1

Mathematical Logic. Addison-Wesley Publ. Co., Reading M a s s . (1967). VIII + 344 pp.

Smullyan, R. M.

1

Theory of F o r m a l S y s t e m s . P r i n c e t o r n University P r e s s , P r i n c e t o n (21961), XI+147 pp.

Wang, F

1

Proving T h e o r e m s by P a t t e r n Recognition 11. Bell S y s t e m Technical J o u r n a l 40 (1961), 1-42.

-

-

C E N T R O INTERNAZIONALE MATEMATICO ESTIVO (C. I. M . E . )

Djurio

Kurepa

ON SEVERAL CONTINUUM HYPOTHESES

C o r s o t e n u t o a V a r e n n a ( C o m o ) dal 9 a1 1 7 s e t i e m b r e .

ON SEVERAL CONTINUUM HYPOTHESES by Djuro Kurepa (Beograd) 1. Continuum hypothesis. The classical Cantor's continuum hypothesis states that for infinite s e t

S

the cardinality of the s e t

PS

is the immediate follower of the cardinality

of a l l kS

every

the subsets of

of

S

S. i. e.

1.1. Continuum problem. The general continuum problem consists t o evaluate the number k P s for any transfinite s e t S

.

2. Conditionned c o n t i ~ u u mproblem. The conditioned continuum a s k s 2.1 F o r any s e t S and any set property

the following:

T, let

P,S

all the subsets of S , each possessing the property 2.2.

Determine the cardinality

2.3.

F o r what properties

kP$

of

Q

be the s e t of

n;

.

TC is

f o r some cardinal numer ( s ) x. F o r each S and each property yb it makes s e n s e to put t o preceding problem. 2.4

In particular , S might be a structured se, e . g. ordered, a graph, a

topblogical space, etc. and the property connected to the structured s e t

S

7C might b e a r also a character

.

2.5. Examples. 2.5.1. ( ) 1 $ks)'

If

5%

of all

means

"to be of cardinality 211, then P S denotes the system X

the 2-point-subsets of S; the cardinality of

and the question whether

(2) S equals

- 58 -

D': Kurepa for every

is linked t o the choice axiom, the identical relation x = x2

sfinite cardinality

.

2.5.2

If

x

family

then for any topological space

P%S: = F(S) of

connexion t o kF(S) one h a s

e.g.

t h e r this number is necessarily with

.

being equivalent to the choice axiom (A. T a r s k i )

m e a n s n to be closed",

one has the

tran-

all

F - s e t s of the space S

the problem of De Groot, a s

a 2-power, i. e. of the form

S

; in

to whe-

2X ;

help of the continuum hypothesis the problem is settled by affirmati-

ve for every regular accessible aleph. 3. Cantor's continuum hypothesis. 3.1.

In

1883 Cantor stated ([I]

( I ) 2kw0 = k i l ;

so

Hilbert published of H i l b d t

I s .

far

no t r a c e of

a uproof"

3.2.A system

C

provided in 3.3.

of (1) (Luzin)

of axioms the adjunction

C

itself

of

. K. ~ E d e lel, 2) proved that in s o m e

(1) will provoque no contradiction

succeded in

which the negation

of

(1) holds and another one

1963 t o exhibit a model

(1)

Consequently, the independence of (1)

a s e t theory in which

1925

t h e r e is no contradiction.

Finally, P. Cohen l l - 3 1 s e t s in

proof was found. In

(Luzin in particular). One started to

great s u c c e s s was when

of a theory of 3.4.

Cantor's

of (1) ; but, the doubts on the c o r r e c t n e s s

'proof w e r e published

doubt on the provability

, p. 192, 244) to have a proof for

is is

in

holding. proved and

which

one has

(1) does not

hold. 3.5. Function

N(q)

.

Now, if for e v e r y ordinal

and if we do not a s s u m e

O(

we write

N(o) = 1 , what is the value

of

N( q ) ?

By

D. Kurepa

a Zermelo-Fraenkel theorem the number

.

3.6.

peut w

0

-

en

Kurepa [52]

pour

N(o)

particulier,.

premier

on

w

to

p. 657 hint

N(o)

poser

N(o)

.

N(o)

is

any

the following:

nous croyons qulon

nlimporte quel ordinal > o peut

wq

.

0

(8) we read

z k W O = kw.

to

not cofinal

non cofinal avec

= 1, wl

ou

N(o) = l e

ordinal inaccessiblen.

. This

3.6.2 T. 1202 ,

where an

formulated under be increasing

analogous statement for

of

Easton

strictly:

,

N(q)

the assumption that the function

98,p. 17 )3.6.3

assumption was proved by Easton

(not necessarily

by the statement end of

and non cofinal

analogue en posant

prendre

'

>o In

l1D1une facon

is

Therefore one might a s s u m e that in particular

ordinal number 3.6.1.

N(a)

'5

and any D<' E

increasing )'

the following statement

(cf. Mathias [I] ord-+N

.

In

(5)

is ord

particular,

(2ublished in [89] ,

is confirmed not to led to a contradiction:

. Hypothesis:

F o r every infinite cardinal

n

the

number

an

is the f i r s t inaccessible cardinal > n (I1 P e r ogni numero cardinale infin nito n, 2 6 il primo numero' inaccessibile >nn)

.

The function (3.5.1. ) was introduced in 1937 (cf. Kurepa C157 , formula (2))

.

The existence of the function (3.5.1.)

is implied by the axiom of choi-

ce; probabljr ,the converse holds too. 3.6.4.

Here is a s e r i e s of nice continuum hypotheses.

Let n

be a

natural number; '.then f o r every ordinal q

(the case n = l yields the Cantor's 3'. 6.5.

A

general continuum

hypothesis)

one has

.

,great variety of continuum hypotheses is obtained in the

D. Kurepa

following way:

F o r every ordinal number

3.6.6.

a

l e t m(q)

be a positive integer; then

Hypothesis.

F o r el erg-

diadic

gp(G{O,.i'Jone h a s

Ord -+

ordinal mapping

of the preceding continuum hypotheses in acceptable.

Each

3 . 7 . On factorials.

--

Fo" any ordinal (cardinal) number c>--'inal (cardinal) numbers m the

,ar-dinal number of

dinallty

11

all

such

n

that

let m

< n.

.. . =

2n

.

F o r natursal Inirlgers w e have

a s well a s ( 3 . 7 . 3 . ') where

such

P ( n ) denoiea the s e t of a l l uniform mappings

that

be ? h e s e t of a l l the If

the permutations of

, then f o r every transfinite cardinal n

( 3 . 7 - 1.)

In

.

n

is defined a s

any s e t

S 3f

car-

D. Kurepa

(3.7.5)

o<S
nite cardinals

n,

if

w e r e holding, t h e r e would be by

an

What would yield

(3.7.6.)

.

15.5 €j

holds f o r e v e r y o r d i n a l n u m b e r . A s t o t r a n s f i -

(3.7.3)

(3.7.6.)

,$

for

= kP(n)

(3.7.1)

.

n=kw ? It would b e 1

:

c o n t r a c t i n g t h ~:,ontinuurn

hypothesis. Ana?ogously, one p r o v e s t h a t t h e

identity

imply

(3.7.6. )

would

and

i n p a r t i c u l a r that

nal

number

4.

<

s e t (2) (12)

the f i r s t inaccessible

t h e s e t of which

all

PO

the

d2generated

subsets

of

(O,<), i. e .

t h e c o m p a r a b i l i t y r e l a t i o n is t r a n s i t i v e ; the e m p t y

(I)), e.g.

equals

linearly and

. Of by

course,

if

(1)

i s degenerated

o r d e r e d o r a n antichain , t h e n

the question concerning t h e c a r d i n a l i t y of

is exactly t h e continuum pyoblem. 4.2.

cardi-

let

is c o n s i d e r e d a s d e g e n e r a t e d ( d - s u b s e t of

is

wo,

If

) be

s u c h orles in

i

ordinalO(>

.

is a n y o r d e r e d s e t

PD ( 0 ,

c ,>i, w h e r e

for every

.

Ordered sets 4.1.

2k% = 2k

We - f o r m u l a t e d t h e following.

Q-hypothesis. E v e r y infinite

t r e e (T, ,<) s a t i s f i e s

the

set

D. Kurepa

(Kurepa 161, p;133).

Remark. Every finite t r e e and the empty t r e e satisfy the

4.2.1.

preceding inequality, As a matter chain that

C

fact, if (T,,<) is any finite t r e e , then t h e r e is a

of

in (T,

kC(resp. kA) be

chain

A* in

well

as

an

> , k c * (resp. kA*)

. Certainly,

(T, <)

. Since

6)a s

kc

antichain

for

kA >, kT;

if

kPA

< k PDT.

Therefore, the requested inequality

Q(T)

T

that

Now, Q(T) By case

such

PA C P D T , one

the

positive

integer

is satisfied a l s o a similar

kA

kA 4 2 , 3 , 4 3

argument on permuting

Number

1s the number

b.

