Infinitary Logic: In Memoriam Carol Karp

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Series: Departmentof Mathematics, University of Maryland,College Park Adviser: L. Greenberg

492 Infinitary Logic: In Memoriam Carol Karp

A Collection of Papers by Various Authors Edited by D. W. Kueker

ETHICS ETH-BIB

IIIITIlUlLIqlllllLJqtllLLIlllllLIklll O0100000567778

Springer-Verlag Berlin.Heidelberg 9 New York 1 975

Editor Prof. David W. Kueker Department of Mathematics University of Maryland College Park Maryland 20705/USA

Library of Congress Cataloging in Publication Data

Main entry under title: Infinitary logic. (Lecture notes in mathematics ; 492) i. Infinitary languages--Addresses~ essays~ lectures. 2. Model theory--Addresses~ essays~ lectures. I. Karp~ Carol, 1926-1972. II. kueker~ David W.~ 1943III. Series: Lecture notes in mathematics (Berlin) ; ~92. QA3.L2$ no. h92 [QA9.37] >lO'.Ss [511'.3]

75-34~6~

A M S Subject Classifications (1970): 02 B 25, 02 H 10, 02 H 13

ISBN 3-540-07419-8 ISBN 0 - 3 8 7 - 0 7 4 1 9 - 8

Springer-Verlag Berlin 9 Heidelberg 9 N e w York Springer-Verlag N e w York 9 Heidelberg 9 Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9 Heidelberg 1975 Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

CAROL KARP

CONTENTS

INTRODUCTION

by E.G.K.

P a r t A.

BACK-AND-FORTH LOGICS

Part

B.

Lopez-Escobar

by D a v i d

CONSISTENCY LANGUAGES

Part

Part

C.

D.

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

ARGUMENTS W.

Kueker

PROPERTIES

by J u d y

AND

Green

FOR

FINITE

PROPERTIES

BACK-AND-FORTH

TECHNIQUES

IN

LANGUAGES

by E l l e n

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

ON A F I N I T E N E S S LANGUAGES

CONDITION

by J o h n

Gregory

73

OF C O N S I S T E N C Y

INFINITE-QUANTIFIER Cunningham

17

QUANTIFIER

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

APPLICATIONS

AND

INFINITARY

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

CHAIN MODELS:

I

FOR

INFINITARY

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

Each part has its own complete

125

table of contents

143

ACKNOWLEDGEMENTS

The origin of the present

volume

is explained

I wish to thank the editors

of Springer's

Logic series

its preparation.

generously

for suggesting

given advice and suggestions

of the book.

I am also grateful

in the Introduction.

Perspectives E.G.K.

i_~n Mathematical

L6pez-Escobar

during the planning

to Pat Berg who carefully

entire volume.

David W. Kueker

has

and editing typed the

INTRODUCTION

BY E,G,K, LOPEZ-ESCOBAR

INTRODUCTION Carol

Karp died on August

cancer w h i c h had had always taught tive

been more

remained

in her one p u b l i s h e d to be brought

students

not be able

dents

are:

Gauntt

lege,

Fullerton),

and even

her

during

to c a r r y i n g

was p u s h e d

Her

teaching

illness

she

with her usual

new m o n o g r a p h

early work was

she r e a l i z e d

against

out her a d m i n i s t r a -

forward

the p l a n n e d

that

on infin-

collected

it very much

R.J.

(Ph.D.

(Ph.D.

J. Gregory

1972,

apprehensive

to complete

all now have

their

their

1969,

needed

(Ph.D.

1969,

but her fears

degrees.

Her

stu-

State

Col-

now at S.U.N.Y.,

Camden)

of the conduct persists

studies;

Ph.D.

now at St. M a r y - o f - t h e - W o o d s

and w a r m p e r s o n a l i t y

that her d o c t o r a l

now at C a l i f o r n i a

now at Rutgers,

them as to us the m e m o r y spirit

To her,

the end she was r a t h e r

would

1974,

battle

up to date.

they

(Ph.D.

but

after a brave

long years (I)

unfinished.

book,

were unfounded:

J. Green

tod,

but u n f o r t u n a t e l y

languages

1972

in a d d i t i o n

Her research,

determination,

Towards

for three

than a duty,

all her classes

tasks.

itary

lasted

20,

and E.

College,

Cunningham

Indiana).

of her life,

as a lasting

Buffalo),

To

exceptional

inspiration.

THE PLANNED MONOGRAPH The new book was tives most

in M a t h e m a t i c a l

intended Logic.

of her own r e s e a r c h

nately nerable

the n e c e s s a r y to virus

treatment

infections,

to o r g a n i z e

the m a t e r i a l

chapter

to be based

was

work

for the Carol during

Springer

Karp was the

in 1972

in the habit

chapters.

Generalized

(1)Her biography and bibliography follow this introduction.

of doing Unfortu-

left her exhausted

so that theme was barely

lectures

Perspec-

summer months.

for the b e g i n n i n g on her

series,

and vul-

time

for her

The final Recursion

3 Theories at the M a n c h e s t e r

Summer School

(1969, unpublished),

and

from her notes it is a p p a r e n t that she viewed it as the greatest c o n t r i b u t i o n of the monograph.

The gist of the programme was to show

that the most natural way of g e n e r a l i z i n g r e e u r s i o n theory was through r e p r e s e n t a b i l i t y formulas.

in formal theories with infinitely long

The p r o p o s e d outline of the whole book was as follows:

CHAPTER I.

