Models of ZF-set theory (Lecture notes in mathematics 223)

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich

223 Ulrich Feigner Universit~t Heidelberg, Heidelberg/Deutschland

Models of ZF-Set Theory

Springer-Verlag Berlin. Heidelberg. New York 1971

A M S Subject Classifications (1970): 02 K 05, 02 K 15, 02 K 20, 04-02, 04 A 25

I S B N 3-540-05591-6 S p r i n g e r V e r l a g B e r l i n • H e i d e l b e r g • N e w Y o r k I S B N 0-387-05591-6 S p r i n g e r V e r l a g N e w Y o r k • H e i d e l b e r g • B e r l i n

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PREFACE

This given

set of notes

is part and parcel

in 1970 from February

tics at the State University as an aid to the people

at Utrecht

attending

as the course developed.

those

In fact~

complete

anddetailed

some few places

there

of lectures

and they were lectures.

of Mathema-

intended

proofs

scribblings.

are given

are only short

looking

In most

in these notes but in

indications

This occurred when a result was only slighty

only

They were written

in spite of their official

aspect, they are no more than prelecture oases

of a series

up to June at the Department

to the proof.

touched

in order to

round up the presentation. It was the aim of these

lectures

some of the basic techniques

and results

Set Theory.

This theory

to compress

it into one series

theme the construction the construction ungrounded

is given, Chapter

some basic

of lectures.

concepts

the generalized

and L@vy's

continuum

Mostowski-Specker

and of P.Cohen

Many of these

explanations attention

are included.

and obscurities

of

models

principles

As an

of reflection.

proof

for the axiom

and the axiom of

the methods

in a general

Although

models.

hypothesis

several

sets,

(containing

of Zermelo-Fraenkel

consistency

applications

as our

L of constructible

iII and IV contain

As an aid to the reader

mistakes

model

relative

Chapters

print.

of

of models

We have chosen

and Cohen-generic

constructibility. applications.

of the theory

I), the axiomatization

GSdel's

an exposition

a wide field and it is not possible

of Ggdel's

like x = {x})

(chapter

II c6ntains

of choice,

covers

of Fraenkel-Mostowski-Specker

sets

introduction

to provide

setting

of Fraenkel-

and various

have not yet appeared informal

discussions

we have attempted

in

and

to reduce

to a minimum we should be glad to have our

drawn to any indiscretion

the reader may discover

in the

text. To finish D.van kind

Dalen

to HelSne

Heidelberg,

I wish to express

and the D e p a r t m e n t

invitation

to Rode V r i j e r and

this preface

of ~[athematics

to spend a y e a r at this and K o R a s m u s s e n

Keller

July

f o r typing

5, 1971

my gratitude

to

at U t r e c h t

for the

Institute. Thanks

are due

for correcting the manuscript.

several

misprints

CONTENTS

CHAPTER

!. P r e r e q u i s i t e s

A) R e c u r s i v e

Functions ........................................

2

B) F o r m a l T h e o r i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

C) A r i t h m e t i z a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

D) S y n t a c t i c a l

Models .........................................

E) Z e r m e i o - F r a e n k e l F) The P r i n c i p l e CHAPTER

6

Set T h e o r y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

of R e f l e c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

ii. C o n s t r u c t i b l e

Sets

A) The A x i o m of C h o i e e B) The C o n s t r u c t i o n

and the C o n t i n u u m

Hypothesis ...........

23

of the m o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

C)

Z~ F - F o r m u l a e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

D)

Z~ F - d e f i n a b i l i t y

of s y n t a c t i c a l

notions ..................

32

E)

&>F _ d e f i n a b i l i t y

of s e m a n t i c

notions .....................

33

F)

z>F _ d e f i n a b i l i t y

of the

G) P r o p e r t i e s

of t h e c l a s s

constructible

model .............. sets . . . . . . . . . . . .

35

and a L e m m a of G . K r e i s e l . . . . . . . . . . . . . . .

39

I) T h e o r e m of J . R . S h o e n f i e l d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

J) R e m a r k s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

H) T h e o r e m of H . P u t n a m

Additions CHAPTER

III.

A) The

to c h a p t e r s

L of c o n s t r u c t i b l e

34

I and

II . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fraenkel-Mg§towski-Specker

Independence

Models

of the A x i o m of F o u n d a t i o n . . . . . . . . . . . . . . . .

B) The F r a e n k e l - M o s t o w s k i - S p e c k e r

Method ......................

52 57

Independence

of the A x i o m

D) The

Independence

of the g e n e r a l i z e d

Continuum-Hypothesis

f r o m the A l e p h - H y p o t h e s i s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Anti-Chain

of the A x i o m

of C h o i c e

of C a r d i n a l i t y

in ZF ° . . . . . . . . . . . . . . . . . . .

G) A F i n a l W o r d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additions CHAPTER

IV.

to e h a p t e r

III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Cohen Extensions

59

(AC) f r o m K u r e p a ' s

Principle .......................................

F) T h e U n d e f i n a b i l i t y

47

of C h o i c e . . . . . . . . . . . . . . . . . . . .

C) The

E) The I n d e p e n d e n c e

45

61 67 73 75

of Z F - M o d e l s

Introduction ..................................................

76

A) The F o r c i n g

78

(Ramified

Relation

languages,

in a g e n e r a l

setting ..................

s t r o n g and w e a k

forcing,

complete

sequences) B) C o h e n - g e n e r i c

sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

VI

(Definitions

of val~ and the extension

the forcing technique:

~

~ZF~

~£,

Hauptsatz of

Digression).

C) The Axiom of Choice in generic extensions .................... ( ~[a]

~ ZF + (AC), Symmetry properties,

96

the Independence

of the countable axiom of choice (AC ~) from ZF). D) The Power of the Continuum in generic extensions ............. (Independence of the Continuum-hypothesis

103

from (AC), indepen-

dence of V = L from (AC) + (GCH)). E) The Independence of the Boolean Prime Ideal theorem (BPI) from the Ordering Principl~ .................................. (Equivalents

of the (BPI), the Order Extension Principle,

Historical retrospect: universal

relational

Mostowski~s model, homogeneous systems~ the independcnee

Orderextension principle

of the

from the Ordering principle).

F) The Kinna-Wagner Choice Principle ............................ (Formulation of that principle (KW-AC) + Ordering principle~ (KW-AC)~

110

independence

G) The Independence

(KW-AC),

122

Proof for ZF

independence of (AC) from

of (KW-AC) from the Ordering principle).

of the Axiom of Choice (AC) from the

Boolean Prime Ideal theorem (BPI) ............................

128

(Outline of a proof that (BPI) holds in Mostowski's model, Construction of the generic model ~ [ a 0 , a l , . . . , A ]

and

proof that (BPI) + 7 (AC) hold in that model, Application: definitions

of continuity).

H) The Axiom of Dependent Choices ...............................

146

(Bernays' and L@vy's formulation of that axiom, proof for ( ~ a ) ( A C ~) ÷ (DC ~) and for ( ~ ) ( D C results:

~) ÷ (AC)~ Independence

Mostowski's model of ZF ° + (DC ~) + ~ (AC~I)~

Jensen's model of ZF + (AC ~) + 7 (DC~)~ Feferman's model ~[ae,a1,...] results).

for ZF + (DC m) + ~ (BP!), List of further

I) A F I N A L W O R D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

REFERENCES

168

.....................................................

CHAPTER

I

prerequisites In this short chapter we list those topics, mostly taken from the lower predicate

calculus and Peano Arithmetic,

our readers to be already familiar with.

that we expect

For more details the reader

is referred to [15] S. FEFERMAN:

Arithmetization

setting~ [60] E. MENDELSON:

Fund. Hath.

Introduction

Company, [TS] J.R. SHOENFIELD:

of metamathematics

49 (1960) p. 35-92.

to Mathematical

Princeton-New York-London Mathematical

in a general

Logic~ v. Nostrand 1966

(3rd printing).

Logic~ Addison-~lesley Publ. Comp.

1967. All our considerations

about formal languages,

their syntax

and semantics are carried out in a certain underlying

Metatheory.

Our metalanguage will be english enriched by some mathematical and logical

symbols such as: ~, &,~,~,~,V,~,~

(negation, universal

conjunction,

disjunction,

quantification,

implication,

existential

equivalence,

quantification and equality}~

It is understood that the use of these symbols is known, a semantics (id est: denotes

set theoretical "membership".

set theory as formulated

in the above

The universe of sets as given by our naive set theory

will be called sometimes and relations

is not given. The symbol 6

We adopt naive set theory as our metatheory

(id est: Zermelo-Fraenkel meta-language).

interpretation)

"the real world" and objects~ operations

in it shall get sometimes the adjective

:'actual" in

order to distinguish them from the corresponding objects, operations and relations of some "object-theory".

Hence E is the actual member-

ship relation and ~ ~ m0 is the actual set of (actual) natural numbers.

A) RECURSIVE FUNCTIONS We adopt the usual definitions cursive

(id est: general recursive)

Mendelson [60] p. 120-121).

of primitive recursive and refunctions of ~ into ~ (see e.g.

We recall briefly the definitions:

A function f is said to be primitive recursive

iff it c a n be

obtained from the functions Z(x) = O, N(x) = x+l, U~(xl,...,x n) = xi, the initial functions, by any finite number of substitutions and recursion.

If f can be obtained

application of the ~-Operator,

in this way but with some finite then f is called recursive.

A relation R(xl,...,x n) between natural numbers primitive recursive

(recursive)

is said to be

iff its characteristic

function

0 if ~ R(x,,...,x n) XR(Xl,...,x n) : 1 if is primitive recursive

(recursive,

R(xl,...,x n) respectively).

A subset A of ~ is called a recursively

enumerable

(r.e.) set

iff B is either empty or the range of a recursive function. set B of ~ is recursive

iff both B and its complement

It is well known that A is r.e. form ( ~ y)(R(x,y))

(recursive)

R iff f can be obtained from the initial functions stitutions

in the relation

together with

function XR of R by any finite number of sub-

and recursion

respectively).

in the

R (one can allow R here to

be prilm~tive recursive )° A function f is primitive recursive the characteristic

~-B are r.e.

iff "x 6 A" is expressible

for some recursive

A sub-

(and application of the u-Operator,

B) F O R ~ L

THEORIES

A formal theory T is a triple (L,!,~) where ~ is a formal language, C is a set of consequence operations and V a set of sentences of ~ closed under i, called the valid sentences of ~. Here we shall discuss only those theories ~ where ! is the firstorder predicate calculus and ~ is an elementary language.

Such

languages L can be defined as abstract algebras whose elements are just the well formed formulas,

|70] H. RASIOWA-R.SIKORSKI:

see eog.

The Mathematics

Monografie Matematyczne,

of Metamathematics;

vol. 41, Warszawa 1963.

Let K an alphabet consisting of one sort of variables v0, vl,...,Vn,...(n E ~),countably many constants c n, countably many primitive predicates ~n and logical symbols ~ , ^ , A ( n e g a t i o n conjunction,

universal quantification)

v , ~ ~, ~/(disjunction, quantification)

implication,

are definable

the other logical symbols:

equivalence,

in terms of -],^~).

"countable': means "finite or countably infinite". of K the set At K of atomic formulae,

existential Remark that From these signs

the set T K of terms of K and

the set ~K of all (well formed) formulae are obtained as usual. Certain formulae of ~K are called logical axioms

(in K) and in

order to be definite we take those defined in Mendelson |60], p.57. The set C of rules of inference consists of the "modus ponens" and the "Generalization"

(cf. [60] p. 57).

For any ordered pair (~,A) as just described~

, where ~ is a first-order language

and A is a set of sentences

from ~, one can

define the ~Foof relation PRF(~,A): PRF(~,~)(¢,S)

holds iff S is a finite sequence of formulae

~0'''''~n from ~ whose last element ~n is ~ and for i < n, ~i is either an element of A or a logical axiom or there are j,k < i such that ~k ~ ~]• ~ ~ i (application of modus ponens) or there is a k < i such that ~i ~ A ~k (application of generalization). U

We shall write A ~ ¢ for (~ S)(PRF(~,A)(~,S))

and say that ¢ is

syntactically derivable from A by means of logical axioms and the rules of inference modus ponens and generalization.

Define PR(A) : {~;~ ~¢},then V e PR(A) is the set of valid sentences of the elementary theory (~,A). C) ARITHMETIZATION The method of aritl~netization of the syntax of an elementary theory

(id est: first order theory)

times called "gSdelization".

is due to K. G~del, hence some-

The method consists

every formula ¢ of an elementary language ~K natural number y(~)

in adjoining to

(with alphabet K) a

, sometimes written as U¢q,

such that every

natural number which is in the range of y is uniquely "readable" as a formula of ~K' i.e. has a unique grammatical

structure.

The

definition of y is by induction on the number of symbols in ¢ 6 ~ K (id est: the complexity of ~) and is carried out in detail in Feferman [I~]. It follows that the sets of g6delnumbers

of formulae

# from At K and from ~K are primitive recursive and similarly the set {y(t); t E T K} of g6delnumbers

of terms is primitive recursive.

The proof relation PRF as given above can be put into the form of a number-theoretic

function prf and it can be shown that prf is

primitive recursive Definition:

in ~, see Feferman [15] p. 44.

A first order theory ~ ~ (~,~) there is a r.e.

is axiomatizable

iff

subset A of V such that V e PR(A).

A is called a set of axioms for T. If there is such a finite subset A, then ~ is called finitely axiomatizable. W. Craig has proved that if a first order theory T has a r.e. set of axioms it has a primitive recursive "On axiomatizability

set of axioms

within a system", J.S.L.

(see his paper:

18 (1953) p.30-32).

Let ~, e ).

For a rigQrous formulation

of Prf e see Feferman [~5]~or p. 104 in: [63] R. MONTAGUE:

Fraenke!'s Addition to the axioms of Zermelo; Fraenkel-Festschrift of Mathematics",

"Essays on the Foundations

J e r u s a l e m 1988, 2nd Edition,

p. 9 1 - 1 1 4 . Now Cou~ can be defined to be the following formula: nVV xy Come is a sentence

(Prf (x ^ nx,y))

(of Peano-Arithmetic)

expressing that the set of

formulae represented by ~ is consistent. The following theorem is a generalization of GSdel's second underivability theorem 1957).

(1931) which is due to S. Feferman

Let P be the axiomatic

system of Peano-Arithmetic

(Thesis, (see

[1~]

p. 50). A formula ~ of P defines the set A of natural numbers in P iff ~ has one free variable and for every n e ~: p ~ ¢(n) if n E A, and:

P ~ 7¢(n)

Theorem

if n ~ A. Here n is the n th numeral of P.

(GSdel-Feferman):

Let A be a consistent

set of elementary

formulas, and let e be a primitive recursive

function which

defines A in P. Then P + Con a is not relatively in A

(see Feferman [15] p. 90).

interpretable

D) SYNTACTICAL MODELS A semantical model

(or mostly simply:a model) of a 1St-order

theory T is given by a (meta-mathematical)

set ~ in which some

relations are defined such that the set V of "valid sentences of T" are all "true" in J. In contrast to this a syntactical model of T is given by a transformation

into another ist-order theory T*.

Here is required that the transformed

"valid sentences of T"

are all "valid sentences of T*". In the case that T and T" are axiomatized by A and ~* respectively this means that the transformed axioms A of T have to be derivable from A*. The notion of a syntactical model is very old and due to many authors. A, Tarski has called them "Interpretations", is by the procedure mentioned above "interpreted" [8~] A. TARSKI-A.

MOSTOWSKI-R.M.

ROBINSON:

since

in T*, see:

Undecidable

theories;

(studies in Logic), North Holland Publ. Comp. Amsterdam 1953.

[87] HA0 ~ N G :

Arithmetic

translations

actions of the Amer.

Math.

of axiom systems~ Soc. vol.

Trans-

71 (1951) p. 283-

293.

The classical notion of "Interpretation" one has been generalized

of one theory in another • n s " by to "parametrical interpretatlo

Petr H~jek: [29] P. HAJEK:

Syntactic models of Axiomatic Theories~

Polon. [30] P. HAJEK:

Bull. Acad.

Sci., vol 13 (1965) p~ 273-278.

Generalized

Interpretability

Note to a paper of R. Montague;

in terms of models Casopis pro pSstov~nf

matematiki vol. 91 (1966) p. 352--357. H~jek has defined his generalized notion of Interpretation only for finitely axiomatized theories and used the name "syntactic models" for it. W e shall follow H~jek but give the definition for arbitrary r.e. axiomatized theories

(the definition can obviously

be stated even more generally by writing E [2 instead of ~2 ~ etc.). In order to simplify the notation we assume henceforth that the alphabe t K of a first order theory T consists of only one sort of variables

v0, vl,...,

signs ] , ^ , A letter).

a sequence of primitive predicates wi and

(hence K does not contain any constant or any function

This assumption

is no loss of generality.

In the following definition let ~l and ~2 be axiomatized I st order theories such that {Vn; n E ~} is the set of variables of T, and {Wn; n 6 ~} the set of variables of ~2 and let I be an index set such that {~i; i e I} is the set of primitive predicates of ~,. n i will be the number indicating that ~i is ni-ary. Definition: Let ~i

m

< ~ 1 ~ I > and ~2 ~ < ~ , ~ 2 > be I st order theories.