4.3.1.

any

kC< kA,

< kPDT

then kA i. e. 2

<

is

. . ... .On

.

the other

hand,

A, C

one

proves

Q(T) for the

In

(1) ?

particular,

is the number

t r e e (T,,<) ?

Ramification hypothesis

The affirmative answer

is

[R

HI.

called ramification

hypothesis and is

i6)p.

130, proposition

. The ramification

4.3.2.

principle f o r

hypothesis

is equivalent to the reduction

t r e e s : E v e r y infinite t r e e has the s a m e cardinality a s one of it

degenerated subtrees (D. Kurepa on p. 138)

C1 (anti-

a s one verifies directly.

connected t o the general Suslin problem (v. Kurepa

1

suck

Let

b(O,<) reached in

b(T, <) reached in

p1 : r73

,
.

kA < k c 4.3.

for

(T,

holding for e v e r y kA satisfies ( k ~ )< ~ 2

kA=1, 5, 6 , .

this relation is satisfied for

in

every chain

has

( k ~ ))I kT ~

A

. It

fi] p.

130 (Prof. P2); equivalence

is interesting that one h a s the following

PPP2

D. Kurepa Theorem

4.4.

. The

kPD (T,Q = ( k ~ ) +for every infinite t r e e

(4.1) conjuction

of

hypothesis

(Kurepa [92] )

the general continuum

In particular, infinite

tree;

on

incompatible with of

reals

preceds b

in

the

and

well

5.

2kW0 =

Without

if

akwl

continuum

(R;

if

+

2

contradicting

the

some ordinal o( satisfying

every infinite t r e e

As a matter

of

for every

and

well

Q(X)

ordering of

a c t b ea s b

of

and

the relation Q(W) would

.

(T,<)

every

W one has

of cardinality

fact, the numbers

for

kwd

infinite t r e e .

N(Ql)>q+l , then for

we

have

b T, kT a r e either equal o r consecu-

(T,,< ) s u c h that

kX

>

; consequently,

kw

d-subset

PX C P D ( O , 6 ) and

therefore kP

D

(0, ,<) >kPX >,

prove the relation

(6.1. )

Notations.

kw,-

> kw

'

= kw,,

means means

O(

or

X

= kwd

, we a r e not

.

.

the imediate predecessor

the immediate follower

of

O(

of O(

.

kT =

6+1

f

kX means cardinality of

do(+

2

. If a = q , kT

and this proves the relation (6.1. ) able t o

a

then

tive (D. Kurepa [6]p. 105, theor. 1) ; in either case t h e r e ,is some

K

is

hypothesis we a r e not able to prove

+

If for

any

R ,

C H

kPD(T, <) ,< (kT) 6.

is

of

KG

Q(T)

t r e e s the equality means

W

(kr)

imply

W

6' )

ordering and

RH

f o r non

G C H ; e.g.

k PD(R,
hypothesis and of the yamification

GCH and the

the contrary, the

is equivalent to the

.

the

R

the set

identity

.

D. Kurepa

BIBLIOGRAPHY

Cohen, P.

The independence of the continuum hypothesis. Proc. Nat. Acad. Sci. Usa, 50(1963) ,f143-1148;

[I].

I1 P a r t , Ibidem 51(1964) 105-110.

[2]

H .Set theory and the continuum hypothesis,

New York-Amster-

dam , (1966) 4+154.

~ b ' d e l , ~ . [I].

The consistency of the axioms of choice and of the generalized continuum hypothesis (Proc. Nat. Ac. Sci. USA 24(1938)556-557). The consistency of the axiom of choice !and of the generalized continuum hypothesis with the axioms of s e t theory Princeton, N. J., (1940)4+69.

123.

"Uber das Unendliche, Math. Annalen, 95(1925)161- 190.

Hilbert, D. Kurepa,D.

[6] [7]

Ensembles ordonnes e t ramifies. These, P a r i s , 1935. Publ. math. Belgrad, 4(1935)1- 138.

.

L1hypothese de ramification, comptes rendus 202,1936. 185-187.

&5]. L1hypoth&sedu continu et l e s ensembles partiellement ofdonnes, Comptes rendus, P a r i s , 205 (1937)1196-1198. [51].

Sur une hypothese d e l a theorie des ensembles, Comptes rendus, P a r i s , 256(1953) 56+-565 (seance du 09.02.1953.

l52].

Sur un principe de l a theorie des espaces abstraits, Ibidem 655-657 (seance du 16.02.1953).

[89]

.

Sulltipotesi del continuo, Rendiconti del Seminario Matematico, Torino, 18, 1958-59, 11-20.

[92]

.

Sur une proposition de l a theorie d e s ensembles, C. r. Sci. P a r i s 249 (1959) 2689-99.

Mathias A.R.D. Sierpifiski,W.

A survey of recent r e s u l t s in s e t theory university, 1968).. Hypothese du continu, Warszawa-L*ow

II+76 (Stanford

1934, 6+192.

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.

A.

I. M. E. )

MOSTOWSKI

MODELS O F S E T THEORY

Corso tenuto a Varenna

dal

1 9 a1 1 7 S e t t e m b r e

1968

MODELS OF SET THEORY by A. Mostowski

(University of Warszawa)

Lecture

I. Aim of the lectures: t o outline various methods used recently

in construction of models for axioms of s e t theory.

No completeness

in pursuing this aim is attempted. In the introductory lecture I we describe three s y s t e m s of axioms for abstract s e t theory.

In all these systems t h e r e a r e

two primitive notions: "class" and

"membership".

We define s e t s

a s c l a s s e s which a r e capable of being members of other classes: x

is a set if and only if there is a c l a s s

y

such

that

x

& y.

We also define atoms a s objects which have no elements. The distinction between s e t s and c l a s s e s was noted already who distinguished between the "consistent sets" and

by Cantor

"inconsistent s e t s " i.e. terminology.

s e t s and (proper) c l a s s e s in the modern

As we shall s e e t h e r e a r e axiomatic systems in which

the existence of proper c l a s s e s is assumed and other systems in which their existerice i s excluded. Set

-

theoretic formulae:

different l e t t e r s (the variables).

Let

xl, x2,.

. be

We shall often replace

a sequence of "x" by any

other letter and omit subscripts. elkxi), xi t x . , x i = X. a r e formulae; the J J v a r i ~ b l e xi is the unique f r e e variable m the f i r s t of them and the (i)Expressions

A. Mostowski x ,x. a r e the unique free variables of the remaining two. i J (ii) If F and G a r e formulae, then s o is the expres-

variables sion

(F)

] (G);

i f it is free in

the variable xi F

(iii) If x.

1

i s free in this formula if and only

o r in G . F

i s a formula, then so i s

is free in this formula if and only if

I

(The symbol general quantifier. from as

1

(x.jF; the variable

J

if j and

F.

xi is free in

i s the Shefferls stroke and

(xi) i s the

We define the propositional connectives different

in the usual way; also the existential quantifier i s defined

- I ( x ~ ) IF.).

We denote by i s a free variable of

i

F r ( F ) the set of those

for which

xi

F.

Predicative formulae a r e those formulae in which all quantifiers a r e limited to sets or atomes. We obtain a precise definition of this class of formulae by replacing the rule where

k

(iiil) If

I?

i s not in

F r ( F ) and

(iii) by the following:

i s a formula, then s o i s k

(xj)(xk) :(xj

xk)

-+ F]

+ j.

The above expression can also be written a s (xj)(xd [(xj t xk)

I (FI F$

Definitions:

x

i s a set

(in symbols

S(x)): C ~ ( X ) & ( E Y )Y); (X€

x is a proper class (Pcl(x)):Cl(x)&7 S ( x ) ; x i s an atom; (At(x)): (y)(y

g x)

We now list axioms of the different systems which we shall consider

.

The first

9 axioms a r e common to them all:

Axiom for classes: Ext.

r Cl(x) & Cl(y)& (z) [ (z E x )

Cll.

Pcl(x)

4

(Ey)(y e x);

(z E

yg + (xZy);

A. Mostowski

C12.

(y g x) +Cl(x);

Fund.

( y c x)

-+

(Ez)

{

( z E x ) & ( t )[ ( t E z)--t(t

# x)]] .

Remarks. Ext is the familiar axiom of extensionality. C l l says that proper c l a s s e s have elements, i. e. , a r e not atoms. C12 says that whatever has elements is a class.

Fund is the axiom

of foundation and says that each c l a s s which has elements contains a t least one minimal element, i. e . ,

one whose elements a r e not elements

of the given class; of course the minimal element may be an atom. Axiom f o r sets.

fib) v ~ t ( x ) ] & [ ~ ( y ) V ~ t ( y ) ]-+ (Ez) ( ( ~ ( z )& (t) / ( t G Z )

Pair.