Partial isomorphisms.

Type of

structures. CHAPTER II. CHAPTER III.

Infinitary formulas. LaB L - m o d e l

theory:

consistency

properties. CHAPTER IV. CHAPTER V.

Admissible

sets.

L A - model theory: cofinality

CHAPTER VI.

~

countable case,

case.

Implicitly r e p r e s e n t a b l e predicates.

THE PRESENT VOLUME AND CONTRIBUTORS As the notes for the new m o n o g r a p h were so sparse, the editors of the Springer series decided that it would do more justice to her name if a separate book were w r i t t e n in her m e m o r y rather than comm i s s i o n anyone to complete her outline.

Furthermore,

in view of

Carol Karp's strong interest in the d e v e l o p m e n t of m a t h e m a t i c a l gic at the U n i v e r s i t y of Maryland,

lo-

it was agreed that the volume be

p u b l i s h e d under the M a r y l a n d section of the Springer Lecture Notes in Mathematics. Since in the section headings references

of Chapter I, there were many

to the a u t o m o r p h i s m results of David Kueker and since he

is also at the U n i v e r s i t y of M a r y l a n d he has kindly agreed to be both editor and a c o n t r i b u t o r to the book. The contents of Chapters III and IV and V had not been

subdivided

into

sections.

clear that the results tant part

of those

w).

to include

results

At the time

she agreed

to w r i t e

exactly results

The general

(cf(e) = w); Tarski

would

were

to work

contributing

did give

course

orginal

izations

at M a r y l a n d theory.

liberal

of r e c u r s i o n

crit e r i a

the 1968

she was

infinitary Kreisel

if it held Gregory who appears

results

of

were

Lee

Karp was not some indemodels. latter

languages

Lee

in the

Sr.

Ellen

Sr.

Cunningham

is

Now a l t h o u g h

However,

to r e t u r n

of r e c u r s i v e

the final

in the

gave rise

60's.

a "generalized set.

formulation

definability

The p r o b l e m of the theorem

interesting

finiteness

in order

theorem"

Karp gave the p r o b l e m

to make

for

results.

in 1969 that the a n s w e r was negative. thesis;

general-

with her general

(or compactness) to many

to

sets but of

so as to obtain

consistent

Karp

on infinitary

Karp w a n t e d

of proof

(and thus

scarce.

of 1970

definition

notions

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

in Gregory's

Karp

(cf(e) =

published

student,

even more

of the finiteness

formulated

Carol

and the

for the

w-chains.

Basically

by Xreisel

languages

showed

w-chains

led to it by the m o d e l - t h e o r e t i c

put forward

formulation

however

the use of general

in the Fall

theory.

was to be p r o o f - t h e o r e t i c viewpoint)

VI were

proof-theoretic

using more

Lew

of her results.

on Chapter

lectures

languages

languages

of models

out the theory

logic and r e c u r s i o n G~del's

a theory

to be another

She had o b t a i n e d

to

it was

to be an impor-

the new m o n o g r a p h

Karp had asked her

an account

The notes

on the

through

some of the p r e l i m i n a r y

Cunningham,

were

fragments,

later changed

to initiate

Festschrifft.

concern

be the results.

in her thesis

models

applied

right

results

planned

pend e n c e

were

Green's

and their

sure what

thesis

available,

Green thus also agreed

cf(e) = w)

had also

for

from the notes

in Judy Green's

chapters.

of the contributors. (especially

However,

it a v a i l a b l e

In

and asked to John The proof he has

5 agreed to i n c l u d e the r e l e v a n t

sections of his thesis

in this vol-

ume.

A REVIEW OF KARP'S RECENT WORK The first book on infinitary languages had had a very favorable reception,

nevertheless,

In the first place, last eight years; austere.

Karp was not c o m p l e t e l y satisfied with it.

the subject had p r o g r e s s e d e n o r m o u s l y in the

secondly,

the style of the book was perhaps too

She was c o n v i n c e d that a freer style would be a great

improvement;

she also w a n t e d to make sure that no librarian would

dare catalogue the new book in the dead languages

section

(as appar-

ently h a p p e n e d to the earlier work at a British university). change in outlook went beyond her

monographs.

This

She always looked

for a coherent view of logic and n a t u r a l l y as the subject d e v e l o p e d what made good sense in 1957 need not do so in 1972.

It is thus not

s u r p r i s i n g to find that her view on the role for infinitary languages changed during her lifetime. At first, a l t h o u g h she o b v i o u s l y enjoyed w o r k i n g with infinitely long formulae, them.

she did not appear to have a very high opinion of

For example in a 1960 r e s e a r c h proposal to the National Sci-

ence F o u n d a t i o n she states:

From the point of view of metamathematics,"formal" calculi based on languages with expressions of infinite length and having infinitely long formal proofs, are of no value. But in recent years, it has been noticed that algebraic results can come from the study of formal systems ..... From this point of view, it is very profitable to consider infinitary formal calculi. It is p r o b a b l y fair to state that at the time when she was w r i t i n g her thesis Karp c o n s i d e r e d h e r s e l f to be p r i n c i p a l l y an "algebraic logician".