The set of formulas of ~2:

6(wl,...,Wk), ~(wl,''',W k, Wk+l), ~i(wl,...,Wk,...,Wk+n.), i E I, l is called a translation from ~i into ~2 iff the following three conditions are satisfied: (1)

If wj is a bound variable in ~i' then j is odd;

(2) A2 ~Vw/--Vw k e(w~,---,wk); (3)

A2 ~ n w , ' ' ' A W k [ O ( w 1 " ' ' ' W k )

~VWk+l

$(w,'''''Wk+l)]"

Since Ocharacterizes the parameters we shall call 0 the "parametercondition"~ ~ defines (in dependence to the parameters) the objects of the model, hence we shall call ~ the "model-condition". The open formulae ~i will be the images (in ~z) of the primitive predicates ~i

(i 6 I) of T,. Condition (I) is required only in order to avoid

confusion of variables in the next definition (othe~lise we would have to care about a suitable changement of the bound variables in 9.. l It seems to be convenient to introduce a replacement operator rep. Therefore let T ~ (0,¢,{~i ~ i e I}) be a translation of ~, into ~z and let h be the following mapping from the set of free variables v0, v], v2,.., of ~i into the set of free variables we, w,, w2,... of ~z given by: h : vj ~ w2(j+k) (k is the numbe~ of free variables in 0 and is therefore uniquely determined by 7). The definition of repT(F) for F e ~l is by induction on the length of F. Remark that if vj is free in ~i, then h(vj) is a free variable of repT(~ i) (this is guaranteed by the

curious condition

(1) from the definition above).

Definition (of the rep-0perator with respect to a translation T): (i) If F is an atomic formula of the form vj, = vJ2' then repT(F) ~ (W2(k+jl) : W2(k+j2)); (ii) If F is an atomic formula of the form ~i(vj l

,...,v. ), then ]n i

repT(F)

. ); ~i(wl .... ,Wk,W2(k+jl ) .... ,W2~k+]ni)

(iii) If F is of the form F, ^ F2, or 7F,, then repT(F) ~ rep~(F I) ^ repT(F2), repT(F) e -h?epT(F ,) respectively; (iv) If F is of the form A

vj F',

then

repT(F) -- A w 2 ( k + j )[$(Wl,...,Wk,W2(k+j))

~ rePT(F*)].

The action of repT can be described briefly as follows:

in

the formula F of ~i every ni-ary primitive predicate ~i is replaced by ~. and all quantifiers are restricted to $. Remark that the l image of a sentence F under repT is again a sentence if and only if 0 is a sentence, id est: if the translation T is without parameters. Definition. A translation T ~ (0'$'{~i; i e I}) from T ~ (~,,Al) into ~2 e (~2'~2)is called a syntactic model of T, in ~2 iff for every ~ 6 ~I the following holds: ~2

~ Aw,""

AWk[~(w1~''''Wk)

~ repT(¢)]

T is called parametric if k ~ o~ parameterfree otherwise. It is easily seen that the notion of a syntactic model (~ la H~jek) is a generalization of the classical notion of Interpretation, because, if J ~ (¢,{~i; i ~ I}) is an interpretation of ~l in ~2 (in the sense of Tarski [85], and F any sentence, provable in ~2, then (F,¢,{~i;

i E I}) is a (parameterfree)

syntactic model of T, in ~2"

Obviously the following holds: if T is a translation from ~I into ~2 and ~ e L a logically true formula (i.e. ~ ~ ~), then --1

rep~(~) is logically true. This enables to prove by means of the deduction theorem and the finiteness-theorem the following lemma. Lemma. Let T--~(0,$,{~i; i ~ I}) be a syntactic model of TI-~(L,,A,) in T 2 -~ (L 2,A 2) and suppose that A, ~ @. Then A 2 ~

Awl...

Awk[ §(w~,...,wk) ~

rapT(@)] •

The importance of the notion of a syntactic model is contained in the following theorem. Theorem. Let ~l e {~l'~l ) and ~2 e (~2,~#

be first order theories

and suppose that ~2 is consistent. If there exists a syntactic model • of ~l in ~2' then ~, is consistent too. Proof. Assume that T is inconsistent. Then there is a formula F -i of ~i such that A, ~ F ^ 7F (and in particular A, ~ F). Hence by the preceeding lemma:

F A w I .. " A w k |8(w I ,''''Wk ) ~ repT(F ^ ]F)] ' and

(o)

~2

(oo)

~2 5 A w , ' O " Awk[8(w1'''''Wk) ~ rep (F)].

By (2) of the definition of "translation" we get from (oo): (+)

~2 ~

~/w,''" Vwk[O(wl ....

'Wk) ^ rapt(F)|"

From (o) we get using clause (iii) of the definition of "repr""

~2 F

Aw,..,AWk[

(8 v rep~(r)) ^ ( q e ~ q r e p

(r))].

We a~e interested only in the second member of the conjunction and get:

(+) and (++) show that ~2 would be inconsistent too, a contradiction to our hypothesis. Theorem (S. 0rey). Let T, and ~2 be axiomatie 1st order theories, ~2 be reflexive and ~2 contain Peano Arithmetic. If for every finite subsystem D of ~I we have, that D has a parameterfree syntactic model in ~2, then T, has a parameterfree syntactic model in ~2"

10

For a proof see Feferman [15] p. 80. A theory ~ is called reflexive iff the consistency of every finite subtheory ~ of T can be proved within T. Further results on syntactic models of set theory will be contained in the following chapters. E) ZERMELO-FRAENKEL

SET THEORY

We call ZF (set theory of Zermelo-Fraenkel) theory with identity whose alphabet,

a first order

formulae and axioms are

defined as follows: The Alphabet of ZF: One sort of variables x0~ x , , . . . , X n , . . . ( n ~ ) (x,y,z,...

stand for these variables)~

a binary predicate

e,

and logical symbols: ] , v , V

, = and brackets.

The Formulae of ZF: xey and x = y are (atomic) formulae~ and if and ~ are formulae, then Vx(~)

~($),

[#) ~ (~) and

are formulae.

We follow the us~ai conventions which allow us to omit brackets in some cases (see Hilbert-Ackermann

p. 74).

The axioms of ZF: (0) Null-set: V x A y [ 7yex]. (I) Extensionalit~ AxAy[

Az(zex

(II) Axiom

(Axiom der Bestimmtheit): ~ zcy) ~ x = y],

of pairs:

AxAyVzlA

ncu~z ~ u

(III) Axiom of unions

: xv

u = y)l,

(Sums):

Ax VyA~lzcy " Vu~ ~xl, (IV) Axiom o f

Vx (V)

infinity:

Vy~y~x

Power-set

AxVyAzlzey

^ Az(z~x

~

VuCu~x

(Potenzmengen-Axiom) :

~ Au(u~z

~ uex)l ,

^ u ~ z ^ AvCv~,

~ v~u)))

,

11

(VI~) Axioms of substitution Ax~y¢(X~y)

~ A a VbAy[yeb

(vii) Axiom of regularity Ax[

VyysX ~

In the schema

(Ersetzungsaxiome) ~

Vx(Xea

^ #(x~y))].

(Fundierungsaxiom)

Vy(yex

^ Az(-]zsy

~ 7zsx))].

(Vl¢) the formula ~ is supposed not to contain any

free occurence of the variable b. (VI~) implies the axioms of subsets (Aussonderungsaxiome). The system of axioms consisting of axioms

(0) - (VII) is called ZF. The subsystem

(0) - (VI%) is called Z ~ . Hence

ZF ~ ZF ° + "Axiom of regularity". Sets, totalities

and classes have always played an important

r61e in mathematics

though this has not been recognized explicitly

It was Georg Cantor

(1845-1918)

ever used properties mathematics

who has formulated e x p l i c i t l y

the

of sets. According to the methodology of

he investigated the interrelations

ties between these various properties.

and interdependen-

This lead to the socalled

Set Theory. [ 8 J G. CANTOR:

Gesammelte Abhandlungen Springer-Verlag

(Edited by E. Zermelo),

Berlin 1932, Reprinted 1962 by

G. Olms, Hildesheim. Cantor's

Set Theory can be viewed as a consequent

transfer our comtemplations

attempt to

with respect to the domain of finite

sets to domains

including infinite sets.

Exempl& gratia:

the principle,

that the image of a finite set is

again a finite set, yields the axiom of replacement

(Vie). On the

other hand there are principles which cannot be translated,

such

as m < m + 1 for finite cardinals m. Hence it is not too much surprising that between these extremes there is a large class of assumptions

for which it is hard to sayw~ether they are true only

when restricted to finite sets. A very classical example for this situation is the axiom of choice: (AC)

Ax(Ay(yex~

y ~ @) -~ V f ( F n c ( f )

^ Ayexf(y)gy))

There are other problems which do not have an a~alog~e in the domain of finite sets, such as the continuum-problem.

But as in the case of

12 the axiom of choice (AC) our questions are wether the Cantorian principles, which are abstracted from the domain of finite sets, are sufficent to decide them. The original Cantorian set Theory [as based on the "Ideal Kalk~l", whose axioms are extensionality and comprehension ~x(AyycX

~ ~(x))] was contradictory (take Russell's predicate

xex). Since 1908 many axiomatic systems for set theory have been given which seem not to give raise to the Russell-antinomie. Roughly they can be divided into two groups. The first group are type-theoretic foundational systems, such as the Principia Mathematica PM of Russell-Whitehead, the systems NF ("New Foundations") and ML ("Mathematical Logic") of Quine, the systems of N. da Costa (Indagationes Math.

27 (1965)), J. Houdebine (Th~se,

Rennes 1967). The second group are Extensions of Zermelo's set theory (1908):

[88] E. ZERMELO: Untersuchungen ~ber die Grundlagen der Mengenlehre; Math. Annalen 65(1908) p.261-281. [89] E. ZERMELO: Uber den Begriff der Definitheit in der Axiomatik; Fund. Math. 14(1929) p.339-344. such as Johann yon Neumann~ System (Hath. Zeitschrift 27(1928) p.669-752, and J. f~r die reine u. angew. Math.(Crelle)154(1925) p.219-240), the system of Bernays (contained in a sequence of 7 papers in the J.S.L.)vol.2,6,7,8,13,19, a modified version is contained in his book, Amsterdam 1958), the version in G~del's monograph (Princeton 1940), the ZF-version of Bourbaki, Skolem, Thiele, Sonner and others. One of the nicest foundational system has been given by: [~]

W. ACKERMANN:

Zur P~iomatik der Menge~lehre; Math. Ann.131

(1956) p.336-345. Since we will not discuss this system~ we refer the reader to the following articles:

[~]

A. LEVY:

On Ackermann's set theory~ JSL 24(1959) p.154-166.

[56J A. LEVY - R.L. VAUGHT: Principles of partial reflection in the set theories of Zermelo and Ackermann; Pacific J. Math. 11(1961) p.1045-1062. [~]

R. GREWE:

On Ackermann's set theory~ Doctoral Dissertation, University of California, Los Angeles 1966.

13

[~8]

R. GREWE:

Natural JSL

[71]

W.N.

models

34/1969)

REiNHARDT:

Topics

Thesis,

of Ackermann's

p.481-488. in the Metamathematics

University

Reinhardt's Coincides

Set Theory~

of Wisconsin

Abstract:

with ~F"

of Set Theory

19677

"Ackermann's

See also

Set Theory

in the AMS-Notices,

vo!.13(1966)

p.727. Zermelo's plus the

original

system

Z contained

only the axioms

AxVy[Az(ZCy The "Fundierungsaxiom" duced by A. Fraenkel Zermeloschen

~Grenzzahlen

We assume

subject,

and the "Brsetzungsaxiom" (in his article: Math.

the whole

system

that the reader

of ZF-set

theory.

such as J. Rubin:

(0) - (VII)

(Fund.

Comp.

Amsterdam

Ordered

is similarly

is the open formula

are defined

defined

of f and pr2(f) respectively.

for ZF-formulae by Church's

(Holden-

(North-Hol-

~.This

is the cartesian

pairs

than pr,(s)

on the second

yields

of ZF the set of logical

conversion

pr,(f)

written

use abstraction

a definitional

terms

extension axioms

is

principle:

x c {y; Cry)} ~ ~(y/x) where y is the free variable tained

from ¢ by substituting

currences

of ¢ and ¢(y/x)

is the formula ob-

x for y at all places

of y in ¢. It is supposed

s}.

coordinate.

of f, sometimes

freely

is

pr, (s) = ( X ~ V y < X , y } e

thlt f is a function,

the range

We shall

the intersection,

x × y

~ la Kuratowski:

as projection expressing

In such an extension

enriched

x N y

and

If s is a set of ordered

pr2(s)

of ZF.

of x,

of s to the first coordinate:

Fnc(f)

{x; ~(x)}

Set Theory

1968).

pairs (x,y)

= {{x},{x,y}}.

on this

conv~entions:

is the power-set

Rg(f)

on

p.29-47)

for the m a t h e m a t i c i a n

x - y the difference

the union,

is the domain

in his article

16(1930)

with the classical

- A. Mostowski:

For sets x,y,P(x)

Dom(f),

p.230-237).

are many textbooks

x U y

the projection

Math.

is familiar There

Set theory

We shall use the following

product.

86(1922)

der Cantor-

it ZF.

Day 1967) or: K. Kuratowski land Publ.

(VI¢) are intro-

Zu den Grundlagen

Annalen

und Mengenbereiche"

and has called

development

~ zcx ^ ~(z))].

Mengenl~hre,

Zermelo has adopted

(x,y)

(0) - (V)

"Aussonderungsschema":

of free oc-

that x is free for y in ~.

as

14

Let ~ and B denote define

Ordinalnumbers

(~ l a v .

Neumann).

Re(x) = U{~(RB(x)) ; 8 < ~}. For x = ~

k n o w ~ v. Neumann's The M i r i m a n o f f - r a n k

"Stufen"

the following Lemma:

p(x) of a set is defined

V

= Re(~).

to be the first

V e and, the r a n k - f u n c t i o n

p have

properties: _c V6,

1) ~ ~ B ~ V 2) p(~)

= e,

3) p(x)

= U{p(y)

4) If xEyeV e 5) x~V

one gets the well-

and we shall write

such that g e Ve+ 1. 'The "Stufen"

For any set x

+ I; y c x},

or

x _C ycVe,

then xeVe~

o O(x) < e,

e

6) Ve+l

= p(ve),

7) V~ = U{V

; e < X} if % is a limit

8) If A is any set of ordinals, dinal B. 9) x C V --

then

ordinal, U V a = v B for some orseA

p(x)'

10) x ~ Vp(x)+l

.

Proof by induction. Note that V = U{ve; sets,

e an ordinal}

is in ZF ° equivalent

The class ~ J v

can be used to construct

This shows the relative consistency the lemma in section D).

F) THE PRINCIPLE

ReI(C,~)

/~x~

~/x~

ZF in Z ~ .

to Z ~

obtained

and a class-term

from ~ by restricting

in ~ to C ~ id est~ by replacing

change

") insert:

a parameterfree.)

of ZF with respect

~ of the ZF-language

be the formula

or

phabetic

V is the class of all

(see

OF REFLECTION

For a given formula quantifier

, where

to the axiom of regularity.

by A x ( X C C

+ 4) or

of bound

variables

syntactic

model of.

Vx(xec

C let

every

each occurrence

of

^ ~), respectively

may also be needed).

(al-

The rigorous

15 definition

(by induction on the length of ¢) of R e I ( C ~ )

given ana!oguously Definilion:

can be

to our definition of repT(,#) in section D.

Let ~ and C be given. % is convenient for C iff the following is provable in ZF:

(.)

Ax, cc

A x ~c[%(xl

. . .

. . . . .

xn)

~ Rel(C~%)]

n

A partial ordering ~ between ZF-formulae

in pr@nexform is defined

by 01 ~ 0z iff there are finitely many quantifiers V x, ~ . . . ~ V x k such that 02 is obtained from ~l by putting in front of 01 negationsymbols

and these quantifiers ~/x i in a certain order.

known that every formula is equivalent to a formula set of free variables) Definition:

Lemma:

in prenex form.

~ in prenex form is hereditarily (,)

It is

(with the same

convenient

for C iff

holds not only for 0 but also for every 9 ~ 0-

Let 0 be a ZF-formula in prenex form and let {Xi; i e m} be a nested sequence of sets (i.e. X i C Xi+!). rily convenient for X : .U X. 1C~

i

If ~ is heredita-

for every Xn, % is hereditarily convenient too.

Proof (by induction on the length of ~): If 0 does not have quantifiefs, then the lemma is immediate. 9

is obvious.

The case where ¢ has °the form

Now suppose 0 has the form ~/x~(X,X, .... ,Xn). By

assumption 0 is hereditarily

convenient

holds for ~. By induction hypothesis for X. Now (.) follows immediately Theorem:

(R.Montague

- A.L@vy):

for each X n. Hence the same

~ is hereditarily

convenient

for ~/x~ and the lemma is proved.

Every instance of the following

schema of complete reflexion is a theorem of ZF: (CR*)

f~Vs[Lim(8)

^ e < 8 ^ x'eVsi .... XneVs/~ [ % -- ReI(V 8,%)]]

Here e and 8 are Variables ranging over ordinals that 8 is a limit ordinal~

and Lim(8) expresses

i.e. 8 ~ 0 ^ 8 = US. It is understood

that xl ~ .... x n are precisely the free vamlables of #. Proof:

It is sufficient

to prove the theorem only for the case

that 0 is in prenex form.