Lit = x, "(t =

sum. S(X) Pot.

(EU

~ ( u &l

(2)

S(x) -+ (Ep) [s(P)&(z)

Emp.

(Ex) [ ~ ( x )&(y) ( y e x u

Inf.

(EX) ( ~ ( x & )

Remarks.

Y)]!)

;

( Z E u) g ( ~ t [(z ) E t ) & ( t EX^ {) { (215p ) z ( Cl(z) & (t) [ ( t ~z ) 4 t E xd))l

(EU)(U EX)&(V)

[(vex)

+ ( ~ w ((w )

EX)

di

These axioms state the existence of the unordered

pair of any two objects (which may be s e t s o r atoms), of a union of s e t s which belong t o a given set, of the power s e t of a given set, of t h empty s e t and of at least one infinite set. will be the usual one: the union,

3 x, y )

The notation for these s e t s

for the unordered pair,

P(x) for the power s e t and

0

i r x for

for the empty set.

The

uniqueness of these s e t s follows f r o m Ext. F u r t h e r axioms will fulfill a twofold role: f i r s t they will determine the number of atoms; secondly they will express the idea that an image of a

s e t under an a r b i t r a r y mapping is again a set.

The f i r s t question does not s e e m t o be v e r y important for

A. Mostowski

the abstract set theory and it is nowadays customary t o dismiss it by simply assuming that with the exception of 0 t h e r e a r e no atoms We shall follow this custom and adopt the following axiom:

altogether.

Noatoms:

(x E y)&(xf 0)4 S(x).

The second aim is much m o r e important. It is achieved in many different ways according t o the system of axioms which one adopts. The system of Zermelo-Fraenkel.

In this system we f i r s t

of all assume that t h e r e a r e no proper classes: Nopcl:

Cl(x)+S(x).

This axiom together with Noatoms allows u s t o simplify previously given by omitting everywhere the expressions

the axioms "S(x)".

The idea that an image of a s e t is again a s e t is expressed in ZF

by the following axiom schema: SubstZF.

in which

F

(x)(E!y ) F

-+

(a)(Eb)(yl { ( y e b) 5 (Ex) [(x E a ) & ~ ] ]

is an a r b i t r a r y formula such that b is not i t s free variable.

System of ~ 8 d e l - B e r n a y s . In this system we assume that every predicative formula determtines . a , c l a s s and e x p r e s s the idea that the image of a set is again a set not by means of a schema but by a single axiom involving the notion of. a class. F i r s t we define the ordered pairs:

1 Fi,xj {] .

<xi, xj> = xi,

A c l a s s whose elements a r e ordered p a i r s is called a relation; it is called a function

(in symbols

Fn(x)) if it satisfies

(w)(u)(v){[ ( < p , v > ~XI&( < u , w > E x u We denote by (Ez)

x"a the c l a s s t

[ (z e a)&(d z,, y> 6 )])

the condition:

4

such that

(v = w)3

(y)

.

(y E t ) . ~

provided that such a c l a s s exists.

A. Mostom ski

The

axioms which we admit in the system GB a r e now a s

follows: Clex

(Ex)(Cl(x) & (u) S(u)+ k u E x ) P F]{)

GB' SubsbB.

Fn(x) & S(a) 3 S(xUa).

Remarks.

In Clex

( c l a s s existence scheme) F is a GB predicative formula in which the variable x is not free. The c l a s s

1

whose existence is stated in Clex

{u: u =

U]

i s denoted by u:F GB is the universal c l a s s consisting of a l l sets.

x''a

e. g.

is called the axiom of substitution; note that the

SubstGB class

);

exists by virtue of the class existence scheme. System of Morse. In this system we leave the axiom of

substitution unchanged and extend the c l a s s existence scheme by allowing a r b i t r a r y formulae

We denote these axioms by Clex

F.

Comparison of the s y s t e m s

ZF, GB and

M

and Subst

M'

M.

The following theorems a r e easily established: I.

ZF

Proof.

is interpretable in

We interpret c l a s s e s of

11. GB is a subtheory of 111.

GB. ZF

a s s e t s of GB.

M (obvious).

GB is an inessential extension of

ZF.

By this we

mean that each preaicative formula provable in GB is also provable in

ZF. Proof.

by GB.

If

F

is predicative and provable in

I (and i t s proof),

F

is provable in GB.

In order t o establish i t s provability

ZF, then

Let F be provable in

in Z F it will be sufficient

t o show that every model of Z F can be extended t o a model of GB in such a way that s e t s of the new model be identical with s e t s of the old model.

Let therefore

M

be a model of

ZF

and let

M' be

A. Mostowski

the family of sets here

F

i t t M: (t, a l ,

...,an)

i s a predicative formula with

satisfies

n+l

M

in

F

)

;

free variables and

al,...,a

a r e elements of M. Interpret classes a s elements n M' and the membership relation a s The resulting model satisfies

.

of

all the axioms of GB and contains a submodel isomorpkic with

it G M:

and consisting of sets

t

x

1.

All the sets of

M

M' have

this form which proves the theorem. We shall show l a t e r that

M

is an essential extension of

GB. Relativisation.

Let

A(x) be formula in which

variable; A may contain free variables other than formula (xi)

we may obtain a

F

by

F

and called the formula F

and

A

ieFr(F). relation

F r o m every

formula by replacing each quantifier The new formula i s denoted by F (A)

relativised t o

We assume that formulae

A.

do not have any variable in common. It is

( X ~ ) F ( ~ ) if

.

i s a free

new

...] .

(xi) [A(xi) -2

x

x

i

6 Fr(F)

The

1

and

symbol

-

( F ( G ) ( ~q ) ( F ( ~ )G ( ~ ) ) (, ( x ~ ) F ) ( ~ ' =

obvious that

((x~)F)*

.= denotes

F

x i A

if

here the logical equivalence

(not the connective). Functions whose values a r e classes.

We insert here a note

about functions whose arguments a r e s e t s and values a r e classes. Of course this notion is important only for systems

GB and

M.

A function all of whose values a r e non vojd classes and

whose arguments a r e s e t s can be defined a s a relation. of such a function fx =

{ y: < x, y>Ef]

f

.

for the argument

of a function, then we define

a relation

ad

x

If we want t o admit f

a s a pair

The value

is simply 0

a s a possible value

a, r where r i s

A. Mostowski a relation and a is a class which contains Dom(r) a s a subclass. The value of x

f

for the argument

is an element of We write

is

x

r

or

X

Dom(r) o r an element of "f:a

-;L~"

instead of

domain a whose values a r e subclasses of

0

according a s

a-Dom(r).

"f i s a function with

y".

The elementary operations

on functions such a s superposition o r restriction of the domain t o a subclass can easily be defined for such functions.

i

domain a whose values for by

Fii.

f+(i/u)

f..

If

f:a-rz,

in a

ijka

A function with

a r e explicity given will be denoted

and

u s z,

then we denote by

the uniquely determined function with domain

a

uli) whose

restriction to a i s f and whose value for the argument i i s u. Semantical notions defined in set theory. The basic semantical notion i s that of satisfaction: a sequence

x

satisfy a formula

F

in

x.

f

of subclasses of a class

If we want t o discuss this relation in

s e t theory, we have t o replace formulae by certain sets. The following theorems a r e provable in

IV.


1

integers

There is a smallest set

,< 2

i j

i, j and is

then s o do <3,

ZF: Frm

a r e elements of

7

such that whenever

< a, b 2)

and

( 4,

such that F r m for

a, b

<' i, a>>

,0 , i >,

arbitrary

belong t o

Frm,

for arbitrary i

in cd

(the set of integers). We write 1 write We call

for 0

i

e

rx. and

1

x?

J

Ei =

for < 3 , c a , b , > a n d Frm the s e t of

By F r m

for

< 1,

< i, j >-.> and similarly

for ( 2, c i, j%>

r(xi)al

for

<

.

We also

4,..

" ~ 8 d e lsets" of formulae.

we denote the subset of Pr ~ b l d e ls e t s of predicative formulae.

Fm consisting of

A. Mostowski V. F

There is a function

( x i ) ) = i

= Fr(a)

y

5.ex?

Fr(

,

F r defined on ) = Fr(

J F r ( b ), F r ( ' T x i ) a = F r ( a ) 1

-

F r m such that

5. ='z = l i , jjand 1 J

1 it .

We call

Fr(l>lb7) =

F r ( a ) the

set of f r e e variables of a. We call a formula

S

with

3

f r e e variables a predicative

satisfaction formula for one of our t h r e e systems

ZF, GB, M

if

the following formulae a r e provable in these systems: Fr(x)); (i) S(x, y, z ) 4 ( x E F r m )&Cl(z)&(yE z Pr (ii) S( % l ( x i P , 24 i, y>], z ) (YE z);

Gig xJ. 1 S( ' xi x?