Her i n c l i n a t i o n towards algebra was never c o m p l e t e l y

f o r g o t t e n and she always

seemed able to draw results c o n c e r n i n g

Boolean

algebras

She had

also

in the

1966

someone

added

mulae;

the

However, finite

operation Karp

tled

of

Holland

problem

languages

quantification

In the

languages On the rules

all

of LaB

one h a n d and

~

required

with

for e x a m p l e

results

in B r o u w e r i a n in

about

em-

algebras.

implicative

In

algebras;

she had

obtained

is best".

of d e f i n i n g

substitution

infinitely

sequences natural

most

people

infinitely

and w o r k e d

long

for-

of s y m b o l s

extension

of the

complications choose

or as

the

arise

latter.

long

formulae

as t r a n s -

the

required

theory

out

(it a p p e a r s

of the

in the

second

long

formulae

infinitely

is that LaB

of c o m p l e t e n e s s .

(admitting

sequences

obvious

chapter

a > wI had

of

Lw~

that

On the 6C < ~

laws

shown to other

of

of h e r

the

the

LaB hand

rule

the

obvious

she had

and

was

y-dependent

of

For the

problematic.

give

that

~
w I) did not

6 < a

case

variables

calculus

situation

that

been

of l e n g t h

of the a x i o m s

a complete

whenever

a complete and

individual

predicate

we do get

Karp

of

has

In the

conjunctions

extensions

~i ~ with

to o b t a i n

distributive

some

but as t e c h n i c a l

L

axioms

such

are

the r e s u l t s

is the m o r e

first-order

axiomatizations. ~,

were

Boolean

calculus

over the

of the

case

what

languages.

structures,

well-ordered

symbols

nature

the n e x t

inference

there

infinitary

book).

the

B) we h a v e

other

results

two w a y s

former

of

the

algebras

her

"For me,

concatenation

infinitary


, dk E A

..,a{+l>

are s i m i l a r

such that

is some

and c o n v e r s e l y ,

we can find such an

So

9

there



ck ( B is

The m a p p i n g

then d e f i n e s

an i s o m o r -

~. this a r g u m e n t

slightly.

Call a f u n c t i o n

f

from

22

a finite if the to

subset

domain

of

of

A f

is

if

f

(or,

Using

(~*)

morphisms

b

we can such

construct

that

It is c l e a r

of

that

isomorphic

"partial

isomorphisms"

are

~

~

I

with

the

of

fn

onto

this

is s i m i l a r

and

a E A

partial

iso-

and

a (dom(g)

fl !

"'"

method only

of

onto

showing

on the

B,

and

~

and

~

submodels

two

iso-

is t h e r e -

c o u n t a b l e mod-

of a f a m i l y

(**).

for a r b i t r a r y

be m o d e l s

written

that

existence

condition

property

of

A

of p a r t i a l

~ .

this

isomorphisms

f0 ~

maps

satisfying

partially isomorphic,

set

a chain

depends

expresses

Let

f c g

into

becomes

is some

that

g : Un(w

els are

Definition.

such

~

E ran(g)).

an i s o m o r p h i s m

definition

isomorphism

there

of



partial then

g

and

(*) t h e n

b E B)

(or,

partial isomorphism

a

Condition

is any

morphism

B

{a~ .... ,a{}

.

(**)

fore

into

The

following

models.

for a l a n g u a g e

~2 ~ '

if there

of

onto

~

of

L.

~

and

is a n o n - e m p t y

submodels

of

back-and-forth property:

for any g E I

f E I such

and

that

a

E A

f c g

(or,

and

a

b

E B)

there

E dom(g)

is some

(or,

b E ran(g)).

We w r i t e

I:

~ m2 ~

to i n d i c a t e

with

the b a c k - a n d - f o r t h

"2"

is to be

The a b o v e parts.

First,

or

there

B,

found

in the

argument since is the

property.

that

I

(The

is a set of i s o m o r p h i s m s

explanation

generalizations

for

(*) did

Cantor's

theorem

not d e p e n d

following

in the

now

on the

observation

of the next

splits

section.)

into

countability

on d e n s e

subscript

linear

two of

A

orders.

23

Any two dense

i.i THEOREM.

partially

linear orderings without end-points are

isomorphic.

Secondly,

there

is the

following

result

which

holds

in c o m p l e t e

generality.

1.2

If

THEOREM.

Furthermore, g

of

~

onto

~

n ( ~},

of

I

for all

k.

Then

since

Remarks. morphism

(2)

(I) of

If ~

class

are

first the

[6]),

the

class

iants

(this

we

define

9

fn

and

class

that

last of

of

uses any

P~

f = f0 ~ fl ~

and

see

and

.

two

the

class

models groups

of

is an i s o m o r -

is an

isomorphism.

if

f

is an iso-

of b a c k - a n d -

models of d e n s e

in a c e r t a i n linear

of a t o m l e s s

of a c o m p l e t e with

and

is a d e n s e are

"''

and

literature

class

[I]),

of the

.

countable the

A- P~

B

In fact,

~ ~2 9

and

b k E ran(f2k+2)

onto

in the

torsion

applications

isomorphisms

{f}:

u-saturated





satis-

theorem.

there

proof

and

quantifiers,

Let

where

~

because

I SI _< K

variables.

3y~(x,y)

infin-

I= ~[~].

That

can a s s u m e

Next

~

to the

in

and

we

~.

Then

quite

K

(ii) as follows.

(iii) are obvious.

Assuming

By the b a c k - a n d - f o r t h

equivalence

there

are

functions

(iii) we

properties

of

f : A2n+I---+A,

such that

if then

L-inequivalent,

n-tuples which are

n E ~.