If ~ is without quantifiers~

is obviously true for 8 = e + ~. The case 0 ~ q ~ trival.

the theorem

is also very

Hence let us suppose that ~(xl ~...,x n) has the form

V x g ( X , X , ~...~Xn).

By the induction hypothesis

a limit-ordinal

8 such that 9 is hereditarily

there is for every convenient for V 8.

Let F(y)xl ~...~x n) be the formula saying that y is the set of sets x

16 of minimal rank such that ~(x,xl ~...,Xn). sequence

of ordinals

Define a countable

80 < 81... < 8 n < ... in the following way:

Let 80 be the first ordinal > s such that 9 is hereditarily convenient

for Vs0.Having

82n+i = Min{y;

defined

8 i for i ~ 2n, put

A XngV82n [ r(y,x,

82n < y ^ A x l e V B 2 n " "

.... 'Xn

) ÷

ycV~l }.

Now let ~2n+2

be the least ordinal > 82n+I

rily convenient

for V82n+ 2. Obviously

such that ~ is heredita-

8 = lim 8 n new

is a limit

ordinal and

v8

: n eUw

v 8n

: n eUw

vs2 n

(see the lemma in section E). In order to show that V 8 is convenient for ~x~(X,X,,...,Xn) , let a,,...,a n g V 8 are elements

of partial universes

Vsi's be contained

be given.

Hence the ai's

Vsi and let these finitely many

in V82 k. If ¢(al ~...,a n) holds,

then Vx~(X,a,,...an )

holds too. Hence Vx[Vy

xey ^ F(y,al ,...,a n ) ^ ~(x,a, ,...,an)]

holds too. It follows

from the definition

an object a in V82n+ 2 such that V82n+2 ~ V8~ a ¢ V~ that R e l ( V s ~ ( a ) ) hereditarily

,it follows

holds

convenient

because

Since

from the induction hypothesis~

(remark that by the construction The direction

are restricted

if Rel(Vs,#)

that there is

for V2n~ hence by the previous

V 8 too). Hence Rel(VB,#). free variables

of 82k+I,

~(a,a, ,...~a n) holds.

ReI(VB,¢)

÷ ~

~(a) is

lemma for when the

to range only over V 8, is obvious~

holds in VS, then there is an object a a V 8 such

that R e i ( V s ~ ( a ) )

holds in V S . By the induction hypothesis n with a e V8 a n ) holds for every sequence (al ,...,a n) ¢ Vs,

~(a,a,, .... Hence V x ~ ( X , a , , .... a n ) holds too for every sequence s + 1. We shall show that ord(e) the first ordinal

we obtain (in both

= s + I. Let s0 be

such that so ~ Ms0+1.

8 6 MS+I, which is 8 6 Def(Ms).

so C ~ { M ~ ;

8 e so} = Ms0.

In

order to show that so 6 Mso+l holds, we have to show that so is a definable subset of M

se

. Consider the formula ¢(x):

u, Au%V

U^

U E X ~- V E X] (x is totally ordered by 6 and transitiv). subset A of M

O.o

. Since M

so

is transitive

#(x) defines a certain

and e is interpreted by

the actual membership relation E, A must be precisely the set of those ordinals, which are in Mso, But these are by definition at least the elements tPansitivity

~ of so. If y 6 M

fop y > eo, then by the so of Moo, so 6 Meo which cannot happen as it was shown

above. Hence A consists precisely of the elements of So.Id est e0~-A and ~0 is definable

in Ms0 and hence an element Of Ms,+1.

If s0 is a successor ordinal, Since S 6 MS+ I = Mso

say so = 8 + I, then s0 ~ 8 U {8}.

(by minimality of se), hence definable, we

have obviously s0 6 Ms0+i too, q.e.d. Since M0 = 0, hence @ 6 M,, the axiom of Null set is true in L. Since extensionality holds in ~ elements

are constructible

and elements of constructible

too, the axiom of extensionality holds in

L. The axiom of pairs is also obviously true in L. Lemma: The axiom of sum-set holds in L. Proof:

Let a be a constructible

set and s an ordinal o f ~

a 6 M s. Since M s is transitive b = U subset) Me

in M s. The formula

Vv(Y

such that

{x; x 6 a} is contained

(as a

e a ^ x ~ y) with constant a from

defines b as a subset of Me. Hence b E Def(M ) = Ms+ 1 C L.

Lemma: The axioms of regularity and infinity hold in L.

28

Proof.

We have shown,

that every

Since the axiom of Infinity But ~ satisfies regularity

ordinal

holds

in~and

is constructible.

in~1~6, m is i n ~ ,

the axiom of Infinity

holds

of~

hence

in L.

in L. Since the axiom of

L C~and

the epsilon-relations

of L

and~

coincide,

the axiom of regularity

Lem~a:

The power

set axiom holds

Proof.

Let a be a set of L and b be the set of c o n s t r u c t i b l e

in L.

sets of a. Since a is a set o f ~ of a in the sense o f ~ . hierarchy mapping

Me, MI,...

x ÷ ord(x)

of replacement

, b is a subset

Therefore

Ms,...

f r o m elements

and one constant

/~u(U

Define

of b to ordinals.

{ord(x);

bound =. Hence b C M e_. Let 8 = Max{e, a E M e . The formula

ing~).

sub-

of the pQwerset

b is a set o f ~ ( s i n c e

was defined

is true i n ~ ,

must hold in L as well.

the in~

the

Since the axiom

x-E b} has a least upper ord(a)},

then b _C M 8 and

e x ÷ u e a) with one free variable

a from M 8 defines

b as a subset

b 6 MS+ I and b is constructible./~he

x

of M 8. Thus

proof of the following

lemma

I

uses techniques 1961 paper,

which

are essentially

rediscovered

The axiom of replacement

Proof.

Consider

y and parameters x in L there

holds

in L.

¢(x,y,a, ,...,ak ) with

free variables

a~,...,a k which are in L. Suppose

is precisely

one y in L such that #(x,y,!~,..oak).

sets y such that there

are sets x with

in~

Max{e,

ord(ak)}.

ord(a),

most simplest refection (GRP):Let

ord(al),...,

We shall

show that b 6 M~o+I.

way by applying

(in~)

{ord(y);

e. Define

y E b} has 80 =

Then a, al,...,a k 6 MB0

and

This can be done in the the following

generalized

principle: {We; e an ordinal}

that for I : [ J ~ W = ~]{We;

be a nested

it holds

e an ordinal}.

ZF, then for every ~

a supremum

and x 6 a.

element y of b is construc-

tible and since b is a set in the sense of~Y~6, by the axiom of replacement

of construc-

#(x,y,...)

We have to show that b is c o n s t r u c t i b l e . E v e r y

b C MSo.

x and

that for every

Now let a be any set in L and let b be the collection tible

in his

in 1963 by P.J.Cohen.

Lemma:

a formula

due to E.J.Thiele

ordinal

sequence

that W l ~ ~ ] { W e ;

e < k}. Let

If #(x, .... ,x n) is a formula of ~ there

is a 8 > ~ such that

A

/\;./\nLX,,,Xn

of sets W e such

C el W, , "

29 It can be shown exactly in the same way as we have proved the principle of complete reflection

(CR*), that every instance of

(GPR) is a theorem of ZF (from the hierarchy V e we needed only the facts~ that for I a limit ordinal: ~

~

÷

C_ VS)

V

holds i n ~ 6 .

.

Sinee~

Vl = U { V e ;

e < l} and

is by assumption a model of ZF, (GPR)

We shall apply (GPR) for the hierarchy M e and the

formula #(x,y,xl ~...,x k) with free variables x,,...~x k replaced for the constants a, ~...~a k (alphabetic change of bound variables in ~ may be needed.

Hence for the ordinal 8o previously defined

and # there is and ordinal Yo > 8o such that A

A

A ~..~ [x,y,x, ..... Xk e Myo ÷ ( R e I ( L ~ ) " Rei(Myo,~))]. y x, k Since ord(al) ~ 80 < Yo,...~ ord(a k) ~ 80 < Y0 we have that al ~...~a k are elements of My0 ; hence we obtain:

For ¢(x~y,a, ~...~[ k) let us simply write 9(x,y) The interpretation of #(x~y)

in the sequel.

in My0 consists of those pairs

(x~y) 6 My° x My ° such that Rel(M~o,~(x,y)).

By the last formula it

is equivalent to say tha%the interpmetation of ~(x,y) in M consists of those pairs (x,y) 6 M 2 such that ReI(L@ (x~y))~°Hence Yo the formula Vx(X

e ~ ^ ~(x,y))

with one free variable y and constants a~a ''''~!k defines the set b in My°. Thus b E My0+1 and b is constructible, Remark.

The proof given above that (L,e)

carried out informally.

q.e.d.

is a model of ZF is

A detailed proof defines first i n ~

language £$Ttwhich contains stant x for each set x o f g ~ .

a

besides the usual ZF-symbols a conIn order that £ ~ i s

a class of

one has to arrange that all the symbols of the alphabet of £9T5 are sets o f T ~ . K . B o J e n s e n

in his lecture notes (Springer~

"Modelle der Mengeniehre"

gives a detailed solution.

1967)

The satisfac-

tion relation ~ between structures (s,e ~s ) (which are sets off*g) and formulae of £~% can be defined in gT6(see Mostowski's "Constructible sets"). Then the class L can be defined i n ~ transfinite recursion.

by means of

30 In order to show, that < L, 6 ) is a model of (AC) and (GCH) we shall show that a much stronger axiom, the soealled axiom of eonstructibility" implies

V : L", hol~s in (L,£ > and that "V = L"

(AC) as well as (GCH).

So far we have only shown, that

"L" can be defined within any given standard model~TL. shall show that the statement set is eonstructible:"V (pure) ZF-language,

Now we

informally expressed as "every

= L" can be expressed by a formula of lhe

so that V = L is a certain ZF-sentenee.

We

shall prove a stronger faet~ namely that V = L is in ZF equivalent to a E,-formula

Defintion.

(this result is due t o

C.K&~p~]).

(A.L~vy): A formula ¢ of the ZF-language

is a E0-for-

mula iff ¢ contains no unbounded quantifiers; iff ¢ has the form

V

¢ is ~,,

.~ with ~ a 7.o-formula; ¢ is ~I

iff ¢ has the form

A X ~ for a To-formula ~. If a ZFx formula F is in ZF (provably) equivalent to a Z0 (ZI ,HI)

formula,

then r is called a ~ZFc~ZF,nZF

respectively)

formula,

r is A zF iff r is both ~Zr and n zF

Lemma I: If ¢ and ~ are zZF, then so are ¢ ^ ~, ¢ v ~,

A(x~ Proof.

~¢)

By hypothesis:

and

V(×~ z

ZF ~ ~ ~

Vx¢ ,

^ ~).

V y ~ with ~ o .

We have to show that

ZF ~ V x V y $ ~ V z ( V x e z V y c z $ ) . The part "÷" is obvious; for the 'part "÷" take z : {x~y},z a Variable not in $. Hence V x ¢ is ~>F. Now it follows that also ¢ ^ $ and ¢ v ~ are 7.ZF. Next we show that :

The part "÷" is obvious.

Ad ~'~": assume x,y are free in ~. Define

a function F: F(x) = {y; $(x,y~...) ^ p(y) : Min{~; V v ( P ( V ) : ~ ^ ~(x,v,...))}} By the axioms of sumset and replacement (.) follows with : U{F(x); l ZF

x e z}. It remains to show that

Vx(x

s z ^ ¢) is

f o r ¢ -~ Vy~ with SZo. But x e z ^ ~ is Zo, hence V ¥ ( x 6 z ^ ~)

is E, and therefore

Vx( VyX

e z ^ ~) is l ZF as was shown above.

3! C o r a l l a~ r y

2: If ¢ and ~ are A ZF , so are ¢ ^ ~ ¢ v ~, ¢-* ~, A x ( X £ Z-~ ¢) and V x ( x e z ^ ¢).

Lermna 3: If ZF ~ Vz V n A x ( ¢ ( x ) - ~ (a)

Ax(¢

(b)

V __x(~_ ~

where Proof.

÷ ~ ) -- A ~) ~

"~" obvious.

the lemma follows

"+":

yt ez

Vy

it is supposed

x e Unz),

gz

yneYn_ 1

AXeyn(¢(x)-+

~)

... V y neYn_l V x e y n(¢-* ~)'

that y, ''''Yn are not free in ¢,~.

since x e L j n z o

Vy, e z . . . V y n e Y n _ l ( X e Y n )

immediately.

Lemma %: The f o l l o w i n g

ZF are E0 -formulae:

formulae

(2): y : {x,z}, y : ~]x,

y e Dom(f),

(6): y : x-z,

(8): y = Rg(f),

y : f ~ z, (I0): y : f"z, Fnc(f),

(I): x = y,

(3): y = (x, , .... x n ) , (4): y : x n z,

(5): y : x ~ y, y : LJx,

y = Dom(f), (13):

..A

then

(ii):

(7):

y e Rg(f),

(9):

x : y x z, (12): y : f-l,

(14): y = x U {x} and

(15):

Ord(x)

(x is an

ordinal ). Here (x, ,x2 ) m e a n s Dom(f)

the ordered

the d o m a i n of. f, Rg(f)

of f to z, f"z = {f(y); x cartesian (1) and

product

anaioguous

and Fnc(f)

(9),

expresses

(3) follows

(10)~

differenee~

the range of f, f ~ z the r e s t r i c t i o n

(8) are obvious,

to (7);

- set t h e o r e t i c a l

y s z} the image of z under

(2) are immediate~

lenhma 3% (5) and

pair,

f for z C Dora(f)=

that f is a function.

from

(7) uses

(2),

(4) follows

from

(3) and lemma 3~ (8) is

(12) and (13) are proved by means of

lenm~a 3. In all cases use a p p r o p r i a t e

defining

formulas

(see C.Karp

[4z] e g ) Lemma

5: If ¢(y~x, ..... x n) is l ZF and Z F ~ A then

¢ is

Proof o Since ZF ~ n is ~ZF. Lemma

AZF.

~ ~

• ..

x,

A

~(y,x,

xn

~..,X

n).

y

V z ( Z ~ y ^ ¢(z, .. )), n ¢ is EZF ~ hence

6 : S.u b.s t i.t u.t i o n s p rinciple" . Let G~ ~ . . . ,G n be m - p l a c e

functions

such that Yi = Gi(xl '''" 'Xm) is E ZF. Then if (ul ~... ~u n > eR is A~ ~ (G, (x, ,... ~x m) .... ,Gn(X, ~... ~x m) } e R is A ZF too. 1

52

Proof.

By lemma 1 the proof of the claim follows from

E R . V y . . . V y n ( (b,g~b,y>~

there is a function f and a transitive

~{D,e~D,y).

Since

Vx~(X,y)

holds in

it holds also in ( D , g [ D , y > since ~ is z~F~ hence:

ReI(D, V ~ ( x , y ) )

and there is an red such that Rel(D,%(v,y))° Since E0ZF we conclude, as before, that #(v,y) holds.

D is transitive and %

It remains to show that HC(v) ~ Max{M0,HC(y)}. HC(v)

= ~

But

~ ~ = ~ < Max{M0,HC(y)}

since veD, hence C(v) C D since D is transitive,

further ~ = ~ since

f is a bijection from b onto D, q.e.d. Theorem (GSdel):

In ZF + (AC) it holds that for every initial ordinal

> m it holds that P(e) ~ L C L(e+)~ wheme P(~) is the (whole) powerset of e and ~+ the next cardinal after e. Proof.

First remark that every ordinal is constructible

(same proof

as in section B). If yeP(~), y C ~, then ~ ~ e and HC(y) ~ ~. ZF Consider the formula Ord(x) ^ ysL(x). Since Ord(x) is E 0 and ZF ygL(x) is At (see lermma 4 and corollary 11 in sections C and F), ZF the formula under consideration is E z (by lemma 1 in section C). If yeP(a) N L, then

Vx(0rd(x)

x[HC(x) ~< Max{M0,HC(y)}

^ yeL(x)) and hence by T" ~evy's theorem:

^ 0rd(x)

^ yEL(x)] . Since HC(y) < e and

~ e, we get HC(x) ~ ~. But x is an ordinal, hence ~ ~< ~. Thus yeL(x)

for x < e+. The sequence L(o),L(1),...,L(B),...

hence L(x) C L(e +) and therefore ysL(~+), Co__rrpllary i. ZF Coroilary Proof.

~

2. ZF ~

is nested,

q.e.d.

V = L ~ for every initial ~, P(~) ~ L(e+). V = L ~ (GCH).

We know already that

(AC) follows from V = L in ZF. Hence

38

given any set x~ x is equipollent to an initial ordinal ~ and P(e) g L(e +) implies ~(x) : P ( ~ P(~)¢L and therefore P(~)EL(~)

satisfies ZF + (AC) Proof:

Consider

~ Rep(F,

the c o r r e s p o n d i n g

satisfies

of

the a x i o m of choice,

the a x i o m of choice too,

id est:

of s are not F-empty and pairwise that

~'notion in the sense of the model ~' is meant,

id

Define

a* is a set of non-empty~

e y}; F(y) pairwise

g a} disjoint

is a set b~ such that:

the condition:

sets

(with respect

b* N y is a singleton

each y s a*. Now b : {F(x); x e b*} i s a satisfies

a permutation

"F" in front of a qotion N indicates

a* = {{x; F(x)

By (AC) there

defines

repla-

is proved.

a set s such that s is ~'not empty" with respect to

(the suffix

est: Rep(F,N)).

the r e l a t i v i z e d

(AC)).