S(

,j < i , y l > ,

=

(iii) S( ralb7, y, z ) z b s ( a , y

(iv) S(

r(xi)a7

(v) S( 'Txi)2 A formula

.

j, y">] ,z )

, { 4 i, yt.> ,

c j, Y')]

= (ytE

, z) =_ ( y t

Y");

= Y");

[ F r ( a ) , z ) V f S(b, y \ F r ( b ) ,z)]

;

, y, z ) 3 S(a, y, z ) if i f C . ~ r ( a ) ; otherwise:

(dl(u c z ) + S ( a , Y u[
y. 2) 5

with t h r e e f r e e variables will be called

S(x, y, z)

ZF, GB o r M

a satisfaction formula f o r

if the following formulae a r e

provable in the corresponding system: (it)

S(x, y, z)

(ii')

S( r ~ l ( x i j l ,y, z )

yi

S( T-,i€x?

.Y~Z)=

Y ~ GYj;

S(

Gi

,y,z)=

yi = y.,

( i t ) and i

t

(vt)

4

= X?

as

(x E F r m ) & C1 (z) & (y: F r ( x ) .+z),

(iii)

S( r(xi)a7 , y,

In case of the system

Z)

z,

J

and (iv),

z

ZF

(u)

[ (u C

a ) --t S(a, y + (ilu). zj]

t h e r e is no need t o distinguish

between the satisfaction and predicative satisfaction formulae because sequences of c l a s s e s can be identified with sequences

of sets.

.

A. Mostowski The following theorems exhibit essential differences between systems

Z F , GB and

M:

VI.

T h e r e is a satisfaction formula f o r

VII.

There is a predicative satisfaction formula for M

but - provided that VIII.

M

If

-

is consistent

GB

ZF.

no satisfaction formula f o r M.

is consistent, then t h e r e is no predicative

satisfaction formula for

GB, and a l s o no satisfaction formula for GB.

We shall sketch the proofs of these theorems. In o r d e r t o prove

VI

we define

s e t s of finite sequences: Str(z, kl ,k2, dl, d2) =

;) YC

Qu(z, k, i, d) = k if =

z

ie

dl" d2 d;

(in

Z F ) two operations on

:( Y

I d l # kl) V ( ~ I d 2 4k2)

otherwise : (u) [(u

25 y E zd

We obtain a predicative satisfaction formula for expressing in the language relations:

z

is a set,

sequence

f

such

1 p, q

or or

o r finally there is a last t e r m of

f

is

j

x;

such that for each

y g z Fr(x);

for each

i

if

fi

is

if

f

is

f.I =

rx


and

i

in Dom(f)

either

a p

such that

fi =

there is a finite sequence in

fi

k

is

f o r some f. =

Cp)f:: 1

with

Tf.J ) f

z

the

Dom(k) = Dom(f)

Dom(f):

r ~ l ( x p F , then

5P ex1q

, then r

ki ={{]

q

:

UEZ];

ki = f{ ,]: ( ~ € 2 )&

if

by

t h e r e is a finite

E o r it is = x l P P q j, k both s m a l l e r than i such that

there exist

ZF

the conjuction of the following

x g Frm,

that

it is

of Z F

yki, u>) E k]{

z)+ y

E

, then ki=({xp,u,

.)

(VEZ)

(u e v ) & ;

:uez},

A. Mostowski

if if

f

= f

fi =

then

,

r

(xp) f;

ki = S t r ( z , k., ky, F r ( f j ) ,P r ( f 4 ) ) .

J

, then

k.= Qu(z, k . , p, i 3 r ( f j ) ) ;

J

1

y is an element of the l a s t t e r m of

.

k

We proceed s i m i l a r l y in o r d e r t o prove the positive p a r t of VII. The negative p a r t of theorem

VII is proved by using the

well known technique of ~ 8 d e l . We a s s u m e that t h e r e e x i s t s a satisfaction formula n

in

S

for M

F r m with

s

and denote by Fr(n) =

{

a function such that for every s ( n ) is the

0jthe s e t

resulting f r o m the formula with the ~ 8 d e ls e t f o r i t s unique f r e e variable. i S ( s ( x o ) , 0, V) where

V

Let

n

induction on the length of

the G

...,xP

s

that

and

.. ,xP

if

We prove by

~ 8 d e sl e t

p

g, then the

M:

S(g, F i p ~ i V). ,

the r e s u l t that the equivalence

M

s(n,).

M.

is a formula with

G

with the

Using this equivalence for

and hence

c a n be defined in

Gbldel s e t

following equivalence is provable in G(xl,.

n

i s the universal class; we r e l y h e r e of

Then 7 S ( s ( n o ) , 0 , V ) has xl,

by substituting

be the ~ 8 d e sl e t of the formula

0

course on the fact that the function

f r e e variables

n

~ 8 d e ls e t of the formula

T

G= lS(s(no),O,V)

we obtain

G G ~ would be provable in

M

would be incbnsistent.

The detailed proof of theorem

VIII will be given later.

The main idea i s this: we assume the existence of a predicative satisfaction formula for set which is a model f o r

GB ZF.

and prove i n

GB

the existence of a

F r o m t h i s we derive (always in GB)

the existence of a s e t which is a model f o r

GB and

thus infer that

A. Mostowski

GB is consistent.

Thus the consistency of

would be provable in

GB

GB which a s is well known entails the inconsistency of GB.

We conclude with a theorem which we shall need l a t e r and which can be proved by using the s a m e technique a s the one used in the proof of theorem VII.

M.

F o r each predicative formula

following formulae is refutable in

T

the conjuction of the

ZF :

(x)(y) [T(x, y) A (x E F r m p r ) & (y

(is')

is a sequence of

s e t s ) & (Dom(y) = F r ( x ) ) ] , (iil')

[ T(

(i)W (u)

'?l(xi)

I

,

1< i, u > l )

similarly a s above with

(iid

(a)(b)(y)( ( a E F r m

(iv")

(a)(i)w (y)

Proof. Hence the (i")

-

(v")

Let

f

"

we

infer by H

Hence the conjuction of

"

.

=

"

;

Pr

Txi)n3,

rk ~

IT(y x i ) ~ ,y) % (u) T ( a , y u { ~ i , u ) r ) ) ] ] .

( aY]],

) & ( ei F r ( a ) )+

be the ~ 8 d e set l of the formula l T ( s ( x o ) , 0).

induction

that if

h

Fr(H), then

H = CIT(to, 0), we obtain (i")

-

(v")

s ( t ) = to.

is the

whose f r e e variables a r e

is a sequence with domain

If we take in particular

replaced by

1 B E Frrnpr)b-(i$?!Fr(a))-+I~(

~ 8 d e sl e t of the formula 7 T ( s ( t ) , 0 ) is

a predicative formula and if

t

E

)& (b E F r m ) 3 ) TE ( ?;l y ) ~ pr pr L-,T(a, y IF r ( a ) ) v-T(b, Y 1 ~ r ( b ) j ) ) )

(a)(i)w (y) {(a E F r m

(v")

"

C U U ~,

Using

~ 8 d e ls e t of Xij,

(j = O , l ,

T(h, f) H E YH.

leads t o a contradiction.

...,k-1)

H(fio,.

..,fik-l).

A. Mostowski Terminological remarks.

If

S

i s a satisfaction formula

(or a predicative satisfaction formula), then we shall write instead of

S(x, y, z).

we shall replace

F ' x

x

,,...,

xn

F

and

let

x

by

and not simply Let

If F

i s the

z

~ 8 d e set l of a formula

x [y j F , then

(although it would be more exact t o write

in these formulae).

F

be a predicative formula with the free variables be its relativisation to the formula

F(')

x 6 z.

Then the formula (~,E.z)L

... &

( y n E z)-i

z t ~ D i l , ~,.+., ~ >
I = F ( ~ ) ( Y ~ , ,yn),$

-

...

i s provable in each of the considered systems of set theory. Let (y E

Zyr(X))

zl,z2

be classes,

& (z1C X

is an elementary subclass of

(z2 z2

. If (x)(y) ( ( x ~ ~ Prr )m& x ly])], then we say that zl z

and write

L.

zl < p r ~ 2 .

A similar

definition can be given for the notion of extension in c a s e when ranges over arbitrary formulae not just the predicative ones. shall not use this notion however.

x We

LECTURE

I1

The backbone of the whole set theory is the stratification In discussing

of the universe into levels(simp1e theory of types).

this phenomenon we Shall a s s u m e a s known the notion of ordinals and the t t e o r e m on definitions by transfinite induction.

We denote by On

the c l a s s of all ordinals; in the c a s e of the s y s t e m

ZF

a r e no c l a s s e s we think ,of On a s of a formula s o that

On(x) means

Sometimes we shall use the expression

"x is an ordinal".

the s a m e a s

where t h e r e

On even in Z F treating it a s equivalent with On(x).

x

We define by transfinite induction s e t s Ro = 0,

RL

= P(R

Rg :

R,=U~R,,:~

1,

'

r6)

is a

(f

limit number). Note: What we define is in GB (or any stronger system) a function which c o r r e l a t e s with each ordinal f c a s e of

Z F we have a formula

that

[odd

(4

Rt

.