(ii) : >

-elementary

n ( m,

~

(~,ao,. .. ,an_ I) -cow (~,bo,...,bn_ I)

(~,ao,''" ,an_l,a n) -oo~ (~,bo,''" 'bn-l'fn(ao ' ....a n ' b o " " ' b n - I ))"

(iii),

there

is some

a0,...,an_ I ( A

there

A0 ~ A are

IA0J

with

~ K

b0,...,bn_ I ( A 0

such that

for any

with

( ~ , a o , " "" 'an- 1 ) -coco (~'bo'" "" 'bn-l)" Let

B

be the c l o s u r e

and

B

is

ward

induction

the

of

universe

A0

of

on f o r m u l a s

some

brief

applications of Scott's We have tions

survey

already

preserving

al a p p l i c a t i o n s

iff

many

referred

methods.

a result

to a b e l i a n

equivalences groups;

~

so

interesting

see,

and i m p o r t a n t

Borel

theorem There

for example,

-I

({i) holds.

One is Scott's

[8].

that

n]

and

general

JB 1 ~ K

straightfor-

shows

on i n v a r i a n t

to F e f f e r m a n ' s

infinitary

L

of

I= ~ [ b l , . . . , b

has o m i t t e d

to prove

Then

A completely

~

of b a c k - a n d - f o r t h Theorem

~ c ~.

Therefore

b l , . . . , b n ( B.

This

{fn : n ( ~}.

~( x l , . . . , x n)

I: ~ [ b l , . . . , b n] for all

under

original sets

use

[24].

on opera-

are also [i].

sever-

Keisler

69

[13] has used b a c k - a n d - f o r t h techniques stronger infinitary logics, Finally,

2~

~I

countable models

this t h e o r e m uses a c o n s t r u c t i o n L

a

of

L~

W l

n o n - i s o m o r p h i c countable models or has

non-isomorphic

to analyze the

in even

a l l o w i n g linearly ordered quantifiers.

there is Morley's t h e o r e m that a sentence

either has at most exactly

to prove e q u i v a l e n c e

[21].

The proof of

like Chang's proof of Theorem 2.3

-types consistent with

o.

The reader should

consult these papers for a broader u n d e r s t a n d i n g of the possible uses of these techniques.

70

REFERENCES 1.

K.J. Barwise, Back and forth through infinitary logic, in: Studies in Model Theory, MAA Studies vol. 8, 1973, 5-34.

2.

M. Benda, Reduced products and nonstandard bolic Logic 34(1969), 424-436.

3.

J.-P. Calais, Partial isomorphisms and infinitary schrift f~r Math. Logik 18(1972), 435-456.

4.

G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, Dover Publ. Co., New York.

5.

C.C. Chang, Some remarks on the model theory of infinitary languages, in: The Syntax and Semantics of Infinitary Languages, Springer-Verlag, Berlin, 1968, 36-63.

6.

C.C. Chang and H.J. Keisler, Model Theory, Amsterdam, 1974.

7.

M. Dickmann, Large Infinitary Languages, Amsterdam, to appear.

8.

S. Fefferman,

languages, Zelt-

North-Holland Publ. Co.,

North-Holland

Publ.

Co.,

Infinitary properties, local functors, and systems of ordinal functions, in: Conference in Mathematical Logic London

9.

logics, Journal of Sym-

'70, Springer-Verlag,

Berlin,

1972,

63-97.

Isomorphisme local et ~quivalence associ~s ~ un ordinal; utilit~ en calcul des formules infinies ~ quanteurs finis, R. Fra~ss~,

in: Proceedings of the Tarski Symposium, American Mathematical Society, Providence, 1974, 241-254. 10.

F. Hausdorff, York.

11.

C. Karp, Languages with Expressions of Infinite Length, NorthHolland Publ. Co., Amsterdam, 1964.

12.

C. Karp, Finite quantifier equivalence, in: The Theory of Models, North-Holland Publ. Co., Amsterdam, 1965, 407-412.

13.

H.J. Keisler, Formulas with linearly ordered quantifiers, in: Syntax and Semantics of Infinitary Languages, Springer-Verlag, Berlin, 1968, 96-130.

14.

H.J. Keisler, Model Theory for Infinitary Lo@ic, North-Holland Publ. Co., Amsterdam, 1971.

15.

D.W. Kueker, Definability, automorphisms, and infinitary languages, in: The Syntax and Semantics of Infinitary Languages, Springer-Verlag, 1968, 152-165.

16.

D.W. Kueker, L~wenheim-Skolem and interpolation theorems in infinitary languages, Bulletin of the American Mathematical Society 78 (1972), 211-215.

17.

E.G.K. L6pez-Escobar, On defining well-orderings, ematicae 59(1966), 13-21.

Grundz~ge der Mengenlehre,

Chelsea Publ.

Co., New

The

Fundamenta Math-

71

18.

E.G.K. L6pez-Escobar, Well-orderings and finite quantifiers, Journal of the Mathematical Society of Japan 20(1968), 477-489.

19.

M. Makkai, Structures elementarily equivalent relative to infinitary languages to models of higher power, Acta Mathematica Acad. Sci. Hungar. 21(1970), 283-295.

20.

J. Malitz, Infinitary analogs of theorems from first order model theory, Journal of Symbolic Logic 36(1971), 216-228.

21.

M. Morley, The number of countable models, Journal of Symbolic Logic 35(1970), 14-18.

22.

G.E. Reyes, Local definability theory, Annals of Math. (1969), 101-138.

23.