6 F and such that the F-elements F-disjoint

Let a be

By the a x i o m of r e p l a c e m e n t

Define b : {F(y); y e b*},

Theorem

two free

Suppose that

one y such that ~(x,y).

to ~ and a* there

Vu(u

=

proper F - i n c l u s i o n

with respec% to 6F).

the r e q u i r e m e n t s Ad(V):

= ~, f(1)

define x = {F(f(i));

F-choice

to e).

for

set for a, i.e.

50

Ay[Y

e F a ÷~z

(zeF b ^ z EF y ) ] .

~ h i s p r o v e s theorem 2. Definition.

A set x is c a l l e d r e f l e x i v the c o n d i t i o n of r e f l e x i v e

Ay(y sets

iff x = {x},

id est

e x ~ y = x) holds. contradicts

T h e o r e m _3. If ZF ° is c o n s i s t e n t ,

iff

The e x i s t e n c e

the a x i o m of F u n d i e r u n g .

t h e n ZF°

+ . ~ / ~. z

= {x})is a l s o

consistent. Proof.

Let

#(x,y)

(x ~ 0 ^ x ~ 1 ^ universe

be the f o r m u l a x = y).

~ defines

s u c h t h a t o n l y the o r d i n a l s

F be the o n e - t o - o n e F(0),

Hence

(x = 0 ^ y = 1) v

hence

1 ~

of ZF ° + V x a theorem

function

1 and 1 :

x = {x}

a permutation

0 and 1 are ; #(x,y)}.

{1} F. H e n c e R e p ( F , - )

in ZF. The r e l a t i v e

in c h a p t . ! ,

Corollarv:

{(x,y)

s e c t i o n D. p a g e

If ZF is c o n s i s t e n t ,

(x = 1 ^ y = 0) v

T h e n F(1)

Hence

from theorems

consistency

follows

t h e n ZF ° + " n e g a t i o n

the a x i o m of f o u n d a t i o n Our n e x t q u e s t i o n

violated flexive

sets or w h e r e

nals

Both questions

E.P.Specker

Theorem

and

Consider

in all o t h e r

lar we have: Hence

F(0)

c a n be

set of re-

c l a s s of r e f l e x i v e

sets

w i t h the c l a s s of all o r d i in the p o s i t i v e w a y by

t h e n ZF ° + " t h e r e

exists

s u c h t h a t R is e q u i p o t e n t

a set R

w i t h m" is

too.

F(x)

h e r e ~*

{x} is n e v e r

F(1)

{3} eF { 3 } ,

permutation

F of the u n i v e r s e :

= y iff x = {y} for y e m" and F(x) is d i f i n e d

to be ~ - {1}

= 0, F(2)

= 0 e 1),

etc.

= {2},

id est:

and o b v i o u s l y

F({2})

In p a r t i c u -

= 2, etc.

{0} E F {0}o

F(x)

= x

= {0,2~3,4~..}.

in ~* and F is w e l l - d e f i n e d .

= 1, F(1)

1 E F 1 (since

{2} e F {2},

sets

the f o l l o w i n g

cases;

f r o m ZF ° + (AC).

[~]).

= {x} iff x ~ m,,

F o r x e m*,

result.

Zeitschr.math. Logik u.Gr.d.Math.

4: If ZF is c o n s i s t e n t ~

consistent

consistency

a countable

a proper

have been answered

of r e f l e x i v e

Proof:

exists

exists

correspondence

(see a l s o M . B o f f a ,

14(1968)p.329-334

+ "negation

too.

is w h e t h e r the a x i o m of f o u n d a t i o n

there

is in o n e - t o - o n e

3 and G ~ d e l ' s

there

from

of the a x i o m

and ZF ° + (AS)

is i n d e p e n d e n t

in s u c h a f o r m w h e r e

which

F(x)

2 and

= 0 e 1 =

is a s y n t a c t i c m o d e l

of the a x i o m of f o u n d a t i o n ~' is c o n s i s t e n t follows

Let

9, q.e.d.

of f o u n d a t i o n ~' is c o n s i s t e n t

This

of t h e

interchanged.

Further

e ~* ÷ x = { F ( x ) } ={x~ F .

51

We shall show that there F-set of reflexive The unordered F(y)}.

The ordered

(x,y)F f(n+l)

F({F(x),

{n+2})F;

(see the proof

n ~ ~}F

between

n ~ ~}, the F-set of F-natural {n+2}

: {F({n+2})}

Corollary:

On*

sets such that there

between

the following

By t h e o r e m

holds.

Hence

(GCH).

This

permutation

is a one-to-one

of all ordinals

of the universe:

to add

like those

consult

the papers

be strengthened follows

proved

: x

an~

of theorem

is indepentent

3 the

(GCH)

from ZF ° + (ACk +

by adding V : L, since obviously

from V = L. For further

in theorems

of M.Boffa

(AC) Zo ZF ° in th.4 and

in the model

the axiom of foundatibn

results

and R"

as in the proof of th.4.

Further,

the axiom of foundation

a proper

: y iff x : {y} for y e On" and F(x)

2 it is possible

above.

cannot

sets

too.

: On - {1}. Now proceed

papers

the class

exists

Here On is the class of all ordinal-numbers

its corollary

: {F(f(n));

and some F-reflexive

class R of reflexive

: {x} iff x e On i, F(x)

Remark.

n e m}F

then ZF ° + "there

Consider

otherwise~

numbers,

n e ~}

If ZF is consistent,

is consistent

F(x)

The function

{n+2})F;

~F : {f(n);

: ~,

for n e ~.

correspondence

Proof.

are the sets f(0)

of th.1).

: {F((f(n),

in the F-sense

is hence

F(y)})}

in the sense of the model

: f(n) U {F(f(n))}

is a function

the

of x and y is {F(x),

pair in the sense of the model

numbers

g : {(f(n).

between

sets n = {n} F for n ~ I and ~. pair in the model-sense

: {F({F(x)}),

The natural

is a F - c o r r e s p o n d e n c e

(those

consistenc U

3 and 4 and its corollary

already

cited and Boffa's

in the C.R.Acad. Sc. Paris v o l . 2 6 4 ( 1 9 6 7 ) p . 2 2 1 - 2 2 2 , v o l . 2 6 5 ( 1 9 6 7 )

p.205-206~ Boffa's Theorem

vol.266(1968)p.545-546,

second C.R.-paper (M.Boffa): consistent,

contains

Let ( s , < )

the following

be any partially

then so is ZF ° + (AC)

set t such that ( s , < ) In particular

(s,~)

(e.g.

totally

a dense

vol.268(1969)p.205).

and ( t , e )

inaccessible

cardinal

ordered

ordered

+ "there

numbers

set.

If ZF is

is a transitive

are isomorphic". ordered

set

set).

it is known that

totally

fine result:

can be taken to be any linearly ordered

In particular

in ZF + (AC)

"the class

by C (see G.Sabbagh,

+ "There are strongly

of G r o t h e n d i e k - u n i v e r s a

Archly

d. Math.

is

2_~0(1969)p.449 ~

52

456).

U.Felgner

AxVy(~ there

< ~ ^ In(~))

it is consistent

that given

any partially

is a set t of Grothendiek-universa:

(t,C)

~@

has shown that

are isomorphic

M°Boffa-

(see U.Fg.Archiv

A weak axiom of foundation,

and

METHOD

with the existence

of re-

sets.

=U{P(Rs(A));

sets.

Define

that e ~ B ÷ Re(A)

C Rs(A)

transitiv.

It is not provable

that

in the preceeding

R0(A)

= A,

8 < e} for e > 0, and W(A)

One proves

section~)

as in the consistency every

set (s, ~ )

d.Math.20(1969)p.561-566:[i~],

compatible

Let A be a set of reflexive Re(A)

ordered

such that ( s , ~ )

G.Sabbagh [5] •

B) THE FRAENKEL-MOSTOWSKI-SPECKER

flexive

with ZF ° + (AC) +

for V = U

relative

(WF) A x i o m of weak foundation:

e e On}

and that all sets Re(A)

Vx(V

= W(x))

but it is consistent

proof

set is wellfounded

= U{Re(A);

Ve).

are

(see the corollary with ZF ° (same proof

In W(A)

it holds

that

to A:

__VA( A x X

~ ~ ÷

V,(y

e x ^ (y N x : ~ v

2

v Agy All models

considered f r ~ m n o w

this axiom. Ay(y

chapter)

the axiom

Vx(V __

{y}))).

will satisfy = W(x))

^

g x ÷ y : {y})).

Automorphisms mapping

of the universe.

of ~efiexive

be any permutation

~(x)

: {~(y);

The uniqueness follows can be defined: automorphism If w permutes

Filters

and let

from A onto A), ~* of

~ so that

and define

y E x}

: Min{e;

: (~,)-I

x C Re(A)}.

This

determined

and

of permutations

(~,~)*

of rank,

shows that every

by a permutation

of A.

then = ~;.

of A and the automorphism

are isomorphic

p(x),

group

and we need not to dis-

them.

of subsroups.

H a subgroup

Let x g Re+I(A)

from the fact that a notion

as Aut(V,e))

between

mapping

we have extended

A and ~* is its extension,

the group

e T(y).

for V, id est V = W(A),

one-to-one

suppose

T of V is uniquely

(-1),

of V (written

O(x)

of V is a one-to-one

in a unique way to an_ a u t o m o r p h i s m

is done by induction:

acts on all sets of Re(A).

tinguish

sets

of A (i.e.

then ~ can be extended V. This

An automorphism

T from V onto V such that x g y ~ T(x)

Let A be a basis

Hence

on (in this

From now on we assume

=

If G is any

(multipiicatively

of G, and g e G, then g-!Hg

is called

written)

group,

a conjugate

53

subgroup

of G, c o n j u g a t e

Definition.

A non-empty a filter

with respect

set F of s u b g r o u p s

(of s u b g r o u p s

conditions

H • F ^ g ¢ G ~

(ii)

H,

g-lHg

e F ^ H2

¢ F ~

that H1is

H, N H2

filter

determines

a model

Definition

F of s u b g r o u p s

of the m o d e l . ~ [ G , F ] .

{T(y);

y ¢ X}.

denote

the t r a n s i t i v e

shall

Obviously

of ZF °.

(Specker

÷ H[y]

=

let C(x)

: {x} O x U U x

U...

e F)}.

elements

of M and the m e m b e r -

Ad(0):

list

universe

some p r o p e r t i e s

V. We

(In ZF ° + V : W ( A ) ) : ~ [ G , F I

If x and y are

Since

F is a filter:

hence

HI {x~y}]

= G e F and

sets o f ~ ,

H[x]

N H[y]

0 C M,

Ux

~ ~(y)

T(x)

: x and T(y)

:

e ~(x)

for T e HIx].

{z;

Hence

Ux

the

@ e M and ~ s a t i s f i e s

T is

then H|x|

But

identical

= l(Ux)

H[x]

~::z

= ~{z;

e F and H|y] N H|y]

e F.

~ H[{x,y}] ~

s h o w that

For

z e y e x ÷

~ s H[xl: But

Y ¢ H[x]

e y E x) -- V y ( T ( z ) z e y e x)}

~-l(Ux)

C Ux

mapping):

= TT-I(Ux)

(e).

e F. We shall

an automorphism.

similar

from

a set o f ~ .

¢ y ~ x)}.

~ x. Thus:

T(Ux)

follows

e F. But H[x]

hence

Vy(Z

since = y'

¢ y e x)} c U x,

(if 1 d e n o t e s

in~

e F. By (8) is {x,y}

Let x be a set o f ~ ;

T(z)

-~ ( M , ¢ )

in~.

The a x i o m of e x t e n s i o n a l i t y

~HIUxl.

e F.

of ZF ° .

S i n c e H[ ~]

HIM

of {M,e ) .

class. of M, t h e n x e M iff H[x]

the a x i o m of N u l l - s e t

Ad(III):

T"x

of G. A g a i n

C(x)

is the one of the w h o l e

|SZ]p.196):

is a m o d e l

Vy(Z

is a s u b g r o u p

are thus

and F a f i l t e r

~ <M,E >

(e) M is a t r a n s i t i v e

Ad(II):

show

G of A u t ( V , £ )

T"x : x} w h e r e

of x, i.e.

But first we

(8) If x is a s u b s e t

A d(I):

. We shall

subgroup

that

is a m o d e l

Proof.

of H~

of any

Let G < A u t ( V , c )

e C(x)

of~[G,F]

~WLlc,r]

Theorem

H[x]

"model"~IG,F]

prove

¢ F.

¢ F.

= {T ~ G;

closure

M : {x; ~ y ( y

of the

three

of ZF ° .

on G. For any set x let H[x]

ship-relation

H2

a subgroup

that e v e r y

Sets

iff the f o l l o w i n g

e F.

S F ^ HI ~ H2 ^ H~ ~ G ~

H e r e HI ~ Ha m e a n s

Now d e f i n e

of G)

of a g r o u p G is c a l l e d

hold:

(i)

(iii) HI

to H.

C ~(Ux)

C Ux.

÷ s y ~ x)

= {~(z); follows.

Hence

54

Thus

T e H[Ux]

• Now by

(ii)

of the

filter-definition

H[Ux]

e F.

By (IB) U x ~ M. A d(IV): (II)

By i n d u c t i o n

and

(III)

requirements A_d(V):

one shows

already

proved).

and T e H[x] , then T(y) y C x and y e M,

The same holds

We shall

for T - 1 . H e n c e

in it.

of p r o v i n g we

First e

one

G:

If y C x

Moreover,

1 below)

if

and t h e r e f o r e

N M)

P(x)

= P(x)

that

first

N M (as above)

N M e M. The

set P(x)

axiom relativized

the r e p l a c e m e n t

schema

with

to~.

is true

that the A u s s o n d e r u n ~ s s c h e m a

to show that

and

N M

in

holds

no free v a r i a b l e s

if x~ ,...,XrfY ~ M

is a set z e M such that x e z ~ x e y ^ R e l ( M , ~ ( x , x l ,...,Xn))]

shows

by i n d u c t i o n

on the

length

of # that

for e v e r y

x,xl ,...,x n e M ÷ (Rel(M,~(x,x, ,..,x n)) ~ R e l ( M , ~ ( T ( x ) , . , T ( X n ) ) ) ) .

In our p r e s e n t

case,

since

H~xl] ~..,H[x n] ,H[y] we can a s s u m e under

C P(x).

Let ~(x,x, ,...,x n) be a Z F - f o r m u l a

Ax[

T

prove

P(x) N M e M.

lemma

of the p o w e r - s e t

directly

shall

the

T(P(x)

t h a n x,xl ,... ,x n. We have

then there

the

N M.

6] M] . Thus

s a t i s f i e s the r e q u i r e m e n t s

T(P~)

e M (see

N M) C P(x)

H[ x] C H[ P(x)

other

(using

~ e M and m s a t i s f i e s

show that

C_ x; h e n c e

then T(y)

T(P(x)

Instead

Hence

all o r d i n a l s

of the a x i o m of infinity.

Let x be any set.

the m o d e l

that M c o n t a i n s

that

e

x, ,..,x n,y e M, thus

F, h e n c e

H[x,]N..NH[Xn]

all of xl ,..,Xn,Y

automorphisms

~ E H0).

N H[y]

are H 0 - s y m m e t r i e

Consider

the

= H0

e F,

(i.e.

invariant

set

z = {x; x e y ^ R e l ( M , # ( x , x , ,...,Xn))}. In o r d e r

to show tha%z

H0-symmetrie. by

(8) that Hence

H0 ~ H[y]

by d e f i n i t i o n

T £ H0 ~ H[x i]

as in (III))~

and

hence

T(x i)

T e H0 ÷ ~(z)

shown,

since

that

e F which

z is

implies

that

to prove

: xi,

_C z. H e n c e

~-~[G,F] that ~

z g G, then

since

) , . . . , T ( x n)).

But

and t h e r e f o r e T(z)

= z (proved

q.e.d. is a m o d e l satisfies

a lemma

i. If x e M and

e T(y)=y

x g z we have

Rel(M,~(T(x),T(xl

(1 ~< i < n),

Z. In o r d e r

a x i o m we need

of H0. Also,

z is H 0 - s y m m e t r i e ,

So far we have

Lemma

to prove

(ii) H[ z]

T E H0 and x e z. Then x e y and T(x)

"[(x) e z. A l t o g e t h e r :

set t h e o r y

(8) e n o u g h by

z is in M. take

R e l ( M , ~ ( x , x l ~...,x n)), since

e M it is by

T h e n H0 ~< H[ z] , h e n c e

z(x)

e M.

of Z e r m e l o -

the r e p l a c e m e n t -

55

Proof

by i n d u c t i o n

x e M + H[x]

e F. We c l a i m that

H[T(x)I Hence Thus

on the e - r e l a t i o n .

> T H[x]T -I.

take o e H[x] . T h e n

(TOT-1)T(X)

T~T "I e H IT(X)] . It f o l l o w s

definition

that H[~(x)]

x g Ro(A))+, then in Ro(A)~

hence

y £ C(T(x))

T(x)

(i) and

x C M, h e n c e

T(x)

C M. Thus,

(id est of T also

+ y = ~(x).

g F and we get x e M.