In the

R(x, y) such that it i s provable in Z F

.

+(E ! Y)R(X.y)]

a set

This unique

y is denoted by

R

.

xB

it satisfies the equations given above.

I. F o r every s e t

x

t h e r e is an ordinal a such that

xSa.

This r leorem is provable in each of our t h r e e systems. The proof is obtained easily from the axiom Fund. The l e a s t ordinal a such that

x l R a is called the rank of x.

We want t o discuss the problem whether some of the R a t s a r e models of and prove in

In o r d e r t o answer this question we shall formulate

ZF. M

the following theorem:

I1 (Scott-Scarpellini). L e t each ordinal

&-,

AT

for each limit ordinal

is a set,

T

.

f

A= A*

u (A C_

A

:f e o nf

and

Under theses assumptions:

where f o r Af

=U{A+,,:fkf]

A. Mostowski

(a).

F o r each

continuous mapping then

A,)= F

Frmpr

+ On

fF: On

t h e r e is an increasing and

such that if

fF(a) = a and

in

There is a function

Frm

the mapping

Pr (a).

formulated in

f: F r m

X On A O n suah that for Pr a + f ( F , a ) satisfies the conditions

(c) There is an increasing and continuous mapping such that

if

t(a) = a, then

proof.- (a) which occur in

we take

f

G

prA.

If

fH.

I H?

?G

is

F

of connectives

is

F

f~

then we take a s

T(xi)~'l

and

fF the compo-

i$Fr(G),

then

The only c a s e which r e q u i r e s a m o r e elaborate

fF = fG.

proof is one in which

F

h a s the form

I-

(xi)H1

and

i 6 Fr(H).

a : ~ ~ ~ ( On, ~ ) and +

We f i r s t define auxiliary functions b:On

t:On+On

has no connectives, then we take a s

F

If

and

Aa<

. We use induction on the number

F. If

the identity map; sition of

Fr(F) xEA a '

F[X]SA/=F~X~. (b).

each

F in

.-+ On a s follows: a(x) = min b ( r ) = sup

{

r E On:(Eu) [(u .s Ar) & ( A k 7 H [xu { ~ ui $, a {

a(.):

xEA

r

~

~J (

~

)

It follows from these definitions that if

1.,

[(A

b H [ u 14i. u~j]))

x 6

(u) (AkH Ab(r)

r

, the11 (1)

l x*
(*)The subscripts after quantifiers denote t h e i r relativisation t o the formula x d a.

Thus

(x).

means the s a m e a s

(x)

1 (x €

a) 9

...J .

A. Mostowski NOWwe define u p

c(0) = 0, c(r+l) = max(c(r),b ( r + l ) ) and c ( r ) =

for limit numbers r.

) r

= f o c. F H This function i s obviously increasing and continuous. If f (x) = r, F Fr(F) we have c(r)> b(r). and for every x in then fH(r) = c ( r ) = r

(

Finally we put

f

Ar

It follows

+

u,@s(u)~

(ulA ( A ~ H [x+i, r

A ~ + F 1x1

r

( A ~ H [xV(ci. u ] ] )

and similarly

A

~

(b).

[ FX J ~(u),

(ACH [ x u {
We define

f(F,x) = x

(without logical operators), f ( r F I = f(H, x)

if

i

4 Fr(H).

= f(H, c(x)) where

c

If

)F ,:

if

-+(u),

r

(AI=H[

xu{ci.

F i s an atomic formula

x) = f(F1, f(F2, x))

i d F r ( H ) , then we put

and

f( q x i ) ~ ?x)=

f ( ? x i ) ~ l ,x) =

is defined similarly as in part (a).

The proof

i s identical a s in the case (a). (c).

We define

F E ~ r r n ~ ~ ,] t(r) ) = sup If

t ( r ) = r, then f(F, r ) = r

hence, in view of

t(0) = 0, t ( r + l ) = max ( t ( r ) , sup t ( r '):r' 4 r

$

if

f

f(F,r+l):

r is a limit number.

for each predicative formula

F

and

(b),

Ar< prA. Remarks on the Scott-Scerpellini theorem.

We want to

discuss the question whether theorem I can be so reformulated a s t o become provable in ZF o r in GB.

A. Mostowski

The case of t h e system ZF.

Since there a r e no proper

c l a s s e s in ZF, we must replace the c l a s s

A

and the relation

x GA

r

If we do this, then we cannot use the satisfaction formula

by formulae.

since t h e r e is no f ~ r m u l adescribing the satisfaction of an a r b i t r a r y x

in

F r m in the domain of all s e t s satisfying a given formula A.

We come around this difficulty by remarking that the satisfiability of an explicitly given formula

in the indicated domair. can be expres-

F

(A)

sed by the relativised formula

F,

Thus we shall have not a single

theorem starting with the general quantifier "for each F in ~ r m "but a theorem schema which can be proved separately for each explicitely given formula F.

A final change in the wording of the theorem con-

c e r n s the function fF: since there a r e no mappings of On into On in Z F we must replace the mapping fF by a formula which

the s y s t e m describes it

The Scott-Scarpellini iheorem takes thus 'in Z F the following form: Let x and. x, y

A(x)

and

B(x, y) be two formulae with the f r e e variables

respectively; let

C

be the conjunction of the following

F

with two f r e e variables

formulae:

-

F o r any formula

DF be the conjunction of the formulae: (XI

[~n(x)

(E

x, y

let

A. Mostowski

With these notations the following holds: formulae

A(x), B(x, y)

H(xl,.

and

..,xn) there

F o r arbitrary

is a formula F(x, y)

such that the implications

a r e provable in ZF. Notice that the formula x's satisfying over

C

A is the union of s e t s

says that the domain of a l l

{ x:B(x, r)]

with

r

ranging

ordinals and that these s e t s form an increasing and continuous

family while

DF

F

says that

defines an increasing and continuous

mapping of ordinals into ordinals. A similar reformulation of the Scott Is also possible in the case of the system GB.

-

Scarpellini theorem

Again we must express

the theorem a s a scheme because there is no satisfaction formula for

GB. However the assumptions of the theorem and the statement

concerning the existence of a mapping can be expressed as in system

M.

Finally we notice that a theorem similar to the Scott-Scarpellini theorem can be proved of the form

u { A,

:r

(in any of our three systems) for unions s

3

where

s

is an inaccessible cardinal.

Applications of the Scott-Scarpellini theorem. The following theorem is proved in

M; it c&n also be proved in any system obtained

from G'S by adding to it new axioms which secure the existence of a

a predicative satisfaction formula: There is an increasing and continuous function f: On +On

111.

such that whenever a model of

f ( r ) = r , then

Rr < p r ~ ; in particular, R

is

ZF. Proof.

The decomposition

V =

U{ Rr

:r

on]

satisfies

the assumptions of the Scott - Scarpellini theorem. Since all axiom of Z F a r e equivalent t o predicative sentences and a r e valid in V, we infer that if R is an elementary subset of V, then R

r is a model of ZF. We can now supply proofs of two theorems which we announced

in Lecture I: IV.

There is no predicative satisfaction formula for GB

provided that GB is consistent. Otherwise we could repeat in GB the proof of theorem I11 and infer that there is a set a which is a model of ZF.

Hence by

adding t o a i t s definable subsets we would obtain a model f o r GB. Since this proof would be formalizable in GB we would have a proof in GB that GB is consistent which would entail

the inconsistency of

GB. V.

M is essentially stronger than GB,

provided that GB is

consistent. Proof.

T h e r e is a predicative satisfaction formula for

M but none for GB. VI.

Z F is not finitely axiomatisable, provided that it is

consistent. Proof. We denote by of axioms of

K

the conjunction of any finite number

ZF and show using the version of the Scott

theorem which is provable in

ZF

-

Scarpellini

that t h e r e is an ordinal r such that

A. Mostowski

R

4'

K. Hence the consistency of K is provable in Pr consistency of the whole Z F is not s o provable. VII.

Z F while the

GB is finitely axiomatisable.

We merely sketch the proof of this theorem. Call particular instance of the axiom (scheme) t o the f o r m ~ l a F.

SubstGB

which corresponds

It can then be shown that all the particular instances

-

GB formulae: (Eul)(Eu2) l ( u = < u1.

UZ>

)&(ul E

...F*9 where F 1 . . .F g a r e the

F;,

can be derived from the axioms

of Subst

F V the

U 2 o

, (Eul)(u= < ul. ul> 1,

1~ < ?

? ( u ~y),

(UE

(ul E Y)

DU

fill =

=

1

< u , t > C y), (Eul)(Eu2)

y) & (u G z ) ,

.