D. Scott, Logic with denumerably long formulas, in: The Theory of Models, North-Holland Publ. Co., Amsterdam, 1968, 329-341.

24.

D. Scott, Invariant Borel sets, Fundamenta Mathematicae 117-128.

25.

T. Skolem, Logisch-kombinatorische Untersuchungen ~ber die

Logic i

56(1964),

Erf~llbarkeit und Beweisbarkeit mathematischen S~tze nebst einem Theoreme ~ber dichte Mengen, in: Selected Works in Logic by T. 26.

Skolem, Universitetsforlaget,

Oslo, 1970, i03-136.

W. Tait, Equivalence in

and isomorphism, to appear.

L ~

PART B

CONSISTENCY PROPERTIES FOR FINITE QUANTIFIER LANGUAGES

BY

JUDY

GREEN

74

CONTENTS PART

INTRODUCTION

CHAPTER

CHAPTER

I.

II.

REFERENCES

B

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

LANGUAGES

CLASSIFIED

BY

Section

i.

Model

Section

2.

Completeness

Section

3.

Interpolation

Section

4.

(K,~)

FRAGMENTS

OF

L

CARDINALITY

existence

theorems

theorems

theorems

consistency

74

. . . . . . . .

76

. . . . . . . .

77

. . . . . . . . .

83

. . . . . . . . .

87

properties

. . . . .

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

102

109

Section

I.

Preliminaries

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

109

Section

2.

A

properties

113

Section

3.

~i

consistency

compactness

. . . . . . .

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

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

115

121

75

IN T R O D U C T I ON t

This article presents an expansion of Smullyan's principle"

for q u a n t i f i c a t i o n theory

uncountable conjunctions c o n s i s t e n c y properties

theoretic here.

As Smullyan's abstract

for finitary f i r s t - o r d e r languages and [22] can be used to prove 1 and compactness theorems in a model-

completeness

fashion,

[25] to languages which allow

and disjunctions.

Makkai's g e n e r a l i z a t i o n of these to interpolation,

so can the

K

L

c o n s i s t e n c y properties defined

A l t h o u g h Smullyan's c o n s i s t e n c y property is a

property,

Makkai's

"unifying

is not.

However it is

consistency

included in this presen-

tation as a model for our generalizations. Theory for Infinitary Logic

K

Since Keisler's Model

[18] served as an inspiration for these

g e n e r a l i z a t i o n s we use it as a guide. The uncountable the original

languages c o n s i d e r e d here fall into two types,

infinitary languages c l a s s i f i e d by cardinality and

those developed by Barwise

[i] using d e f i n a b i l i t y criteria.

first type gives us completeness the Craig, LK~,

~

Lyndon and Malitz

regular,

and

LK+~,

theorems and the g e n e r a l i z a t i o n s

i n t e r p o l a t i o n theorems K

The

of eofinality

~,

of

for languages while the second

type gives us an uncountable g e n e r a l i z a t i o n of the Barwise compactness theorem, Karp [2],

the c o f i n a l i t y

[16].

~

compactness

t h e o r e m of Barwise and

Thus the unifying p r i n c i p l e

Smullyan found for fi-

nitary logic will be seen to be a unifying principle among finite q u a n t i f i e r languages. Consistency properties

can also be defined for other types of

languages to obtain similar results,

e.g.

another g e n e r a l i z a t i o n of

#Most of the techniques for uncountable languages appeared in the author's doctoral dissertation written under the direction of Carol Karp. The author was partially supported by NSF Grant GP 11263 at the University of Maryland and, during preparation of this article, by the Rutgers University Research Council.

76

the Barwise

compactness

theorem

not stated for, a modified

[9] which is proved using,

version of the indexed

languages

of Karp,

and interpolation

and compactness

theorems

finite quantifier

languages

[17] and the article by Cunningham

in this volume

[7]).

([6],

for chain models

though

for in-

?7

CHAPTER LANGUAGES

In t h i s previous

chapter

article

notation,

we

the

W h e n we c o n s i d e r K

symbols

most

K

of the

symbols

A cardinal 21 < K.

For

strong

limit

consider

if

if

strong

cardinal

If then

L

a basic

limit

iff

symbol

Cl,...,c n

and

~7 ~,

denotes

L

and

we assume

there

are

cardinal

less and

than

at

cardinal.

= U { ( 2 1 )+ I I < K}

K : 2 *K.

h < ~,

SO t h a t

~

is strongly

A cardinal

and a strong

including w.r.t.

limit

cardinal,

is a

inaooes-

i.e.,

a

the

C

set

of c o n s t a n t

is a c o n s t a n t

where

~

is an

symbols

symbol n

of

place

L

C or a

function

( C. formula

obtained

by m o v i n g a n e g a t i o n

inside

i.e.,

ml

X{Ai I i(l} that

in the

terminology

limit cardinal if for all

2 *K

f ( c l , . . . , c n)

the

defined

cardinal.

term of form

standard

is a s u c c e s s o r

K

~,

is a l a n g u a g e

t e r m of the

L K~

is a l i m i t

sible if it is i n a c c e s s i b l e regular

LKh

following.

is a strong

cardinal

languages

We use

a language

K

BY CARDINALITY

the

volume.

language

K

any

CLASSIFIED

in t h i s

including

I

for

all

i

= l~

if

(Ar

= v{~m I m ~ }

(Vr

= A{Tm Ira(C}

(Vx~)I

= 3xl~

(3xm)1

: ?xl~

is the ( I,

m

set of all

f(i)

~ A.. 1

is a t o m i c

functions

f

defined

on

I

such

78

i.