~ M for all y E RB(A)

= T(X).

(ii) of the f i l t e r

of the a c t i o n

But y e C ( T ( x ) )

= H[T(x)]

= tO(X)

if x is r e f l e x i v e

is by d e f i n i t i o n

reflexive.

÷ H[y]

and y ~ M ÷ T(y) implies

e F. Now~

by

= TO(T-IT)(X)

Hence

If x e Re(A)

for 8 < e, then x g M

by

(8), H[T(x)]

e F implies

• (x) g M, q.e.d° Now we r e t u r n presence

to the p r o o f

of the a x i o m

schema

of r e p l a c e m e n t

A

AAwl+(u,v) ^

schema

of S p e c k e r ' s

of s u b s e t s

is e q u i v a l e n t

+ v :

theorem.

In the

(Aussonderung)

the a x i o m

to the s c h e m a

+ AyV

AuAv(U

y^ v

where with

#(u,v)

is a Z F - f o r m u l a .

no free v a r i b l e s

xl ~...,x n

e

other

M and all u , v , w

Rel(M,~(u,w))

implies

t : {v g M;

in ZF,

by the r e p l a c e m e n t a x i o m

Further

t C M~ h e n c e

contains

the

id est T(Z) products.

the p r o o f

identical

Hence

H[z]

mapping:

sets

was

z : U{T(t);

1+ thus

T g G}.

z C M. Since G z is G - s y m m e t r i c ,

(8): ~ e M. Thus schema

constructed

holds.

that n e i t h e r

relative

is a s u b c l a s s

under

the r e l a t i This

finishes

to the set A of

of W(A)

A nor the e l e m e n t s

if the f i l t e r

e A ~ H[x]

e F]

= ~eRe(A)-

of A are always

. But in all a p p l i c a t i o n s

m e t h o d we just w a n t

condition: .Ax[x

Put

theorem.

in the m o d e l ~ [ G , F ]

(iv)

y e M. D e f i n e

t C z. F u r t h e r

such that~[G,F]

This w i l l be the case

. A s s u m e that for n Rel(M,~(u,v)) ^

since as a g r o u p G is c l o s e d

f o r m of the r e p l a c e m e n t

Mostowski-Specker

be such a f o r m u l a

z is a set since G is a set.

: G e F, and by

of S p e c k e r ' s

It is r e m a r k a b l e

tional

of ZF,

T(t) C M+ by !emma

The m o d e l ~ [ G , F ] reflexive

t is a set.

: Z fo~ all T e G~

vized weakened

sets

Let y be a set,

z).

g y ^ Rel(M,¢(u,v)))}

By the r e p l a c e m e n t a x i o m Again

M we h a v e that

e

v = w.

Vu(u

Now let ~(u,v)

than u,v,x, ,...,x

E

of the F r a e n k e l -

to h a v e A as a set i n ~ .

F satisfies

the

following

addi-

56

It is easily (M,E)

seen

(see the proof

and F satisfying

(since HI A] Further

(i),

of lemma

1) that

(ii)(iii)(iv),

for~[G,F]

-~

A _C M, hence A c M

= G e F) by (8), holds.

, if F satisfies

axiom of foundation

(i) .... ,(iv) then i n ~ [ G , F ]

(WF) holds

the w e a k

(if in the surrounding

set-theory

(WF) holds). By definition identical

mapping

Klxl

where

= {r

e G;

"t I ' x

for any set x:

Ix}

=

I x is the identical

of T to x. Remark K[x]

T e HI x] ÷ x"x = x, but x need not to be the on x. Define

on x and "r 1" x is the r e s t r i c t i o n

mapping

that always

K[ x] ~ H[x]

~ G. If H[x]

need not to be in F, but if K[ x] e F then there

ring of x in ~ [ G , F ]

, if the axiom of choice holds

e F then

is a wellordein the surroun-

ding set theory. Lemma

2: Every

T e G acts as the identity

M (] ~J~V~ Proof by induction Lemma

on the M i r m a n o f f - r a n k

3: (In ZF ° + (WF) of subgroups

well-founded

Proof.

contains

(AC) holds

in the surrounding

(i),

(it),

the axiom of choice

set theory

2: H[x]

= K[x|

(by lemma

is contained

in~[G,F]

are just the well-founded

set theory')

one-to-one

f from x onto e is also a w e l l - f o u n d e d

mapping

also i n ~ [ G , F ]

Lemma

4: (In ZF ° + (WF) of subgroups

and hence ~ is in ~ [ G , F ]

sets

and the set,

. + (AC)):

If G ~ Aut(V,c)

of G satisfying

(i),

set x of ~

[G,F]

one fashion

onto a well-founded

K[ x] E F.

= G e F.

2) every well-

of the surrounding hence

(iii),

for each

set theory x can be mapped

e. But obviously

sets of ~6~[G,F]

and F is a filter

relations

. Hence

sets x.

part of ~ [ G , F ] .

then by lemma

of the surrounding

(the well-founded

part

for well-founded

conditions

wellordering

set x of ~ [ G , F ]

one on an ordinal

founded~et

p(x)

If G < Aut(V,¢)

in the well-founded

If x is well-founded,

Since the one-to

+ (AC)):

of G satisfying

then ~ [ G , F ] holds

on the w e l l - f o u n d e d

of M.

can be mapped

and F is a filter

(it) and (iii)

in~[G,F]

then a

in a one-to-

set y of ~ [ G , F ]

iff

57

Proof.

a) Suppose that there

~[G,F]

is such a o n e - t o - o n e

from x e M onto a w e l l - f o u n d e d

f] e F it is sufficient

~((u,f(u)))

Hence,

: 0). greater

implies

iteration

with

mapping

equal

is finite,

of T one comes

and e v e r y

T. Thus D2 t o g e t h e r

group (D2,T~

- D,,

Tm°*(Y) =

= z] }.

^ yn(w)

of T. Since F(y)

~ is a o n e - t o - o n e

sum of t h e s e

If z e F(y)

D2

many

S z = {z,$(z),T2(z),...}

the d i r e c t

application

since

under

Since

such that

e m such that

= w)}

E D,

of s u c c e s s i v e

is f i n i t e

group.

V w Vn[W

Further

number

is a group:

of Y e H0[t]

to find a mo

< n ^ ~n(w)

iterated

too.

a finite

Ho[t]

to look at:

D,

Tn means DI

it is n a t u r a l

Since

= T m ° ~ F(y), : y,

id est o*(y)

also ~T m°

is the

from o(Tm°(z))

~*(Tm°)*(y)

By h y p o t h e s i s

h e n c e o-IT m° = T m *o

identity

e Ho[y]

~ H|y],

thus

(y). on F(y) we o b t a i n

in a q u i t e

: z for z c F(y):

= y.

y C T*(y),

and

since

T* is an a u t o m o r p h i s m ,

we h a v e

66

y £ ~.(y) £ (~).(y) ~ . . . ~ (,~).(y).

(3)

Thus by (i); y C ~'(y). But (3) also yields o*(y) C o*(Ym°)*(y). Applying

(2), we have o*(y) C y. Thus y : o*(y).

deduce y : (Tm°)*(y).

Finally,

From (1) we

from (3), we arrive at the contra-

diction y : T'(Y) and lemma 3 is proved. Let (AC~) be the axiom of choice for families elements

are couples

the (unrestricted) model~,

(= unordered pairs).

axiom of choice

we ~ a l l

model ~ [ Proof:

V, V(u,v

A = Ro(A)

Thus Y _ C ~

(AC2) fails in ~ .

(AC2) does not hold in Halpern's

G,F] .

Let Y : {z;

(V~U)})}.

Instead of proving that

(AC) does not hold in Halpern's

show that already

Lemma 4: The weak axiom of choice

g A : Ro(A)

is a set of ~

^ u ~ v ^ Z : ((U,V > ,

as was noticed previously.

But Y is closed under G, thus H[Y]

= G e F. Hence,

by (8) of section B, Y is a set of the m o d e l ~ . = 2 and distinct elements of Y are disjoint. be a choice set C for Y i n ~ dinality subset

(: sets) whose

I] and C e ~ .

, id est

Suppose there would

/ \ w [w e Y ÷ w N C has car-

It follows H[C]

of the infinite set Ro(A).

Also z E Y implies

E F and F[C]

is a finite

Pick elements u~v e R0(A) - F(C)

such that u ~ v. Let T be the permutation of Ro (A) which interchanges u and v and is the identity otherwise. T*(C)

Then T s Ho[C] ~ H|C], hence

= C, and y ={(u,v ) ,(v,u )} g Y. Suppose(u~v ) e C, then

Y'((u,v )) =(v,u ) E T*(C), hence ( v , u ) (v,u)

E C then one concludes

tradicting

~ C: a contradiction.

similarly that ( u ~ v )

If

e C, again con-

the assumption on C. Thus Y has no choice set in ~ , q . e . d .

This finishes the proof, that in Halpern's model ~ [ G , F ] axioms of ZF°~ Kurepa's Antichain

Principle

(KA) and q(AC2)

all are true.

As a Corol!ary

(J.D.HALPERN):

The axiom of choice (AC) does not follow

from Kurepa~s A n t i c h a i n - P r i n c i p l e Remark. ciple

Since (AC2) fails in Halpern's m o d e l ~ ,

(0) fails in ~

(BPI) + (0)

too, since Z F ° ~ ( 0 )

(via compactness-theorem

lus, e.g.) where

(BPI) is the

p.37-IIi)

÷ (AC2). Further Z F ° ~

of the lower predicate calcu-

Stone

(Trans.AMS voi.40(1936)

has shown in ZF ° that (BPI) is equivalent to the "Repre-

sentation T h e o r e m for Boolean Algebras": ,•

the ordering prin-

Boolean Prime Ideal theorem "Every

Boolean algebra has a prime ideal".

(B,U

(KA) in ZF ° .

, I>

"Every Boolean Algebra

is isomorphic to a set-algebra (C, U, N, - >"

67

The statement (SPI):

"Every

follows

infinite

set algebra has a non-principle

from the (BPI).

Tarski

has asked, w h e T h e r ( S P I )

provable.

Halpern has shown,

that

provable.

Halpern

in the model above the

while

the

(BPI)

Lemma

5 (U.Felgner~M.Z.

shows that

fails

(KA) implies every Proof.

111(1969)):

erdered

such that there

onto K with the property

which

chains.

selects

function

theorem

(LW) holds

mot m t h e o r e m

from o~f

(LW).

mapping

potency

by abstraction.

p.442~

is

By Zermelo's

or 65(1908)p.107-

too. proved

Since

ZF ° ~ (LW) ÷

D: in ZF ° the that

(AC)

(LW)--~ (KA)

is

IN ZF ~

is a one-to-one number

(or equinumerous),

function

mapping

x of x is obtained

~ equipotent

(AC) with

we are still

x ~ la Frege-Russell-Scott:

A z(Z

~ x ~

of sets y of lowest by abstraction

x on

from equi-

of the axiom of choice

to be the least ordinal

p is the Mirimanoff-rank

and

- {~}.

(AC) but the axiom of foundation~

adequately

x : {y; y ~ x ^

Definitions

C of U K

just one element.

in chapter

Fel~er

In the presence

the term x can be defined

x consists

by Ks but K is

q.e.d.

OF CARDINALITY

of the cardinal

x. If we do not have

where

- {~}

e K is isomor-

Z? ° .

x = y, iff there

able to define

f from P(s)

antichain

on P(s)

model ~ L

our result

Further,

P(s) of

are pairwise

f(t)

isomorphically

One says that the sets x and y are equipotent in symbols

Principle

says that

The powerset

from each chain

in Halpern's

F) THE U N D E F I N A B I L I T Y

y. The notion

is not

(Math.Ann.65(1908)p.

(Math.Ann.59(1904)p.514-516,

(PW)~ we have strengthened is independent

set.

A maximal

g defined

128) the set s can be wellordered, Thus

Antichain

for ~ ~ t e P(s),

is represented

a choice

well-ordering

ordered

is a one-to-one

disjoint

function

is

(SPI) holds

is a set K whose'elements

that

a set of pairwise

Thus we get a choice

implication

(LW) which

a theorem of Zermelo

28) there

phic to t. Thus P(s)

÷ (BPI)

set can be well-ordered.

Let (s, ~ ) be a linearly

disjoint,

Kurepa's

in ZF ° the statement

linearely

theorem

in ZF ° this

ideal"

in it.

s is a set of chains.By 261-281,

prime

p(y) ~ P(Z))}

function

(see chapt. I,sect.E).

rank equinumerous

in axiomatic

Here

with x (see D.Scott:

set theory,

BulI.AMS

6--1(1955)

68

[~]

Dana SCOTT: The notion of rank in set-theory; Summer Institute

Summaries

for Symbolic Logic, Cornell Univ.1957,

p.267-269)~ We remark,

that even in the absence of both the axioms of choice and

regularity but in the presence of either the weak axiom of foundation in the form "there is a set A such that V = ~ R ~ ( A ) " (UoFg.,Arehiv d. Mat~.20):

or the axiom

~'the universe V can be covered by a well-

ordered sequence of sets s~, ~ an ordinal". ZF ° without any additional

We shall show that in

covering axiom (like foundation,

etc.)

there is no adequate definition of the term ~o This result was obtained first by Azriel L@vy

[50]

A.L£VY:

The Definability

of Cardinal Numbers;

of Mathematies~'~ GSdel-Festschrift~

in: "Foundations

Springer-Verlag

Berlin

1969~p.15-38. Also R.J.Gauntt has obtained this result [~Z] R.J.GAUNTT:

Undefinability

U.C.L.A,-set

(independently):

of Cardinality;

Theory Institute

Proceedings

of the

1967. To appear in 1970.

In the presentation of the proof we shall follow mainly R.J.Gauntt but in few details A.L~vy. When one considers the question of whether one can define in ZF ° --

the eardinality operation x~ the following possibilities (a) x is definable

turn up:

i n a set theory ST: there is a term t(x) of ST

with the only free variable x such that ST ~ % % [ t ( x )

= t(y) ~ x ~ y]

(b) x is relatively definable

in a set theory ST: there is a term t(x,

z) of ST with the only free variables ST ~ X Obviously

Ax ¢[t(x'z)=

(a) entails

(b) (L~vy [50]

If we take ZF ° + foundation ST, then (a) holds. namely,that Theorem

t(Y'Z)~

z and x such that x ~ y]"

eonsiders further possibilities).

(id est ZF) or ZF ° + (AC) as ~et theory

We shall prove a strong undefinability result,

even (b) does not hold for the set theory ZF ° .

(L@vy,Gauntt):

If ZF ° is consistent~

then so is ZF°plus the

schema

(*)

"VxAVy[@(y,a,x)^

Ab(a

~ b ~ ¢(y,b,x))] .

69

Proof.

If ZF ° is c o n s i s t e n t ,

"there

is a p r o p e r c l a s s A of r e f l e x i v e

c l a s s of all o r d i n a l s ) ,

there

sequel

Each ordinal

(the

of Chapt. III,

mapping

of ZF ° p l u s the s c h e m a

(').

(in a u n i q u e way)

and n £ ~ (this

Define e -

On o n e -

a Fraenkel- In the

0 iff n ~

follows

as

8 + n where

from Cantor's

0 (congruence modulo

normal-

2) f o r

^ n £ m, and d e f i n e ~ -= I iff n = I m o d u l o

for ~ = 8 + n ^ Lim(8)

^ n E e. The o r d i n a l s

0,2,4,... ~m,m+2,e+4,...

and the o r d i n a l s

m + l , ~ + 3 .... For e a c h o r d i n a l end

(see the r e s u l t s

now construct within this universe

e can be w r i t t e n

= 6 + n ^ Lim(8)

w i t h On

of A are c a l l e d atoms.

8 is a l i m i t o r d i n a l form theorem).

is c o n s i s t e n t

model ~L

the e l e m e n t s

sets e q u i n u m e r o u s

is a f u n c t i o n G (a c i a s s t e r m )

t o - o n e o n t o A. We w i l l Mostowski-Specker

( c a l l e d ZF V) ZF°+

s u c h t h a t for e v e r y x t h e r e e x i s t s y E x w i t h

e i t h e r y ~ x = ~ o T y ~ A" sect.A). ~ence

t h e n a l s o the t h e o r y

if ~ -- o t h e n A 8 1 8

~,

congruent

congruent

{G(~),G(e+I)}

2

o are thus

I are

1,3,5,...,

is a p a i r of a t o m s

{G(S),G(6+I)

--- 0 ^ a ~e 6 ÷ { G ( ~ ) , G ( e + I ) }

~|. Definition.

F(~)

The f o l l o w i n g (1917)p.33

= {G(6);

definition

(6 m 0 ^

8 < e) v

is due to D . M i r i m a n o f f

Ker(x)

(L'Ens.Math.vol.17

= C(x) ~ A = the set of atoms closure

t h e k e r n e l of x; M i r i m a n o f f

We n o w r e s t r i c t i.e.

8 4 e)}.

and p.211).

Definition.

(read:

(8 m I A

the u n i v e r s e

V = U ~ ( U y R y ( F ( e ~) ) ) . .

of all x for w h i c h

0

of x.

u s e d the t e r m

to e l e m e n t s

That

"

~( e )

"noyaux~').

of sets b u i l t up f r o m F ( e ) ' s ,

is, t h e r e s t r i c t e d

V ~ V..(x ~ y ^ Ker(y) L ~ y

-+

in the t r a n s i t i v e

universe

consists

c F(e)).