< u1

(EU)(U=


< u2,

u3>>

< .I>

4U2,

U3>>

u

>

2 & (
=

U1.

U2>

) &

) & ( C u Z . ul> E Y). ( E U ~ ) ( E U ~ ) ( E U ~ )

ul>> C- Y)

) & ( < u 2 >< u l 9

U~)>EY)]

.

(Eul)(E~2)(E~3)

.

These axioms state the existence of the following classes:

E = ~ < u , v > :u ~v

E ~ .3I = ~ < u . u >: u e v ) .

V

Dom (Y), Y x V,

Y,

Cnvl(Y), Cnv2(Y)

E

t h e r e is a c l a s s

u

f o r which

H(u).

Details of this proof

in G8de11s monograph.

Natural models. of

H

and I by means of the operations enumerated

above and consisting of all can be found

Y, Y A Z .

and our theorem reduces t o

the statement that f o r every predicative formula Y constructed f r o m

v -

Z F , GB, o r

If

r

is an ordinal and R

is a model

M, then we call it a natural model of the corresponding system

In theorem I11 we established (in M) the existence of a "tower" of natural models f o r

Z F which is ordered by the relation

-
We now show

that the existence even of a single pair Rr' Rs such that ~~dR Pr s We do cannot be established in ZF provided that ZF is consistent. t h i s by proving in Z F the following:

A. Mostowski VIII. (Montague a model of

-

Vaught).

+0

r there is an (Exl)(xo

x xl)

r # 0.

and in

r

then

R

Rs

1.f

is

t t r , then

and

R eR and hence t r Thus the formula

s

x

r' t +l#r.

Using the fact that ZF

0.

R x. t Rt in R and hence it must also

Thus there is an

check that all axioms of

je:

Rs, it i s t r u e in R

such that

is satisfied by

be satisfied in R and therefore

is a limit number

(Ex)(xZx) is true in

Since the formula R

R,(~,R,;

ZF. Proof. We f i r s t show that

hence

If

r

in R

such that

R E x

t

is a positive limit number we easily

a r e valid in

.

The verification is

Rr

evident in all caseswith the exception of the axiom scheme Subst

ZF

which requires a separate treatement. Let. F be a formula with

k+2 free variable and let

5 E R:~'~)-

'i~ 'f be such that

Rr

+ xi)^! x j

in R

let

f(x) be the unique

y

in

4,y

.

Let

x and x P and l e t a be an element of and satisfies the condition

R

such that

be variables

Rr' R s

' F[ J

The s e t

. R~

For

-x

!= F[~U,&,

which do not occur in Im(f, a) = b

~ [ ~~ vpa>, ,i

( q,

b>]]

belongs t o where

H

is the formula

It follows that t h e r e is a s e t R r k H [ ~ u ) < p a> , , < q, b>i] Thus axiom

Subst ZF

be also proved in

M.

is valid

b'

in

Rr

such that

and we easily prove that in

Rr'

Theorem

b1 = Im(f, a).

IX can obviously

x,, F

Rs

A. Mostowski

<'

In o r d e r to obtain m o r e information about the relation in the c l a s s

R 's

of a l l

We c a l l an ordinal t h e r e is a sequence and that

R

f(t) t
f

r

extendable if f o r any ordinal

of o r d e r type

s

such

for

prRf(o

( c ) of the Scott-Scarpellini

that t h e r e a r e extendable ordinals.

a r e elementary extensions of

R

On

so

and

Rr

<pr~s

so

then R s

that because of the extendability of such that

s

Rr<pr Rs

from a certain

s

0

>r

>r

r

R m l f ) and R r(prRm(f). follows :

-(

s

0

such

such

which

<

pr'

hold.

r

R Pr s

( l e a s t ) extendable

r

with the properties : whenever We notice

t h e r e a r e a r b i t r a r i l y high

and our assumption s a y s that each such

on can further

>

Rr

< Pr R t f o r s o m e t > s. be extended t o an

F o r a r b i t r a r y finite sequence m ( f ) the l e a s t ordinal

such that

us a s s u m e that t h e r e is a

such that t h e r e e x i s t s an ordinal

X

r

M.

R (pr~t

does the relation Let

R 's

F o r each extendable ordinal

a r b i t r a r i l y high ordinals

Proof.

theorem it follows

and well o r d e r e d by the relation

(Ryll Nardzewski).

but f o r no t > s

>

such

consisting of

The next theorem is provable in

s

t, t '

Moreover t h e r e a r e ordinals

that t h e r e is a sequence of o r d e r type

there are

f(0) = r

that

each p a i r

s

.

F r o m part

IX.

P

we introduce the following definition:

f

Rt

.

of s e t s we denote

that all t e r m s of

f

by

belong t o

Consider the formula T(x, f) defined a s

s s

A. Mostowski

we

shall show that this formula has the charateristic pro-

perties (if')-(v")of the truth predicate (cf. Lecture I, theorem IX). Since we know that we can refute the conjunction of (i") - (v") we shall have the proof that our assumption leads to a contradiction a s soon a s we veri= fy that T has the properties ( i f ' )- (v").' Of these, ( i f ' ) , (ii") and (iv") a r e obvious In o r d e r to verify the remaining two we prove a lemma:

R R , thenlS< Pr t pr t '

I f so < s < t a n d n r 5 R R r S,

Proof of the lemma. F o r each s >s such that R 0

mentary extension of R

S

is an ele-

we denote by s 1the least ordinal )s

such that

The existence of s 1follows from s' our a s ~ u m p t i o n s.Now we s t a r t from given ordinals S, t and construct two Rs, is an elementary extension of R

sequence f and g saticfving the inductive equations f(O)=s, f(n+l)' = (f(n)I1,f ( u ) =sup {f(n): n c u 3

sup {g(n): n < u ) ,

R

=

g(n) <,,R~(,)

U{

Since R

We obviously have R

<pr

: n

G On

-

X

I- Y; =-

]

.Let x t F r m

V we infer that R s +

we prove similary that Rti=

< R and f(n) p r f ( m )

f o r n.4 m; moreover V =

Rg(n)

S

, g(0) = t , g(n+l) = (g(n)I1, g(u)=

Rt

v k x [y]

XprV

U Pr

x [y]

;

on)

R ~ ( ~ n,

and y

(F

V

=

RF r ( x ) S

b x

. lyj

;

and therefore

whence ( R . ~ = x

[Y])

L (RtkX

The lemma is thus proved. Verification of condition (iii"). Let us assume that

[Y3)

.

A. Mostowski

This assumption is equivalent t o the statement that either R

I-

m(f)

+..,x1

I-

the lemma R

m(f)

pr Rm(f ) F r ( x . )

'fi

Fr(x2)]

. Eince by

for i= 1, 2 we infer that the

1

previous statement i s equivalent t o the disjunction of the statements Rm(f

I Fr(xi)) *?x:

[f

I

Fr(xi)]

for

i = l,2

and this disjunction

is precisely the right hand side of (iii"). Verification of condition (v"). We a s s u m e that x E F r m and i

F r ( x ) . Let us consider the

statement T ( ' (x.) 2

P

'

, f);

this statement says that f is a sequence with domain F r ( x ) and that x [f 0 ( < i ,

j= Rm(f)

u>j]

for every u in R

m(f)'

We hLve t o

rf u)-=i,

prove that this statement is equivalent t o the following: for every u, R m ( f ~ { < i ,u s ]

x

u1 .1

.

Obviously the first statement is implied by the second because m(f w ((
>2

NOW we m(f) ' assume the f i r s t statement and choose an a r b i t r a r y u. By the l e m m a

<

Rm(f)

p r R m ( f ~<{i , u)

) = m(f) whenever u is in R

f

) and hence the f i r s t statement implies

@(xi). [gwhence we infer that the second statement is Rm(fu{< i. u > ) ) valid. Theorem IX i s thus proved A theorem similar t o M can also be proved for other transfinite sequences of sets, for instance for s e t s L shall discuss later.

which we

A.

Mostowski

In connection with theorem M we discuss briefly the ordinals which a r e not extendable. Let us call a

function f whose do=

main is an ordinal r and which satisfies the condition

<'pr Rf(,, f o r a r b i t r a r y m E n E r a chain of length r s t a r = f(n) ting i n f(0). An ordinal s i s not extendable if and only if t h e r e is R

an ordinal r such that t h e r e is no chain of length r s t a r t i n g at s. The l e a s t such r is called the height of s. The height of an extendable ordinal could additionally be defined a s On. We don't have an exact

charscterisation of ordinals

which a r e heights of non extendable o r d i n a l s . However we can exhibit a r a t h e r l a r g e number of examples of such ordinals.

->0

We call an ordinal r

is a predicative formula F with

s E On and r Rs

C

F

e

[ (
R- definable ordinal i f t h e r e

an

Fr(F) =

$

0

f

such that whenever

Rs, then r is the unique element of R x,f]

S

while 0 is a unique such x if r

such that

4

R

.