Model

Existence

Theorems.

We d e f i n e

three

types

of c o n s i s t e n c y

properties. (i)

For

languages

Lwl~,

(ii)

For

languages

L,

erties,

hence

(iii)

For

called

K

prove have

the

same

from

when K

on the

language

we t a l k

when

being

show

to f o r m

case we w r i t e

of

the

C

third

~

prop-

properties.

type

singular

is a c o l l e c t i o n

we

add

use

~,

will

also allow

these

of one

us to

languages

The

considering

consistency

as a c o u n t a b l e

style L

so that

a new

C

C

de-

is c o u n t a b l e

In the

of s m a l l e r

of

set of

of

and has

properties.

to

construction

cardinality

properties

union

of s e n t e n c e s ,

is e q u i v a l e n t

language

M.

consistency

of sets

a Henkin

to each

language

we are


~,

LX+ ~.

or

~,

Let

L

there D

K

has no f u n c t i o n symbols

is a

submodels L

if

1

cf(K)

cf(K) = ~) d e f i n e

and

relative

(L

symbols

if

to K = ~i )

~< K

if = ~.

by r e p l a c i n g

such that

K

o

which

is r e g u l a r

and

free

K

~

is regu-

are sentences

do not o c c u r

and

For e a c h f o r m u l a

the q u a n t i f i e r

If

formula

func-

in

ef(K) > ~ , 9 @D

of of

o

or of

LX+ W LK~

(LK~ as

follows: ~D = 9, (q~)D

:

for

~

atomic

I (~D)

(A~) D = A{~D I ~ }

(v~) D = v{~D I ~ }

If

~

(Vx~(x)) D = A { ~ D ( d )

I d~D}

(3x~(x)) D = v{~D(d)

I d~D}.

is p r e s e r v e d

(oD--+~D).

Since

K

~i ~

in the usual way.

be a set of c o n s t a n t s

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

cardina!ity

is

e).

:

Assume

of

if

(~

PROOF.

lar and

~

K> ~

2" ~

where

is a l s o

assignment

Keisler

existence

languages,

and a l s o

as usual).

eonverse

on the m o d e l

finite-quantifier ~

the

I:~

w" Karp's student Judy Green extended i by d e f i n i n g c o n s i s t e n c y p r o p e r t i e s and p r o v i n g

investigations

corresponding

ume,

(written

finite-quantifier

In his guages

models

some

K

are

as are LKK


~.

If

~

2K

has

~

E~K ~ "

automor-

phisms. If

3.6 C O R O L L A R Y .

~K then

~" ~

where has

PROOF: 2
w,

and

of strict power

K,

automorphisms.

By the p r e c e d i n g

automorphisms.

corollary,

But an a u t o m o r p h i s m

an a u t o m o r p h i s m

of the s t a n d a r d m o d e l

automorphisms.

-I

At the o p p o s i t e

extreme

are t h o s e

the d e c o m p o s i t i o n of the d e c o m p o s i t i o n

~ ,

and thus

chain models

~

has

~

has ~

is

2K

w h i c h are r i g i d

141

(i.e., have only one automorphism).

Among the interesting results

is the following,

proven by an a p p l i c a t i o n of Scott sentences:

3.7 THEOREM.

~

is rigid,

If

then

~

is

chain model of strict power

a

is definable

madeZs by a sentence

of

L

up to isomorphism

< >

which

~

among all chain

KK

The proof of this t h e o r e m makes n e c e s s a r y use of infinite quantifiers, let

~n

and thus it does not hold for

= != ~,

s'uch tha't

of

then

X

and

R'

= R.

%X,'(,R,SI,... ,Sn>i:

for a]].

~',

Sl,...

n

I,~1'

$'hen R.

~

-s c a l l e d a

A set

R

X~finite) of

X

X-a.i.

is c a l l e d

iff it is

X-a.J .d.

X-a.i

is i X-a.i.d. ( s e e

i'ff it,

X-a.i.; def'.ncs some set

(ahso!ut~ly im~lieit~p

des

bz some

below).

if the pr~incipal r,e l a t i o n persistence

defJnJ:tJon

Note

symbol

R_

L hat

.~. ~

or

Note tI:at each e l e m e n t u n i q u e l y Jeter'm~nes

is specified.

requr-._ement; it a t t e m p t s

definable,

~Qndizion

R

(2) J s a

to give a pr'edicazive c h a r a c t e r

to

the d e f % n i t i o n . For the d e f i n i t i o n or

X-tee),

replace

d e f i n i t i Q n of

X-$.i.d.

"R' - R"

X-s.i.i.d.

X-re).~ r e p l a c e , X-s.;i,i.d.

of

"R'

= R"

(X-genZ 1 , i s

(inv~r'iantly imp;~icJtl ~ d e f i n a b l e ,

ir~ c l a u s e

(2) by

"R'

~ X = I{."

For the

(seml-invar'i~ntly imp!ieit$y

definable,

by

"R' _~ R."