,,.,

- - ~Atoms

A

F(6) No%ice

that

class.

For each permutation

(restricted)

e a c h F(e)

universe

is s set and U ~ F ( s ) f on F(e),

as follows:

= A, w h e r e A is a p r o p e r

define

f(x)

o v e r the e n t i r e

70

f(x)

= x

for atoms

f(x)

= {f(y);

This de£inition universe,

y e x} for sets x.

is welldefined

then x C Ry(F(8))

definition assumed

x not in F(s),

of Ry(a)

if x is in the restricted

for some ordinals

see p.53].

that f is defined

since

By induction

~ and y [for the

hypothesis

for all y e R6(F(8))

it is

for ~ ~ y and

all 8. Definition.

A permutation

f on F(e)

id est,

that 8 ~ 0 there

exists

y < e such that y m 0 a n d

A permutation

6 ~ 0 it holds

is called

id est:

x is symmetric

~ there

fixed,

is a finite

symmetric

the transitive Digression.

of F(~) rence.

model

are called

In the definition

of the notion

Hence

"nice". support

we avoided

the filter formulate

F a collection

sets can be collected and in which to systems

classes

etc

But there

are already

proper classes. classes

and

It is possible theory

to

in which

(~ l a v . Neumann-Bernays-G6del) to totalities~ "set"~

totalities

"class",

"totality"

using an idea of !.L. Novak-Gal

In such a set theory

the filter

the use

and in the defini-

of proper

of ZF-set

(in which the predicates

K[~],

K[a]

diffe-

the use of a filter of subgroups.

can be collected

3_~7(1951)p.87-110).

the groups

permutations

set x we avoided

subgroup

extension

tO classes

y of

just defined

is one important

of those totalities.

"system" .... are primitiv) Math.

x and every element

C. The admissible

be totalities

a type theoretic

leaves

pointwisel).

sets x which are here-

of the m o d e l ~

is done since the permutations the groups K[a]would

T which

of x is symmetric).

of a symmetric

of a finite

tion of the m o d e l ~ This

(id est:

in section

there

permutation

are those

closure

Nofiice the similarity

with Fraenkel~s

set a of atoms

fixes x (not necessarily

Sets of the m o d e l ~ 6 ~ ditarily

iff

: {G(B),G(B+I)}.

such that each admissible

Definition.

admissible

for all 8 < ~ such that

that

f({G(6),G(8+I)})

pointwise

for all 8 < e such

= {G(y),G(~+I)}.

f on F(e)

it fixes pairs,

Definition.

semi-admissible

pairs~

f({G(~)~G(8+I)}) Definition.

is called

iff it preserves

F~ eZc.

But since

(Fund.

one can talk about in the discussion

7~

above r e f e r e n c e permutations

is made only with respect

f of the sets F(~) we could restrict ourself to men-

tion only p e r m u t a t i o n s "subgroup",

to one single class of

of certain type.

The use of the notions

"filter" would make only linguistieal

Further remark that a p e r m u t a t i o n

only elements w h i c h are in some F(e). of a p e r m u t a t i o n

on A is a set. This

tion of a symmetric class-variables The formulae:

Hence the "essential" explaines

set we have q u a n t i f i e d

are thus ZF-formulae.

Thus

(thus

on A are not needed).

and "x is h e r e d i t a r i l y

~ :

part

that in the defini-

only over sets

to range over p e r m u t a t i o n s

"x is symmetric"

differences.

of the class A of atoms moves

symmetric ~'

{x; x is h e r e d i t a r i l y

symmetric}

is a c l a s s - t e r m of ZF. The following to those of section

lemmata are easily proved.

The proofs are similar

B.

Lemma 1. (In ZF V ): If f is s e m i - a d m i s s i b l e

and g is admissibles

then f-lgf is admissible. Lemma 2. Lemma

(In ZF V ): x E ~

~ (x C ~

3. (In ZF V ): No two disjoint equinumerous

^ x is symmetric). infinite

sets of atoms are

in~.

~ucb Proof.

Suppose the lemma

and y and a one-to-one

is false.

Then there a r e ] i n f i n i t e

function g, m a p p i n g x onto y, i n ~

x,y and g are symmetric~

there are finite

that every a d m i s s i b l e

permutation

fixed~

fixes x (resp.

y,g).

G(~+I)

e x. Now pick ~ e 0 such that G(~)

g maps x onto y~ g(G(~)) (G(~),G(8)

interchanges

identity otherwise. tical m a p p i n g (G(e+I),G(8))

Lemma 4.

c ~(g)

= g

maps ~ onto itself.

Since

e y. Thus permu-

but is the

~ acts as the iden-

= (~(G(e))~G(8)) e pointwise

=

fixed.

g w o u l d not be one-to-one~

(In ZF v ): Any p e r m u t a t i o n

Let a be the finite

))

since ~ leaves

a contradiction~

Then every a d m i s s i b l e

= G(y)

e g. Take an admissible

Since x and y are disjognt,

finitely m a n y atoms, Proof.

e x - (a U b U c).

the atoms G(~) and G(e+l)

on y. Thus ~((G(~),G(8)

G(B),

such

b,c) pointwise

e x - a for e m 0, then

e y and g(G(~+l))

} e g and ( G ( ~ + I ) , G ( y ) )

tation ~ which

g(G(~+l))=

= G(B)

. Since

sets a~b,c of Atoms

~ leaving a (resp.

If G(~)

sets x

Hence q.e.d.

on F(~), which moves only

is i n ~ .

set of atoms moved by the p e r m u t a t i o n

permutation

T which

leaves

a pointwise

~.

fixed

72

Lemma 5. (In ZF V ): For each x and semi-admissible x

Proof.

~

~ ~

-~(x)

1 of chapt.

III,

Lemma

6o For each ZF-formula

(i)

2 and proceed

section

the following

quantifiers

are theorems

~ semi-admissible

Lem~a

The proof

÷ [~(xl ..... x n) ~ ~(~(xl)~..,~(Xn))] ÷ [Rel(~(xl

obtained

I, page lemma

is (with respect

mappings

set x in the restricted

These

universe

is essential

Len~a

~(xl,x2~x3)

8o For each ZF-formula the following

Proof.

hence

VxAaVy[~(y,a,x)

Suppose

~(xl~x~,x3)

is provable

that the lemma

and a set x i n ~

in the restricted

Ker(x)

C F(~), where

Cleary~

DI

e ~.

Suppose

C F(~).

Case 2. Ker(y)

~ F(e).

S~o

define

There

of ad-

are sets! segment

with three

free variables~

Ab(a

is false.

~ b ~ ~(y,b,x))]).

Then there

there

above.

is a ZF-formula Since x is i n ~ ,

is an ordinal

~ such that

Define

- F(~).

Case 1. Ker(y)

Then D2

groups

only an initial

as required

y is the

id est R e l ( ~ ] ~ ¢ ( y ~ D ,

If case 1 holds,

B, p.54)

(definite).

~ m 0 can be choosen.

DI = F(~+~)

y g~,

(in section

in zFV:

^

universes

is

"hyper-classes ~' of all

from A onto A, but take only the groups on the sets F(~).

all

5o

theorem

F(~) of the class A of atoms

Rel(~,7

14). The proof

to e) a model of ZF °.

2, 5 and 6. Do not take the

permutations

For every

from ~ by restricting

(see chapt.

is like the one of Specker~s

one-to-one

..... Xn)) ~

is the formula

on the length of ~, using

lemmata

missible

of zFV:

..... ~(Xn)))].

7. (In ZF V ): ~

using

as in the proof of

55-56.

Rel(~,~(~(xl)

to the c l a s s ~

by induction

B, page

~(x~ ~...,x n) with n free variables~

(ii) ~ semi-admissible

Here R e l ( ~ , ~ )

~,

.

Use lemma 1 and lemma

lemma

permutation

D2

(unique)

cardinal

of Dl, where

~x)).

= F(~+m.2)

- F(~+m),

is a semi-admissible

where

permutation

~.2 = m+~. w of the atoms:

73

w(G(~+n))

= G(~+~+n)

~(GC~+~+n)) ~(GCB))

= G(~+n)

= G(8)

for B < ~ or ~+~.2 < B.

Thus ~ fixes each element of F(e) and takes Di onto D2. Hence ~(Di ) = D2 ~ w(D2) Rel(~¢(y~Dl

= DI, ~(x)

= x and ~(y)

,x)) ° R e l ( ~ , ¢ ( ~ ( y ) ,

Rel(~,¢(y,Dx

= y. Then

~(D,),

~(x)))

°

,x))

Hence y is also the cardinal of D2. Thus R e I ( ~ , D , lemma

G(~)

~ D2) violating

3. If case

2 holds~

E Ker(y)

^ G(8)

define

there ~ F(~).

a permutation

8 ~ ~ such that

Pick an ordinal

• on DI U F(y+I)

G(y) and interchanges G(8-1) w i t h G(y-1)

is an ordinal

which

G(8+1) with G(Y+I)

Y,Y ~ 8, Y > B, and interchanges

6 < ~, T fixes all elements

of F(e).

and hence

T is semi-admissible.

Rel(~,¢(y,Dl,x)) Thus ~(y)

Thus

is the cardinal

and is a one-to-one

of ~(D~).

in~

= x. T moves Ker(y) Hence by lelmma 6:

Since y is the cardinal cardinality

of D~

and are there-

. But by lemma 4, • is a set of

function

would be equinumerous

T(x)

~ F(~) ÷

~ ReI(~,~(T(y)~T(DI)~x)).

and ~(y) ~ y, D~ and T(D~ ) have different fore not equinumerous

and

iff 8 m 0~ and interchanges

iff B ~ 1. Since e ~ 0, hence G(6)

T(y) ~ y. Clearly

G(~)

in the sense o f ~ .

in~,

Thus D~ and T(D~)

a contradiction.

Lemma

8 is thus

proved. The t h e o r e m of L @ v y - G a u n t t

follows d i r e c t l y

from lemmata

7 and 8.

G) A FINAL WORD

The main

idea behind GSdel's

construction

of the model ( L , ~ )

ZF + (AC) was to make all sets of the model definable by means

of a certain

complexe

defined)

wellordering

of the language

model-class ~in

language.

The natural

induced

choice

fails

is to guarantee

tely many sets of "indiscernible" why a function indiscernible

f defined elements

should choose

and not the other element.

that~

of ZF°-models

contains

Then there

of the

infini-

is no reason

set of sets of mutually

from each set just the one

This was made precise

the groups G of p e r m u t a t i o n s "atoms ~' (reflexive

sets.

on a infinite

(inductively

a wellordering

L. The main idea behind the c o n s t r u c t i o n

which

of

(or nameable)

on some infinite

by i n t r o d u c i n g

set A = Ro(A)

sets) and the filter F of subgroups

of G.

of

74

The symmetries

of the model ~

~ C ~ a n d , x = {T(y); discernibles

B of these

In Fraenkel's

{a2k,a2k+l}

are e.g.

sequence

model

(see this chapter,

sets of indiscernibles.

sets of indiscernibles

part to Russell's

by F. If x is in

T e G} for every y c x, then x is a set of in-

in~.

C) the sets

are determined

The set

is the set-theoretical

of pairs of (mutually

section

counteP-

indiscernible)

socks. The "classical" indiscernibles "reflexive

was to take an infinite

sets"

acts on them. right

filter

way for obtaining

and to take a certain

The choice

of the right

of subgroups

of choice holds atoms the

sequence

filter

method.

conditions

applications

In the next chapter This method pendence

applies

results

from the (GCH), further

results.

to full ZF-set

"below"

Again

sets by destroying to construct

which contain

indiscernib!es,

~--->(~)~ structible

cardinal~

satisfies that the

and then the model Thus~

in order

to get

the filter F has to

Cohen's

forcing

and yields

of (GCH)

the

ZF models ~

, see J . S i l v e r ' s universe,

theory

it is possible

indiscernible

l~rge

such that the

method.

not only inde-

th~ (AC) but also the independence

the independence

on

{i} of G.

we shall describe

even possible tence of

of sets.

subgroup

F defines

set theory the axiom

model ~ [ G , F ]

of the FMS-method,

never the trivial

group and the

(it is supposed

(i),...,(iv)),

with the whole universe

non-trivial contain

is discrete

or

group which

in all applications

The filter

and the weak axiom of foundation

F satisfies

coincides

of "urelements"

permutation

If in the surrounding

iff the topology

of sets of

nice permutation

form a set, then the corresponding

(AC)

families

is the alpha and omega

of the F r a e n k e l - M o s t o w s k i - S p e c k e r the group G a topology.

those

from (AC) and lots of

to introduce (AC).

in Cohen-models

We remark

that

in which V = L holds

but then one has to assume satisfying paper:

of V = L

the p a r t i t i o n

A large

cardinal

Fun~.Math.69(1970)p.93-100o

it is and the exis-

relation in the c o n -

~5

Additions

to chapter

1) The part K[xl

III

s F then there is a o n e - t o - o n e

some w e l l - f o u n d e d trivially

set, of lemma 4 in section

c F then x is w e l l o r d e r a b l e

be any w e l l o r d e r i n g

in~[G,F]

of x, then w C ~ I G , F ]

can be

thus w e ~ | G , F ] .

are w e l l - f o u n d e d 2) The corollary

; namely

Thus x is w e l l o r d e r a b l e

sets,

let w

. But o b v i o u s l y

are i--I-mappings from x onto some ordinals

in~L

i n ~

KIx]

and there

. But ordinals

Q.E.D.

on p.62 which says that

(PW) holds

mode! ~ q ~

can be s t r e g t h e n e d

by a s s e r t i n g

~while

(AC) fails.

Let ( s , < )

in~.

from x onto

B, p.57-58~

proved as follows°

If K[x]

H[w],

mapping

Proof.

in FraenkelVs

that even

(LW) holds

be a lineari!y

in

ordered

Define R = {(a,b ) ; a,b e s ^ a ~ b}; t h u s H [ R ]

set

e F and

H[ R] ~ HI s] . We claim that for each y s s it holds that HIR]

~ H[y].

Suppose not~ then there are y e s and a T e H [ R ] s u c h that T(y) ~ y. But T(y) {y~T(y)

e s and R is a linear ordering on s, thus either ) e R or ( T ( y ) ~ y )

T((y,T(y)

} ) = (T(y).T~(y))

But ( y , Y ( y ) )

H[R]

leaves

3) It holds

: (T(y),y)

e R ^ (T(y),y)

The same argument

relation

e R. If ( y ~ T ( y ) )

applies

s R yields

: R, since

s pointwise

fixed.

Thus,

< H[w]

and it follows

s R. Thus every • e

if w is any w e l l o r d e r i n g that w s ~

that ZF ° ~ (AC) ÷ (LW) + (PW), while ZF 2) that

(LW) + (AC) [6~]

in ZF ° . Let us indicate

ciple of choice of proper, ski's model,

since

(PW) holds

in it (see Mostowski:

note further

subsets

in ZF ° .

(PW) ÷ (LW) is not Kinna-Wagners

cannot hold

in it and otherwise

(AC) would be

has shown that in M o s t o w s k i ' s

ideal theorem

(BPI) holds

prin-

in Mostow-

Colloqu.Math.6(1958)p°207-208).

that J . D . H a l p e r n

the Boolean prime

is not provable

that o b v i o u s l y

, q.e.d.

~ (AC) ~ (LW) ~ (PW).

one shows that

non-empty

T 2= 1.

y = T(y), a contradiction[

on s, then H[R]

We have shown under

true

e T(R)

to the case ( T ( y ) , y )

Using the model of Mostowski provable

e R~ then

Let us model

(Fund.Math.55(1964)

p.57-66. 4) Finally we refer to some important is applied: A.Mostowski:

H.L~uchli:

E.Mendelson

[61]

papers

,[62]

On the Principle

of D e p e n d e n t

(1948)p.127-130:

[~81.

Auswahlaxiom

in which the F M S - m e t h o d

, and:

in der Algebra;

choices;Fund.Math. B5

Comment.Math.Helvetica

37

(1962/63)p.1-18. H.L~uehli:

The Independence tricted

of the Ordering principle

axiom of choice;

from a res-

Fund.Math.54(1964)p.31-43.

CHAPTER IV

COHEN EXTENSIONS OF ZF-MODELS In this chapter we study Cohen's forcing technique for constructing extensions

of ZF-models.

This technique was introduced

in 1963 by Paul J.Cohen. Using this method Cohen has solved the long outstanding problems of the independence of the Continuumhypothesis

from the axiom of choice and the independence of the

axiom of choice from the ZF-axioms [9]

P.JoCOHEN:

The Independence of the axiom of choice; mimeographed

notes(32 pages), [IO]

P.JoCOHEN:

(including foundation):

Stanford University

1963.

The Independence of the Continuum Hypothesis;

Proe.

Nat.Aead. Sci.USA, part 1 in vol.50(1963)p.1143-1148, part 2 in vol.51(1964)p.105-110. A sketch of the proofs is contained in: [Ill

P.JoCOHEN:

Independence results

of Models-Symposium,

in set theory;

In: The Theory

North Holland Publ.Comp.Amst.1965,

p.39-54. In these papers the constructible G6del's F(e)-hierarchy

closure is obtained by means of

(GSdel~s monograph [~S],of 1940). Dana Scott

has remarked that the constructible closure can be obtained in a much more elegant way using G~de!~s M~-hierarehy

(G6del's paper [2~]

of 1939). The presentation of the independence proofs in Cohen's monograph is based on these improvements: [I~]

P.J.COHEN:

Set Theory and the Continuum Hypothesis;

New York - Amsterdam 1966 (Benjamin,

Inc.).