We c a l l F a definition of r. We now prove in M:

X. ( W i l m e r s ) . F o r every R-definable ordinal r t h e r e is

.

an ordinal whose height i s r+l. Proof. Using p a r t (c) of the Scott - Scarpellini theorem we easily prove that t h e r e a r e ordinals x such that R element a chain of length r

+

+ 1 which belongs t o Ra

b = f(0) and claim that the height of b is r + 1 Since f i s a chain of length r

is a chain

g

of length r

+

contains a s

1. Let a be a smallest such ordinal

and l e f be a chain of length r

obvious that the height of b i s

X

3

r

+

.

. We put

+ 1 starting at b, it is

1. Now we a s s u m e that t h e r e

2 starting at b and derive f r o m t h i s a s s u m =

ption a contradiction. There e x i s t s a predicative formula H with

A. Mostowski

f

Fr(H) =

p and a r b i t r a r y u, v in R

H P is a chain of length (1

P

< 0,

R

such that for an a r b i t r a r y ordinal

0, 1 1

u

>

<

,

I, v

(V

E On)&(u

7).

We shall indicate below how to construct such a formula. Assuming that we c a r r i e d out the construction we proceed a s fol= lows.

-

We easily show that n G Rn+l) c

n in r + 1; since g(n) G R R

g ( r + I).

Now

g

1

g(r+l)

R

g(n+l)

we s e e that g

( r + 1 ) is a chahof length

for a r b i t r a r y

1

(r+l)E

r

+

1 and hence

R g ( r + 1) I = H [ { < o , g ( r + l ) > , T] Denoting by F a definition of r we obtain from the l a s t formula

Since R

g(0)

EX^) EX^) EX^)

(

R g ( r + 1)

is an elementary subset of

can replace in this formula g ( r

+

1) by

infer that there a r e elements c, d of R I

F [{<2,

r>jj

[ F ( x ~ ) & ( x 2 # 0)

and R b

b

H [{do,

R

g(O), i. e. b

g ( r + 1)

by b. We thus

such that k f 0, R

d>,

<

1, c

we

b 'P

+ I>?]-

.

A. The first

of these formulae provesthat

Mostowski

c = r

and the

second that d is a chain of length r + l . Since d 62 Rb and b

<

a ,

we obtain a contradiction with the definition of It remains t o construct the formula

H

. Let

R' ( x , y ) be a formula which defines the relation =

(cf. p

R

C Frm

(f): (

)

79

P='

and let

) & (f

.

a

x =

G ( z, t ) be the formula

c zFr(X))

!=

6, (

xCfJ)--.

The required formula is On(x

1

& i 6

) & F n ( x ) & ( D o m ( x o ) = x l ) & ( i ) ( j ) ( z ) ( t ) 0 j

)

&

+G(

(

j x ~o ) & (

o

j

,

t>fx0)+

)I .

t

z,

>

i

Natural models of G B and of M . These models a r e si= tuated much l e s s densely than the natural models of ZF: we shall prove the following result in the system

ZF

ZF

+

AC

resulting from

b y adjunction of the axiom of choice:

XI. (Shepherdson) anly if

r

is of the form

.

Rr

s + 1

is a natural model of

and

s

is a

.

M

Proof. If r

is a successor ,

R

is a

model of

GB

if and

strongly inaccessible

cardinal; this condition is also necessary and sufficient f o r be a model of

GB

o r of

M

R

r

to

, then

r = s + 1 because there exists a universal

of the model. F r o m the satisfiability of the axiom Subst it follows that if

t

is an ordinal <. s

and

f

a mapping

of

t

A. Mostowski Im ( f, t ) & R s and hence t h e r e is an ordinal

into s

, then

t1< s

such that

Im ( f, t )

t

t'

.

This proves that

s

is

weakly inaccessible. In o r d e r t o show that s is strongly inaccessible we have

l e s s than

s

<

t

t o show that if

.

s , then the cardinal nwmher of

By our assumption

Since we assume that

t

P ( t )

is

and hence p( t )

Rs

Rs.

R

is a model of GB and since t h e r e s + l is a well ordered set which belongs to R and has the same power S

as

P ( t ) , we infer that t h e r e exists an ordinal number in

which has the s a m e power a s is s m a l l e r than

If all

s

P ( t )

. Thus

Rs

this cardinal number

.

s is strongly inaccessible, then we easily check that

the axiom of

M a r e t r u e in

axiom Subst is satisfied because if

Rs + f

.

In particular the

is an9 mapping of

into R s and a belogs to Rs, then Im(f, a) belogs t o R A comparison with theorem

problem: how t o characterize ordinals sfying the relation

Rr

+

VIII r ,

XprR s +

1 ?

Rs

s'

suggests the following s ( r

(

s )

sati=

It i s easy t o s e e

that this problem is wrongly expressed because the relation

in

but not of R is the l a r g e s t element of R r r+l s+J. Therefore we modify the problem and discuss not the relation of question never holds (R

elementary extbnsion but a closely connected relation of elementary embeddability. Definition. t h e r e is a function

f

Rr

is elementarily embeddable in

whith maps

Rr

isomorphically

Rs (with

if

A.

E

respect t o the relation

>

r

Rr,

t r u e of

f

. Since

+

Rr +

strongly inaccessible. Since and

Rs

Rr

onto

is a model of

where

p

s

=

+

r + l

GB, the same is

s

, where

1

is

is the largest element of

Rr

the largest element of

it follow that

Rs + 1

( i. e. f( Rr ) = R s ). s Similary f( r ) = s since

Rr+, maps

f

R

and s R r + l hand the range of f Of

I

such that

i

p

( r

Rg ( f

because an ordinal in r

+ 1

r

is

the largest ordinal of

for reason of cardinality.

ment in

+

be a function which embeds R

+ p and hence r ' + p

R r'

Rr'

is measurable.

Proof. Let into

Rs is a strongly inaccessible ordinal

r

i s elementarily embeddable into

R r + l r, then

.

onto an elementary subset of

)

XII. (Reinhardt). If and r'

Mostowski

1(

+

1 )

the l a r g e s t ordinal

.

Rs +

On the other

s

is strictly contained in

Hence t h e r e is an ordinal Obviously

r + 1 ) ).

Rg ( f )

+

1

s

p

P f R g ( f )

must be the value of

f

for an

argu=

.

F r o m the properties of

f

it easily follows that

f ( x n y ) = f ( x ) nf ( y ) , f ( x V . y ) - f ( x ) V f ( y ) and Rr

f( x - y ) = f( x )

.

-

f ( y )

for arbitrary

x,

y

in

F o r instance the f i r s t equation is established by no=

+

ticing that

x 0 y

is characterised a s the unique element

togcther with

x and y satisfies in

(t)

I[(t E xl

(t 6 x 0 )

together with R s + l '

Rr

& ( t

f ( x ) and f ( y )

c

+

x2)

z

which

the formula

.

Hence

f ( z )

satisfies the s a m e formula in

A. Mostowski Now

F

claim that

we put

t-multiplicative f o r each If and If

x

C

x,

y C r

and hence

= s

y e F,

r G F

then

r - x

-

.

of

F

.

f ( g )

g

F

The intersection unique element of

+

1

x )

is

and

Rr +

.

r

; both these formulae

-

x ) = 0

, then t

g

, it follows that

.

F

is t-multipli=

consisting of elements

satisfies in

Rr

is a function with domain

Rs

f ( t )

n Rg

+

.

( g )

Hence either

prime.

t & r

satisfies in

it is a function with domain

Rr

-

s

p

f ( x ) fl f ( r and

p CE s = f ( r )

and we infer f ( r ) =

Since

g e R r + l

formula which s a y s that infer that

x )

be a sequence of type

Since

From

is not void.

F

f ( r

p

belongs t o F

g

F.

y

Now we show that if cative. Let

f ( y ) = f ( x A y )

then f ( x ) = f ( x ) n f ( ~2) f ( ~ )

F,

f ( x ) or

n

.

r = x d ( r

p 6 f ( x )

and

which is prime and

p € f ( x )

and hence

cannot be true because x or

r

is a filter of subsets of

F

= f ( r - x ) y

either

r : p 6 f ( x )]

.

r

then

x

and

Hence x 5 r,

5

x

p E f ( y ) , i.e.,

it follows that If

<

t

x n y 6 F

hence

{

F =

is a filter of subsets of

a

+

t

, we

the formula saying that

1

can be characterised a s the

which together with

g

satisfies in

the formula ( t E x O ) = ( w ) ( v ) [(
V>E X1)+(t

Hence the same forfnula is satisfies in f ( n R g ( g ) and

f A g ) )

.