All

X-zenE 1

s.ets 9re

defi h @. then

i_

x



of

For let

be a

Z~.en

tion symbols, dJ s The r e l a t i o n is

X-$

( R ~ ~ <X,(>I: ~(Xl,...,Xn)).

without parameters

iff it has a

X-defining

formula. A r e l a t i o n is

X-genA

iff it is both

X-genE

n

similarly,

a

The

X-A

Z0

formulas

is transitive,

The ty:

if

geng I X

r e l a t i o n is both

n

x ( X,

and

X-gen~ n;

n

~(~)

X-E

and

n

X-~

n

.

have an absoluteness property:

and



is an end extension of

if X,

}$ then

<X,(>I = ~(~)

iff

formulas

have an important persistence proper-

~(~)

is transitive,

x ( X,

I = ~(~).

and

<X, (> I: ~(x),

then,

for every end extension



of

X,

I: ~(~).

Thus, X

as m e n t i o n e d earlier, and

X

definition,

0.4.

~(v,v)

is

X-gen~ 1

genZl,

sets are

then

Vv(R(v)+-~ (v,~))

by the p e r s i s t e n c e property

An admissible

set

X

X-s.i.i.d.

For if

is a

X-s.i.i.

just mentioned.

is a transitive

set with the following

a x i o m a t i z a b l e properties:

x,y

( X

implies

{x,y}

( X

and

E 0 - s e p a r a t i o n principle:

if

E 0 - r e f l e c t i o n principle:

if binary

Vy (x 3z (X R(y,z)

R

Ux ( X

is

X-Z 0 R

is

and

X-Z 0

x ( X,

and

then

x ( X,

x n R ( X.

then

-~ 3w (X Vy (x 3z ( w R(y,z).

The c o r r e s p o n d i n g axioms are the universal closures of the following formulas

(where

v, v0, Vl, v Z

are distinct variables):

q,$~.

pair:

Jv2Vg~.(vi.ev~-{v ] i 9.J Vl':O0)

10-separatJon

sehe~la:

[~v2VvI.!vI c y2+-*y1 m v &~) sueh

Z0~rerlectio~

that

there~.s

the admissible

~ fin{re

~

such that

els of the finite

subccllec:ion

((

to be an admissible

in [2] and [15].

formula of form is a. X-S1 Further,

X

we have

where each

X satisfies ih

satisfies

the

some notion of

principle

~

is

formula Z 0.

X-genA 1

and

because

s).) by

X

is equivalent i.s

is under-

X-~

relations

each

X-gen>] 1

reiat~on

is a

X-A 1

'then

on

X,

(In thJ.s paper,

then the to

ZlrrefleetioD

property

for

Z1

for admissible

if

{-reflection

"X-~"

for a given transitive

A theory of recurston

relation.

9'A R'( X

axioms of pair a~d unioh imoly ~ h e eq~:~vaienc6'0f"the genZl-reflegtion.prine~ples

Z1

relation

in the state-

similarly., We cab d~fi~e

The notion, of

of the persistence

to a

principle:.

"X-}]0"

princip'le.

to those

X-equivalent

Zl.-refleetion p~inciple.

Z0-reflection

{-separatioo. pri.neiPle.

such that

Thus,

~ ( X,

is the result of changing

ment of the

important

gent I

A 1-separation

X:A 1 the

~.

sets whJ.ch are mod-

a set denoted

of admissi.ble se=s

Each

.re!abH ~0(x)

that

evenly

because and

H]

every ~,

is

-

E x)

SECTION J, FA.ILURE OFw,WF~

Chefs e x i s t Let ~.

admiesibZe aets

M

be a countable

X-a.i.d.

admissible

set is countable,

follows

that WFT fails for

sisting

09 ~ e

a ~ b

2)

Vuvw(R(u,v) V

se~(~.g'.,,

X ~ M,

X

and the

o(X)

a.

=

~:(~)I~ i ~ o m t a ~ - n ~

admissible

for c6nsidem

fop each & R(u,w)--+v

R(~,~)

(Compare

X;

and

exists

o(X) X-A 1

such

= o(M). set

r

~t con-

"f~1~owin~: 'Sentences '( 1 );;" ( ~), '('3).

i)

the

a d b

F

a

of the example of

X

of

X

~ w)

fop each

The construction 1.2.

f o r whioh~W~ f . i Z 8

It will be shown that an uncountable

that e a c h

3)

X

of H;,

X in 0.7.)

uses the following

Theme ezists an indezed set

LEMMA.

lemma.

{Mz I finite

z [ m I}

of

oountabZe admiaaibZe sees for ~hioh: implies thar

a)

z ! z'

b)

Mz n Mz'

c)

~here ie an indexed eer

=

ie an e~emen~ary aubmodeZ of

Mz

Mz'

Mz ~ z'

{an l n ~ }

e.oh ~ha~:

a d)

and

Mz ~ M

This

o(Mz)

= o(M).

lemma will eventually

be pmoved

in w

(If

M

is a model

of ZFC, then the pmoof of Lemma 1.2 will make oum M0

a model of the

Peplacement

will not be a

axioms

and the axiom of choice.

But

M0

model

of the power Cpntinuation

set axiom.) of proof of Theorem MA

~

Then MA.{ i~..an elemen%ary theqr~

MA

e•

o[

.For each

M0

~Y

(a) and a fact of model

sys~em..gf e].ementamy

An uncountable

sets are~e'ount'able..

Not~ that

A ~ ~i'. de~ine

z ~ i}.

extensio n ~f each of the membepsiof

is admissible.

a.i.d,

O {Mz I 5inite

(the. union of a, directed

elementary

l.l:;

A



of all finitary ~ ~ M R l - s e n t e n c e s

of a.~.d..

.We .now def:ine.,, by induction where .each Put since

~

A ~ : N {A 8

6 < ~}

~s c0untable

(see 0.9

i.s o~ :o~de.r. type

of

Case i.