Since the publication of Cohen's papers [9 ], [10]

and [ ~ ]

the

forcing technique has been modified in various ways by several authors.

Using modified

"GSdel-functions

F" W.Felscher and H.Schwarz

have studied systematically Cohen-generic models

(see Tagungsbe-

riehte Oberwolfaeh April 1965 and the dissertation of H.Schwarz: Ueber generische Modelle und ihre Anwendungen;

Freiburg i. Br.1966).

A topological approach to forcing has been developed by C.RyllNardzewsky and G.Takeuti:

77

[~]

G.TAKEUTI: Topological Space and forcing; Abstract in the J.S.L. vol.32(1967)p.568-569.

A detailed exposition of this approach is contained in: [66] A.MOSTOWSKI: Constructib!e Sets with applications; Amsterdam - Warszawa 1969(North Holland + PWN). That forcing can be understood as a boolean valuation of sentences V

has been discovered by D.Seott, R.M.Solovay and P.Vopenka

-see

the forthcoming paper by Scott-Solovay, or Scott's lecture notes of the UCLA set theory Institute (August 1967) and : [7~]

J.B.ROSSER:

Simplified Independence Proofs; Academic Press

1969. [86]

PoVOPENKA: General theory of V-models; Comment.Math.Univ. Carolinae (Prague) vol.8(1967)p.145-170.

For further litterature on V-models see the bibliography in [@6]. Some of Vop~nka's papers have been reviewed by K.Kunen in the J.S.L.

34(1969)p.515-516.

-We shall present here the forcing me-

thod in a way close to P.J.Cohen, using ideas which are due to D.Scott~ R.M.Solovay and others. The following basic publications will be useful:

[39]

R.B.JENSEN: Modelle der Mengenlehre;

Springer-Lecture Notes~

voi.37, 1967. [40]

R.B.JENSEN: Concrete Models of Set Theory; In Sets, Models and Recursion theory~ Leicester Proceedings 1965, North Holland PublComp.Amsterdam 1967~ p.44-74.

[80] J.SILVER: Forcing A la Solovay; unpublished lecture notes (28 pages). [51] A.L~VY: Definability in axiomatic Set Theory I; in: Logic, Methodology and Philosophy of Sci., Congress Jerusalem 1964, North Holland Publ.Comp.Amst.1965~ p.127-151. The main difficulties which arise when one wants to extend a given ZF-model~

by adjoining some new sets a0,al,.., to g?~, are that

the sets a i may contain undesired information encoded by the interior 6-structure of a i. For instance, the interior E-structure of a i may give rise to mappings which destroy the replacement axiom in the extension. These "new" sets a i which~ when added t o ~

,

$enerate a ZF-model are called "~eneri c sets". The forcing method

78

is a technique

to obtain

generic

sets.

that every

finite

be i n ~ ,

id est, a i has to fulfill

which

can be posed

in~

terior E-structure baum's

completing

a "complete

the main

E-structure

. Then a determination

process

idea is

of a i has to

finite amounts

of a i is obtained

sequeneo

of conditions

of the whole

in a way similar

in-

to Linden-

(see e.g. M e n d e l s o n [ 6 ~ p . 6 4 )

by choosing

of conditions".

In this chapter The extensions

Herein

part of the interior

~e shall not construct

soealled

"endextensions ~'

we are dealing with are those which contain

the

same ordinals!

A) THE FORCING The simplest

general

ZF is provided approach, work~

RELATION

IN A GENERAL

framework

by considering

a straightforward

of Solovay's

Let ~

partially

ni-ary relations

R i (i E I) defined

set in the sense of ~ . copy of 0 ~

introducing

V

limited

Eex%(x):

E is taken (read

system

of ~

in~

on A. We assume

I

~'happens" this

formal

shall in ~

ex-

~ we

to talk about

is done by

E e (intended

interpretation

less than e satisfying

~'Ensemble")

"there exists

in a certain

. Since

Formally

terms

set of sets x of rank as:

~

which has means

Ve separately. comprehension

to

which this copy has to ful-

way all that what language

that A is a

by adding

are expressed

from the french word

V x~(X)

gene-

with domain A and some

£ describes

as a ramified

every v.Neumann-Stufe of

~

in a very detailed £

here a slight

of ZF (see p.25 for the

We want to extend ~

. The language

construct

model

. The properties

fill in the extension press

This

original

e ! relational

language I

structures.

of Cohen~s

Let

be a first-order

a generic

of

approach.

of "stanCard").

(A;Ri)i

Cohen models

ordered

We shall present

~ (M,E M ) be a standard

definition ~:

for constructing

generalization

is due to R.M.Solovay.

ralization

SETTING

and limited

~; the

quantifiers

an x of rank less than ~ such

that ~(x)). Th~ A l p h a b e t h

oftheramifi~d

ianguage

i) One sort of set-variables: are used to stand 2) Set-constants

for these

v0,v~,v2,...,Vn,... variables.

x for each set x of ~ .

(n e ~). x,y,z,..

79

3) Constants

A. for each j e A. ] predicates ~i for each

4) ni-ary 5) logical

symbols:

6) limited

comprehension

V~

for each ordinal

It is possible the following

V

~

~, v

= ( 1,~

, V

operatirs ~ of~

to arrange

= ( 0,3

~i = { 5,i ) and

The formulae

of 2

tion as usual formulae

constitutes

Definition.

limited

) , E~

( = (6,0)

of ~

of a ranked

and

(for

no occurrence set-constant

£

formulae

are defined

comprehension

formulas,

terms,

and of a as follows: set-

then so are 7 %,

of~

such that

8 > ~, (iii)

~ contains

x for a set x of M i r i m a n o f f - r a n k

is limited

of a free variable

la without

free variables

comprehension

no occurrence

comprehension is defined

constants terms

of a limited

spect to the parameter X ~ . I n

no

> ~

(iv)

of a.3~ then

term.

as usual;

is said to be a limited

to the set-constants,

other

(i) % con-

of V 8 w i t h 8 > ~, (ii) ~ contains

of E B with

The notion

Definition.

of all

in

if ~ ~ I then ~ contains

respectively

by concatena-

formula with no free variables

tains no occurrence

above

symbols

(= limited)

than x~ and e is an ordinal

definition

) ,

A.] or variables, then ul g u~ are limited formulae.

(c) If ~ is a limited

the limited

= ( 3,X

that the collection

t e r m of

(b) If # and ~ are l~mited

refer

~

) , x

or constants

and ~i(u, ,...,Un.)

ESx~(x)

in

.

(a) If u, ,u2 ~... are limited

v

= (0,2) = ( 2,~

from these

comprehension

constants

are sets of ~

,) = ( 6 , 1 )

It follows

a class

The notions

symbols

= ( 0,4+i

are obtained

by reeursion.

quantifiers

, and finally brackets.

~ v : (O,l 7 , V

) , Vi

&j = ( 4 , j ) ,

exists).

E ~ and limited

that these

way: 7 = ( 0 , 0 )

) , e

i e I and e for membership.

(not, or, there

a limited sentence.

formuWe shall

of the form Aj (j e A) and

as constant

comprehension

terms.

Remark

that the

term is given with re-

most applications

we choose

k to be

~+1. Let p(x) be the M i r i m a n o f f - r a n k sense of ~

(see p.14).

term t is given by: (a) 6(x) (b)

= ~(x),

6(Aj)= I

(c) 6(Eax¢

(x))

:

of the set x in the

The degree

~(t) of,a constant

80

Abbreviations.

Let u and v be constant

terms or variables;

then

u = v stands

for A (x e u ~ x e v) where x is a variable distinct x For constant terms u and v~ u ~ v will stand for

from u,v.

A~x e u ~ x E v) where ~ = Max{~(u),6(v)}. x limited sentence. Next we define ring b e t w e e n

in ~

limited

a well-founded~

formulas

Read Ord(¢)

in

the the forcing relation ~

formulas

between

Obviously

"conditions"

m2 s + m°e + m we could define Ord(#)

For a limited 0rd(#) where

=

ordering

Ord(~)

to be

to these triples.

formula ~ define ~2.e

+

~.e

m

+

(i) e is the least ordinal

no q u a n t i f i e r

and limited

instead of defining

and then taking the l e x i c o g r a p h i c a l

Definition.

partial-orde-

to ~ an ordinal Ord(~)

as "the order of ~". This then allows to define

by induction on Ord(¢).

to be the ordinal (~,e,m)

localizable

~ by a s s i g n i n g

of 9 ~ . 9~

u ~ v is thus a

V 6 with

such that # contains

8 > e and no constant

term t of

degree ~ ~, (it) e = 3 iff ~ contains

at least one of the symbols

e = 2 iff ~ does not contain least one of the symbols

any ~i but ~ contains

A~, e = 1 iff ~ contains

symbol ~i and no symbol A~ but ~ contains

~i'

at no

a subformula

J

V S u where v is either a constant or a variable w h i c h stands quantifier

term with ~(v)

+ 1 =

in the scope of a limited

A ~ (for e defined

in (i)), e = 0 in all other

cases. (iii) m is the length of ~. Let S be an infinite sup{p(x);x

e S} where

0 ~ = p)(q I~ ¢).

(6) p IF V ~ ¢ ( x )

~ (~u

x (7) p I~ wi(u, ..... Uni) & p

!i-ul

=

&.

&

...

e T)(~(u) < ~ & p IF¢(u)). ~ (~j,,...,Jni

IR

& p

]~

strictly Further e =I .

smaller remark

occurring

=

by

~.

Ri &

).

In i

on Ord(¢)~

side of ~ have

occurring

in the d e f i n i t i o n

e A)(<j, .... j n ~ e

induction

on the right

that the f o r m u l a e

that

u

ni

To see that p I~ ~ is indeed d e f i n e d that the f o r m u l a e

= 1).

on the

of Ord(ul

notice

order

left side of ~.

~ u2) we h a v e

82

The definition given

of p I~¢ for arbitrary

in the Metalanguage

(ordinary)

(and not i n ~ )

by induction

length of ¢. This definition

over a set P and the collection

will be valid

of all formulae

a set in the sense of the meta-theomy sense of the meta-theory).

Z-sentences

Again

¢ will be on the since p ranges

of Z constitutes

(since ~ i s

a set in the

let u,v range over T and p,q

range over P. Definition

of p I~ ¢ for arbitrary (8)

p Ib u ~ v and p l ~ i ( u ,

(9)

p I~

,..,Uni)

< a & p I~ ¢(u)).

~ ( B u 6 T)(~(u)

to this definition

for limited

sentences

iff p I~ ¢ according

section

of the forcing

as above.

~ p)(q Ib ¢).

~ (3 u e T)(p I> ¢(u)).

that

¢.

are defined

(11) p I~V~¢(x)

The rest of this properties

¢ ~ ~(~q

Z-sentences

(12) p l ~ x ¢ ( X ) It is obvious

mata

(unlimited)

is devoted relation

¢ of £, p I~ ¢ according

to the former definition.

to the study of the formal I~. In the following

three

lem-

let @ be an~f Z-sentence.

Consistency-Lemma. Proof.

For no p e P do we have both p I~ @ and p I~7 ¢.

If p I~ @ and p I~ q ¢ for some p 6 p and some Z-formula

then by (9) p I~ ~ ¢ ÷ ~ p I~ ¢ and we get a contradiction metalanguage,

If p I~ ¢ and p < q, then q Ib ~.

Lemma.

Proof by induction tences

in the

q.e.d.

First Extension

on the complexity

# by induction

on the length

¢,

on Ord(¢)

of ¢ (i.e.

for limited

and for unlimited

sen-

¢ by induction

of ¢), see e.g. Jensen [~9]p.94-95.

Second Extension

Lemma.

For every p 6 p there

is a q 6 p, p ~< q,

such that either q I~ ~ or q I~-] ~.

83 Proof.

Suppose that for no q ~ p we do have q i~ ¢. Then p I~7 ¢

by (9). Suppose now that for no q > p we do have q I~7 ¢. Then by (9): p I~ 7(7 #). But applying p I~77

# * ~(3q

(Vq

(9) twice one gets

> p)[~(3 q' > q)(q' I> ¢)] > p)(3 q' > q)(q' I~ ~)

Thus there exists q' > p such that q' I> ~, q.e.d. Remark that forcing does not obey some simple rules of the propositional Furthermore, (I0),

calculus.

Exempl~ gratia,

p may force 7 7 ~ but not ~.

the forcing relation }~ has by definition

(12)) a homomorphism

(v,%) and existential

property with respect %o

quantification

junction ^ and universal

(V,3).

quantification

A

(^ ,&) or for universal p I~¢^ holds.

~ ~ (~q,

quantification ~ p)(~q2

We shall introduce

If we introduce property

(A,~).

~ p)[ql

the homomorphism

for conjunction

For example only

I~ ¢ & q~ I ~ ]

a relation I~f (called weak forcing), which

and universal

property

con-

then one no-

has the property that p I~*¢ ~ p I~7 -I ¢ and the homomorphism ty for conjunction

(5),

disjunction

as usual,

tices that I~ does not have the homomorphism

(clauses

quantification,

for disjunction

proper-

l~'does not have

and existential

quantifi-

cation and is, as we may say, dual to the strong forcing relation I~. Definition.

p I~*¢ ~ p I~-7(7 ¢) "p weakly forces p H ¢ ~ (p I ~

Qp

I~7 ~) "p decides

¢" ~"

p II*¢ ~ (p I~¢ ~ between elements p of

the set of conditions

(of 9 ~ )

in t h ~ " g r o u n d m o d e l " ~

and limited £-formulae ~ was given

while the definition of p I ~

was given in the underlying meta-theory.

for unlimited

We shall show in the

sequel that for each specific £-sentence ~ the forcing relation can be defined i n ~ ,

because # is finite and the construction of the

class K~ of p's forcing ~ can be done in finitely many steps. For each specific ~ the mechanism of constructing within ~

K~ can be implemented

but the mechanism is not universally applicable for all

sentences ¢ of £, so that within ~

we do not have the whole rela-

tion I>. This is not too much surprising,

since the definition of

forcing resembles very much the definition of truth, and by the Epimen~des-Tarski ~-language)

paradox we cannot define in ZF (or within the

the notion of truth for £-sentences

p.138, A.Tarski:

Logim, Semantics, Metamathematies

p. 248, Fraenkel-BarHillel: 1958) p.306 and Kleene: Groningen 1967) p.39,42,

[see L4vy [51] (Oxford 1956)

Foundation of Set Theory (Amsterdam

Introduction to Meta-Mathematics

(Amsterdam-

501, see also Mendelson [60]p.151].

However we can define forcing for a ~ingle given sentence # or f ~ some particular family of senten-ees w i t h i n ~ . Lemma I: Let ~(x,,...,x n) be an unlimited formula of £. There is a class K# of the model ~J'~ whose elements are the (n+l)tuples (p,ul,...,u n) such that p l ~ ( u l , .... Un) , where the ui(l ~ i ~ n) are constant terms. According to our remark on page 79 the constant terms u i are considered as certain special finite sequences of symbols which are in ~ -

for more details see Easton's thesis, Annals of math. Logic,

vol 1(1970). Proof by induction on the length of the formula ~. Since the atomic formulae are all limited formulae,

the lemma is true for

atomic ~. If ~(xl ~...,x n) is ~,(xl .... ,xn) v ~2(x,,..,x n) and the classes K~, and K~2 satisfy the lemma for ~i and ~2 respectively~

87

then K@ : K~I U K~2 is the required class for ~. If @ is V y ~ ( y , x l , .... x n) and if K~ satisfies the lemma for ~(x0,xl,...Xn) , then {(p,z); ~y(p,y,z) is V ~ ( y , x ,

E K~} is the required class. The case that

.... x n) is similar to the previous

7 ~(xl ,...,x n) and if K~ satisfies

{(p,z)

; p c P ^ ~Vqep(p

one. If @ is

the lemma for ~, then

~ q ^ # and ~ ~

are equiva-

lent. Lemma L: Let ~ be a complete sequence of conditions an Z-formula.

and ~(x,,...,x n)

If ~ [Q ul = vl ,...,~ I~ u n = v n for constant

terms u, ,... ,Un,V , ,...,Vn, then [~ #(u, .... ,u n) ~ ~ [~ ~(v I ,... ,v n).

This follows by induction on the length of ¢ from lemmata F~ G and corollary H. So far we have investigated forcing relation.

complete sequence of conditions predicates

several useful properties

gives raise to a valuation of the

~j so that the resulting

B) COHEN - GENERIC

sets are generic.

SETS

We shall use the terminology and formalism introduced Definition:

of the

In the next section we shall show that every

in section A.

Let ~ be a complete sequence of conditions.

Define

the function val~ (valuation or interpretation with respect to ~) on the set T of all constant terms of the language £ by induction on their degree as follows: valB(u)

= {val~(v);

Finally define:

v e T & 6(v) < 6(u) & ~ t~ v e u}

89 : {val~ (u);u 6 T}

~

(we shall usually omit the subscript ~ from val~ and

~).