R s + l

E by

vd

A. Now we notice

that the s a m e formula

irj

Rg(f(g)) = s + l by (1Rg(f(g)) and f(g) and hence Since Rg(g)S F, we s e e that pE f(x) f o r e v e r y x in Rg(g). tice that x 6 Rg(g) is equivalent t o t=(Ey)( ( y, xo) d x l ) [ { Rr+l R s + l I= ( ~ y ) ( E x l ) [{< 0. f ( x ) >

.

.c

also sati=

f(nRg(g)).

n

sf'ied i n R

Mostowski

< 1, g>]l

Now no=

and hence t o

1, f ( g ) > ~ ] ; i. e..

to

f(x) E R (f(g)). Thus p is an element of e v e r y m e m b e r of Rg(f(g)), g Rg(g) E F. i. e. , p& flRg(f(g)) = f ( 0 Rg(g)). Hence

0

This proves theorem XIII. Notice that this proof like the proof of theorem XI was c a r r i e d out in the s y s t e m Z F

+ AC.

A. Mostowski Lecture

111 In this lecture we shall apply the Scott

-

Scarpellini

theorem to obtain various families of sets which form models of We call a set in f

A , every in aFr (

e

F

Frm

) the set

a

,

belongs to A.

The

of a determined by

with

P'

set

Fr ( F ) =

SF ( a, f )

and

b F [ f

$

and for every

x>j]j;J2

u +O.

will be called the section

f.

We shall show that all sectionsof

elements

a

of

A

can be obtained by iterating finitely many operations Let us con= sider the following operations:

.

predicatively closed if for every a

Fx e a : a

= F

A

ZF

A.

I. If A

rationa

-

1

A

0

e

Mostowski

and A is closed with respect t o the ope=

1

1 2 , then

is predicatively closed.

A

We shall only indicate the essential steps of the proof. F i r s t we notice that an a r b i t r a r y finite sequence whose t e r m s a r e integers belongsto

. Hence

A

c e s with a common finite domain does the set of all sequences

f

s

if a set

of finite sequen=

d SCd ~ e l o n g st o

o x where

x

A , then s o

s

ranges over

and

i s a fixed one one mapping of a s e t d' onto d. Fr( F Hence if the s e t

f

.

.

{ x ,ka F r ( F'

belongs t o A , then s o does the set

where F ' a r i s e s from F by a permutation of variables.

A

By i t c r a t i n g suitably the operations prove that if a

d

& A.

1

- A

F r o m this we

s

easily infer that if

e

A

a s e t of finite sequences with a common finite domain ranges contained te s e t

df

in

a

where

such that

d

5

a

is in

we

a G A , then

is a finite s e t of integers and

d

12

and d

s

is

and with

A , then for e v e r y fini=

d j ~ Wthe s e t

)

f (. ad' : f

d E s t

belongs t o A. Finally we show that i f integers and f s

r

$

n

ad : ( Eu )

d

,s

ad

[ f U{ < n,

a

" ln3, u>]

cs A ,

d

is

a finite s e t of

then the s e t s]{ belongs to

A along with

.We show this by noticing that in o r d e r to obtain

this s e t f r o m

s

it is sufficient to subtract from every member

A. Mostowski

s

of

the s e t of all s e t s of the form \
{f

=

consider the set

E aFr@) : a )= F

f

I] .

rxo t

one of the formulae r ~ l ( x 0 D ( a ) btlongs t o

A1O -

Using the operation

case when A

3

F

e

A

for

we use

F =

a.

L

=

If

a E A

x:

,

and

F is

rxo = x z

, then

A

we extend t h i s result t o the

7

A

2,

established above we show that if

i = 1, 2, then the same result is t r u e for the

Fi

c a s e s when

DF ( a )

t

has a r b i t r a r y variables. Using the operations

a s well a s the r e m a r k s

D ( a )

where

a s we easily see inspecting the operations

A

F

t>]

r F1

I

and

F :

A I 2 in o r d e r to construct

r(

F =

.

xi ) F :

SF( a, f )

from

Finally

DF( a )

.

The main result of the present lecture is a s follows: 11. let

A

be a family of s e t s (indexed by ordinals)

which satisfies the assumptions of the Scott If in addition

A

a =

u { As

and predicatively closed for every

-

s E 0nj

:

r

Scarpellini theorem. and

in On, then A

A

is transitive

is a model of

ZF. Proof. The verification of most of the axiom is immediate We discuss only the axioms Inf, Pot and Subst which a r e slightly mo= r e difficult to verify,. Axiom of infinity r

n

of ordinals: r 0 = 0,

.

We define by induction a sequence

r n+l = min

s

:

Ar

E n

A

and put

A. Mostowski

r = sup r

then A

n'

satisfies the conditions stated in the axiom Inf.

Axiom of power set. F o r let

s ( x ) = min

: x g

P ( a )

& ( x

At ).

f

.

Then

the formula ( v )

[. ( v

tively closed ,

+

xo )

2 f ) -

Fr( F ) in

AFr(

A k ( x O )( E ! x l ) F f ( x ) = m i n

< 1.

y

$

"

>

PI)]?

At E A

of

A

1

>

(xCa) &

u

and F

and since

A u

is predicaz

ip].

a

A

and let

'1

F

be a

p

be

such that

For

x

in

A

let

[{

( y e A S ) & ( A * ' F

{

t = max( r , sup

f(

40, x>

x ) : ( x C-- Ar u

Using the Scott - Scarpellini theorem we determine an ordinal such

that

u

>

t

and

is

.

3 . F u r t h e r m o r e let

C

and put

s ( x ) :

where

U

7

(v E x l )

. Let

0,

s : ( E y ) [

f

P(a) n A

and hence t o A .

U

Axiom of substitution formula with

a>))

p( a ) .n A

it belongs t o A

in

CP(a ) =

u such that

< 1,

SF( At,

This section is equal to

a sequence

x C A -3(x

Now take an ordinal

determine the section

and x

max (r , sup

s: x g A s $ , t =

A]

r\

a CS Ar

for arbitrary

x, y in

AU the following

equivalence holds :

-

Finally we choose an ordinal

and determine the sectiom. an integer such that (xl)

{

( x1

E

x j

xO)

SG( Au.

I

< j, F

isjnot f r e e in ( Exo

[

v such that

(

XO

a

>y and

G

X.J

AU E Av

p ) where G

j

is

is the formula &F]Z

.

,

)I)-

A. Mostowski This section which we denote by

b

belongs t o

A

v

and satisfies the

condition

Remark. Theorem I1 was proved above on the basis of the system

M

established in

.

There is a version of this theorem which can be

ZF. Similary a s on p. 82 we consider a formula

with two free variables and define a formula (Er)

[ On( r

.

) Ri B( x, r

4 formulae listed on p. 8 2

the s e t

1x

Let

C*

B

A by be the conjunction of the

and of the following formulae:

(The f i r s t of the above formulae expresses the fact that : B(x, r ) ]

is transitive and the second that this

is closed with respect t o the operations

set

.

A

- A obviously,the 1 12 ' second formula should be expressed in t h e language of Z F which

can easily be done by writing down the definitions of

Ai(x, y )

as

set theoretic formulae). Imitating the proof of theorem I1 we can derive from C in Z F all formulae obtained f r o m the axiom of Z F by relativising all quantifiers to the formula A. In other words the formula A define an interpretation of

ZF

in

ZF

+

C

%

.

Examples of c l a s s e s which determine models of Example

1. V =

Example 2:

UI Rr

.

constructible sets. We define f o r an a r b i t r a r y t

set a a' =

: r E On]

ZF.

f

SF(a, f) : (F E F r m

Pr

) & (0 b F r ( F ) ) & (f 6 a F ~ ( F -)

103)j

a ' is the family of all sections of a determined by an a r b i t r a r y formula

F

in

Frm

in a. We now put Lo = 0;

+

Lr

The union

= L',;

1

L =

and an a r b i t r a r y sequence

Pr

=

Ls

u{ L

U{ L

f

with t e r m s

3

: r t s (s is a l i m i t number)

is

: r E On)

called the c l a s s of con=

structible sets. It i s e a s y t o construct a relation which well o r d e r s the c l a s s L. We define it a s the union Xo = 0,

+

X

of F r m

X

u

in

Lr

+

s

: r

is obtained f r o m

1

every

{

=

X

- Lr

: r

E

where

on]

s is a limit number and where

if

by the following construction. we denote by

such that f o r s o m e

Pr

Vi Xr

f

F

U

Fr(F) -

L

in

the e a r l i e s t

{

i'or element

03

The t e r m "earliest" r e f e r s t o a fixed well ordering of the denumerable s e t

Frm

Pr

which we think

If t h e r e a r e many sequences equation is t r u e

then we denote by

f

U

of a s fixed in advance. f for which the above the e a r l i e s t of them in the

lexicographical ordering 44 of finite sequences induced by the r e = lation

r

. Now we define

pairs

< u,

v>

where

Xr

u

+

and

v

a s the union of

Xr,

both bdong to

Lr

of the s e t

+

1 - Lr