Le~ma M: Let u and v be constant terms. If p [~ u e v then there is a constant term w such that ~(w) ~ ~(u), 6(w) < 6(v) and p Ibu = w, p I~ w e v. (for a proof see e . g . A . L ~ v y

[511p.141).

Lemma N: ~ I~ u = v ~ ~ i~ u ~ v ~ val~(u) ~ val~(v). Lemma O:

~

is a transitive set. For each x 6 ~ ,

hence

~

Proof: The transitivity of of val~ and

~.

val~(x) = x,

C ~. ~

follows directly from the definitions

val(x) = x follows easily by induction on ~(~)

using the definition of val(x), the forcing-definition lemma N. Thus the witnessing tained in ~

constants x ensure that ~

and the is con-

as a transitive submodel.

The semantics of £. For each x ~ ~

let r(x) be the least 6(w)

for which val(w) = x. Thus x,y e ~

& x 6 y ~ r(x) < r(y) by

lemma N. Now the formulae Qf £ can be interpreted

in ~

in the

following way: (i)

A term u is interpreted

(it)

u e v holds i n ~

in ~

(iii) The sentential connectives 7, v rare

by val(u).

iff val(u) 6 val(v). and the existential quantifier

interpreted as usual by -, ~ and ~ .

(iv)

V~(x)

holds in ~

iff there exists y E ~

(v)

y, ..... Yni satisfy ~i(x

with r(y)
m. By d e f i n i t i o n

E~x~(x)

= tf.

of ~

~ f is a limited

By our a s s u m p t i o n

thus by lemma

p J~ Fnc(tf) of numbers

^ Fnc(tf -I)

j such that

be any

(sufficient

Dom(p)

C ~ × k. This

that

Otherwise

in ~

that

term

f is one-to-oNe;

p 1~(gj,aj)

all

number

such

j e oec(~)

C

be the

~-set

tf : Eex#(x). that

Let k

occ(¢)

are smaller

C k and

than k and

i 6 m, e E 2, then n < k. e tf for

exist

and n a t u r a l

= b ^ Range(tf)

6~. Let occ(¢)

in ~, where

natural

means:

there w o u l d

of forcing)

^ Dom(tf)

sequence

aj occurs

large)

((i,n) ,e) 6 p for some

We c l a i m

it holds

comprehension

P:

for some p in the complete

if

a contradiction.

step.

c of b. We claim

that

obtained

be a o n e - t o - o n e

f be any o n e - t o - o n e

subset

we have

that b = {aj;

some m e m b e r

in the

3.Step.

all

in ~

Otherwise

proved

some

I st E x t e n s i o n - l e m m a ,

j > m = k + 1.

an e x t e n s i o n

numbers

n, ,n2

q of p (by the d e f i n i t i o n

such that

n~ ~ n2 , m ~ nl ,

m ~ n2 and (0)

q l~ ( an, " 'an2 " ) e tf

Choose

h 6 ~ such that (< n,j) ,e) 6 q implies

that n, ~ h, n2 # h. Define ~(n2)

= h, ~(i)

lemma

to

a permutation

n < h, j < h and

~ on ~ by ~(h)

= i for i 6 ~-(h,n2 }. An a p p l i c a t i o n

such

= n2 ,

of the s y m m e t r y -

(0) yields:

~(q) Ib < an, ,Ah) c tf since occ(~) q(Eex#(x)) q U a(q) ~(q)

is a f u n c t i o n

.Therefore

q U o(q) Hence

C_ k < m ~ nl ,n2 , hence

= Eexs(~(x))

and hence

by the first

[~ (an, ~an 2)

q u o(q)

: Eex~(x)

I~ &n2

o(#)

= tf.

= ~, thus q(tf) By d e f i n i t i o n

a condition

extension

lemma

extending

both

and lemma

B:

e tf ~ (&n, '&h ) e tf ^ Fne(tf) = tf(an, ) = ah'

since

=

of o, q and

^ Fnc(tf-l).

tf is a function.

This

103 contradicts the result proved in the first step. Thus in fact p ~tf(aj)

= aj for all j > m and f must be surjectiv. This

finishes the proof the theorem. D) THE POWER OF THE CONTINUUM IN GENERIC EXTENSIONS If we assume the axiom of choice (AC), then every set x is equipotent with precfsely one aleph M s. If reals then

~

= ~

~

is the set of all

for a certain ordinal ~. Is it possible to

determine this ordinal? It follows from Cantors theorem A x(~ < that ~ > I. G.Cantor has spent many years in order to solve this problem without arriving at the determination of the value for ~. The natural approach to this problem is to determine the cardinalities of various subsets of perfect set has cardinality 2M° compact subset of

~.

Cantor showed that every

( a set is perfect iff it is a

~, non-void and every element of it is an

accumulation point of it). Moreover, the Cantor-Bendixion-theorem asserts that every closed subset of

~

is either countable or the

union of a perfect set and a eountaS%e set. Thus no closed subset of

~ has a cardina$ strictly between Ro and 2 M° . Some further

results of classical descriptive set theory read as follows: (a) Every uncountable ~ - s e t

of reals contains a perfect subset.

(b) Every ~ - s e t

of reals is the disjoint union of MI many Borel sets. It follows that every ~ - s e t has power < M0 or = 2~0 Since Borel-sets are ~ every ~ - s e t notion ~ ,

(Souslin's theorem), hence ~ ,

of reals has cardinality < M, or = 2~

(For the

etc,.., see chapter II, page 44). For a treatment of

these results see: [~]

it follows that

[~8] an~:

A.A.LJAPUNOW: Arbeiten zur deskriptiven Mengenlehre; V.E.B.-Deutscher Verlag der Wissenschaften, Berlin 1955.

Since it was impossible to exhibit a subset of strictly between M1 and 2 ~ (CH)

2~

~

of cardinality

, Cantor conjectured in 1878 that

= ~,,

called the "Continuum-Hypothesis".

David Hilbert listed this

problem as the first problem in his famous list of unsolved problems at the first international congress of Mathematicians in 1900 in Paris. Despite many ~ttempts this problem remained for a long time unsolved. It was however used freely in proofs since it turned

104

out to be a powerful

assertion

W.Sierpi~ski

a large number of propositions

deduced

C1 -

C82)

[~91

W.SIERPI~SKI:

from

Hypoth&se

In the lit erature

du continu.

New York

there are many papers

to other

concerning

H.Rubin has shown, some r it holds

a series Logic

(GCH)

in which

of notes

published around

is discussed

on the

(GCH)

What

Kurt G6del

II of these

it fellows,

(GCH)

lecture

Hence

cannot be decided including Theorem

Proof.

has published of formal

K.GSdei

a survey

has on results

the more philosophic

Problem?

(GCH)

notes.

Since the

provable

question

Amer.Math. vol. SS(1948)p.151.

is consistent

Thus the

here a proof~ (GCH)

that also the

is neither

[ 9 ]-[I~]

B.Soboci~ski

Corrections

in 1938 that the

(AC) to the axioms

P.J.Cohen

Continuum

54(1947)p.515-525,

(GCH)

that

implies

(GCH)

has shown that the

(GCH)

cannot

(AC) cannot the

(AC) -

in the system

if we are

willing

set theory

(CH) or even the (GCH)

the truth or the falsity

with

is not a theorem

nor refutable

arises:

of Zermelo-Fraenkel

then the continuum-hypothesis

ZF + (AC).

in ZF to:

q, then for

of the (GCH):

ZF. But now the following add the

is equivalent

in which he gives

from the ZF-axioms.

of ZF. Thus

contributed

in the Notre Dame Journa!

in ZF. We have presented

24 -

and proved

°

3 and 4 (1962,63).

is Cantor's

showed

ZF, see chapter be proved

(GCH)

and in which he discusses

Monthly

be refuted

•

p and q, if p covers

(GCH)

I~ II, I!I~ vol.

K.GODEL:

•

W.S1erplnskl

AMS.65(1959)p.282-283).

in 1947 an article

the

see page

that the

cardinals

problem of the "truth" [Z6]

statements.

(CH) or the

that p = 2 r".

Bull.

(parts

1934

the (GCH).

e.g.,

"For all transfinite (see H.Rubin,

(there called

Warszawa-Lw6w

•

papers

situations.

1956).

continuum-hypothesis

to be equivalent many

symplified

(CH),

(2nd edition,

generalized

and also often

ZF, is

derivable?

is not provable

of the continuum

on the basis of the usual axioms

to

in

hypothesis

of set theory,

the axiom of choice•

(P.J.Cohen).

If ZF is consistent,

is consistent

too.

not a theorem

of ZF + (AC).

Let ~

Thus

be a countable

then ZF + (AC) + 2 M° > ~2

the continuum

standard

hypothesis

(CH)

model of ZF + (AC).

is

We shall

105

construct

an extension

new subsets

(though

that we will

Since

Define

find

a ramified

x for x 6 ~

hence has only eoun-

~ are all classes on p.79).

Define

from m × ~2 into

of is hope, since

£ which has besides

existential

quantifiers

E ~ (for ordinals

predicate

of ~ t ~ (use e.g. a condition

Z = {0,1}.

(see page

81-82)

the

symbol

a.

x ~ x~ ~ ~ E ~

the standard

p to be a finite

Define

the

V e~ the

~ in ~ ) ,

so that the correspondences

presented

{~ in the usual way

subsets

) M,), there

of m not yet i n ~

, a further binary

and ~ ~ V

function

language

operators

Define £ in such a way~

. We shall

of~.

the limited

comprehension

constants

many subsets

outside

in ~

so many

to add B~-in the sense o f ~ - n e w

~q~ is countable,

(M2)~-

in ~

ZF-symbols,

limited

generically

= HI is violated

these sets have cardina!ity

is countable

usual

2~

adding

(in the sense of the mete-language)

in ~

(~)~

~by

it is sufficient

of m to ~ .

tably many

of

of m, such that

show below that subsets

~

trick

partial

the forcing relation

containing

the following

key-clause:

p((n,9)) This means system

~

in terms = (A,R)

introduced

a generic

a complete

sequence

function

= ~2 y~u

copy of

and defining

B, our Hauptsatz

tells

~

the valuation-

us, that the model

of ZF which contains

a = val(Ea(x,y) a(x,y))

(the superscript

indicates

C ~ x MI

~as

a

~ for

that the concepts

are

in the sense of ~ ) .

By a theorem proved

in section

we have added t o ~ , set. Thus

it remains

by the t r a n s i t i o n from

if we define a~

C, ~C~ is also a model

of (AC),

since

a model of ZF + (AC), only one new Cohento show,

hypothesis is wrong. Since ~TT~ is the ordinal

for ~ < ~ = {n;

n e ~

then a~ ~ ~ and ~, ~ ~ theorem)

case A : 1 = {0} and R = ~,

in S = ~ × ~2. Thus by choosing

in this way is a model

and contains

understood

generic

~

~ of conditions

as in section

~obtained submodel

in section A: We take as relational

the very special

and choose

= 1).

and we get

m29T%

that in ~

in ~ a n d

to

the continuum ordinals

is an ordinal of

are preserved Thus

= y: ^ (n,~)

e

+ ag, ~ a ~

a}

(as in the proof of the preceeding

106

=7%

2 ~° > y y is in ~

(in ~

).

the second infinite cardinal:

~ : m2 7[b ; We shall show

that cardinals are preserved in the extension, which is a cardinal in ~

is a cardinal

i.e. an ordinal

in ~

and vice versa.

Then it will follow that y : m29~ is also the second in£inite cardinal in ~ n

:

y : m2 yt

, and hence 2 M° > M2 in ~

as desired.

To

this end we need some lemmata. Lemma 1. If B i~ in ~

a set of conditions

are pairwise incompatible, (~B

Proof.

C Cond)[B 6 ~ h

such that its elements

then B is ( i n ~ )

countable.

& ( ~ p l ,p2 e B)(p, ~ p2 ÷ p, u p2 ~ ¢tnd)

Cond is The ~'~-set of all conditions.

false, and let B be a set of ~ ,

Suppose the le~ma is

such that p,,p2 E B ÷ (pl = p~ v

p, U P2 ~ Cond) and ~7~L > m. Define B n : {p 6 B; { < n} n_L~_~lBn=

: B and B is uncountable,

B n is ( i n ~ )

Since

number n 6 m such that

there is

uncountable.

There are conditions

q E Cond such that (p 6 Bn; q C p} is in

~still

uncountable,

namely the empty condition q = ~ has this

property.

On the other hand the cardinality of all such conditions

q is bounded by n, since q C p. Thus we may define m to be the greatest natural number such that there exists a condition q such that ~ = m and {p e Bn ; p ~ q} is in ~ a~ondition

uncountable.

of cardinality m having this property.

Let q0 be such

Now choose in

{p e Bn; p ~ q0} any condition pl • Since in B all conditions pairwise

incompatible,

are

the elements of {p 6 B; p D q0} are also

pairwise incompatible. pl-q0

is not empty, since otherwise p, : qo and p, would be

included in all conditions

in {p E B; p D q0}, and hence compatible

with them. Thus we can find ((k,9),e)

6 pl-qe

is contained

uncountabl~ many conditions

in (in the sense o f ~ )

from B* : {p 6 Bn; p ~ qo}. This follows,

such that ((k,~),l-e)

since p, is incompatible

with every p e B*. It follows that {p e Bn; p ~ qo U { ( ( k ~ ) ,l-e)}} is uncountable

iN the sense of ~

and q0 U {((k,~),l-e)}

has

cardinality m+l, a contradiction to the choice of q 0 ~ a x i m a l dinality having this property.

car-

Thus lemma 1 is p r o v e d . ~ ;}~

Lemma 2: If f is a function in ~

, such that Dom(f) 6 ~

Range(f) C x for some x e ~ ,

and

then there exists a function

107 g in ~

such that Dom(f)

x, and g(s) is i n ~ Proof.

Since f 6 ~

= Dom(g),

countable

, there

Range(f) ~ R a n g e ( g )

for every s 6 Dom(f).

is by definition

of ~

a term tf

of the forcing language ~ such that holds in ~ (for x,z E ~ ) :

f e vai(tf).

(*)

~ if + v = w] ^ Dom(tf) = ~ ^ Range(f) C x.

Since

~u 4~w

~

complete

|(u'v)

is a generic

e tf ~ (u,w>

extension,

Thus the following

there is a condition P0 in the

sequence ~ (which d e f i n e s ~

) such that pc forces

(*)

-see lemma P in section B. Using weak forcing and lemma A of section A, this entails: ('')

(Vu,v,w

6~)(~q

> p0)|q l~'(u,v) e tf & q IC(~,w3

Further,

for every u 6 Dom(f)

there is a condition

sequence ~ such that p' i~*(~,f(u)) (~,f(u))

E tf holds

in ~

e tf = V : W] .

p' in the complete

~ tf (thiS follows

since

). Since both p~ and p' are in ~ and

is totally ordered by ~ we obtain that P0 U p' is a condition. Hence,defining g(s) = {y; y 6 x & (9 p' > Po)(P';~



6 R. Hence we have obtained • -

%~.

that

E R~

six generic terms Ai, ~ .... ai6

(i)

~ S~ai4

16

(ii)

~ 1~(~im.. < ~ik v ~ik ~ aim ) for m,k = 1,2,...6 with ( m , k )

(iii) ~ l{n(~im ~ aj V ~

# (4,6) ,

< aim)

for m = 1,2, .... 6 and j ~ c = occ(t) (iv)

3~u b (ai, ,ai ) 6 R ~ (al2. ,a13.) E R.

Since everything which holds weakly)

by some condition

in ~

must be forced (strongly or

in the corresponding

complete sequence ~,

we obtain, that there is a condition p in ~ such that p l~(t) By the restriction

^ (ail,ai2) e t ^ (ai2,ai3) e t lemma, we may assume that p contains

many ordered pairs

(in,i> ,e > (with n 6 m, e E 2) only with i6occ(t)

U {i,,i2,i3}

Define

finitely

117

p1(oec(t),i, ,i2) = P/oee(t)

U {il,i2}

p~(oec(t),i2,i3)

U {i2,i3}.

Then by the restriction (+)

= P/oce(t)

lemma:

pl(oce(t),i, ,i2) I~*T(t) ^

of pairwise

in ~

x

,... ,hn).

, then

disjoint

basic

(for 1 ~ ~ ~< n) and the is any sequence

of A such that h~ 6 br~ for'l ~< ~ ~ n~ then

7[

none

~. Let g ----(gl ~...,gn > be a sequence

of A. If %(gi ,...,gn ) holds

open sets of 2 ~, such that g~ E br~ following

first

b r N A = ~ . Thus

ai (for i E ~), but % may contain

of different

in the

subset of 2 ~, q.e.d.

other than x, ,...,x n and suppose

for x e ~

that

which has p as

Let ~(x, ,...,x n) be an i - f o r m u l a

of the symbols

members

~

that b r N A ~ ~. This

that p (and hence

of A and A is in 9 ~

Continuity-Lemm~.

e b r- It follows

of different

134

Proof.

Suppose that for sets gl ,...,gn E A the sentence ~(g, ,...,gn )

holds in ~

. Let tl ,...,t n be constant terms of £, such that g9 =

val(t v) for I

~ r),

~- ~ if ~ is a variable

and ([r,%] ,_i) ~ r

for:

Vx tX s r ^ A

(Y

x-

(A:(z

(the Kuratowski-definition ted s e n t e n c e

of

/?)

and

y

[r,~]) v