Suppose'there

[0,y

set

R.

are

y, z, y

Define

a. sequence

:

(convention:

NO ~ ~I).

(!)) .. Take

Y~ < ~i

. Then

A ~ ~ ~,

sP ,that

~+i. , There are two. cases

for the d~f,

A h.

A~N ~

= sup z

~ ( ~i'

Ad ( ~ .

A ~ N [0,y~. inition

on

], Sfnce

and R

~

exists is a

A

MAJi.i.

is uncou~ab'Ie

such that

such that

(i.e~., .. y = U~ ~{~+~I ~ ( z }

(:T,

deflnition

i~d

y ( R-M[0,y

A

M[0,u ],

]

~A N [0,y~]

=

df some ~ n c 6 u n t a b l e is c o u n t i B i e ,

finite 'z.c, A,l"y

iS the strict

there

( Mz, ' a h d

le.as%:.upper bound).

168

A s = A-(y~,y) Case

2.

Suppose

~ ~

no

for a choice

A

exists

AI~: Note

that,

if

prove

this,

put

that

A~o N [0,y~]

= A~

.

Fop any

by i n d u c t i o n

=

hypothesis. step

guaranteed

that

equality

and

Consider

A~

equal

< ~i'

and

~+i.

Thus,

A

[0,y6]

Put

We show that no set. able

Suppose set

satisfies tion

step

definition Then

R.

that

r

Since

the s u p p o s i t i o n we chose

of

R',

(~:-(y~,y))

y (M(~l-(y~,y))

To

B ~ (a'~l ]

Then

[0,y~].

either the

~ = ~

same o r d e r

Y6

Therefore,

The d e f i n i t i o n

-

or

type

we chose

6+1 ~ e+l.

n [0,y~].

by

= A~ N [0,yS].

Combining,

] = A o6 N [ 0 , y ~ ]

= A~ ~ N

[0,y~]

of

By

9

is finished. for each

It is uncountable:

N [0,y

X-a.i.

is a

E ~

] = A ~ N [0,y

definition

X-a.i.

and

of case

A~

of order

]

type

i at i n d u c t i o n

N z ~ (~:-(u

N Mz = M ( ( ~ l - ( y ~ , y ) )

of some u n c o u n t -

= A ~ N [0,y~], step

such that

y ~ M[0 ,ya] ,

otherwise,

of an u n c o u n t a b l e

definition

N [0,y~]

A, R' , y, z, y

y ~ R' ,

y ~ M(~l-(y~,y)):

Put

X = MAKI.

~

A~:

e,

A~:

is a

some

N

o

It is closed.

E ~.

A~

sets have

type

(*), the i n d u c t i o n

it has the segment

on

of the c o n s t r u c t i o n ,

A~ N [0,y~]

.

= A ~ N [0,y~].

] = A ~ N [0,ya],

A 6 N [0,y 6] = A 6 N [0,y 6] N [0,y this

A s = A ~~ ~ ~.

B ~ (~,~i ].

N ~E[a,B)

is of order

n A o6 N [O,y ~ ] Ao6 N [0,y 6 ] _ A6

A ~~ N

~ ~ ~

A 6 N [0,y6]

Define

by i n d u c t i o n

Let

These

o

A, R, y, z, y.

N [0,y~]

We prove

A o6 N [0,y

At i n d u c t i o n

such that

A~

= A ~ N [0,y~].

6 ~ [~,~),

i.

such

FI {Ao~ I o~ < 1~i } then

A oB N [0,ye]

(*)

~+i.

A~:

for case

=

a < ~i'

of some

~.

~

y ~ Mz,

At induc-

is a

and

=

N z) ~ M[0,y~],

MA-a.i.

y = sup z.

since N [O,y)

A~:

[O,y~],

by the

169

extension of (b); this contradicts y ~ M(~l_(y ~,Y)) _n M A This contradicts

_n X _n R.

the persistence

Since every possible done.

X-a0i.

y ~ M[0,Y~].

Thus,

R ~ R',

Thus, and

MA _n MA~ _~ X.

property. definition

is some

~,

we are

~0

SECTION 2 EXAMPLE WITH INDIVIDUALS

Theorem

l.l~has

occur

both

proof

since

1.2.

Acutally,

sion

of T h e o r e m

tioned

this

in set t h e o r y some

in the

is o n l y

paper

be an let

and

not

of

Afterwards,

(If if

WFT

M z

I n(~} .

I e(u}

to

Tw

show

fails

is some

zc

that,

for

t

was

= in

of

usage.

ver-

6.8 men-

in this

section

The

each

u ~ ~, 0.9

show

for u n c o u n t a b l e

,n

rest

I a~,

For e a c h

be an a d m i s s i b l e

We w i l l

L(a)

subset

mentioned

a stronger

I : {i

individuals.

in

of L e m m a

of

section.

let

defined

this

(5). that

~

n(~}

( ~,

set c o n t a i n i n g let

Tu

be the

(An

Mz

of Lem-

Tu

A c ~

is a d m i s s i b ~ . such

that

~ -A

TA.

segment

is a f i n i t e

on this

For

O z.)

later

to

of

of the ~,

constructible

then

it turns

out

hierarchy that

LTC({aB i B(z})(a) , 0.9

(4).

Also,

Tu

= U {Tz

I finite

u}.)

2.1.

i)

L

M

proof

introduced

cardinal,

individuals.

Tz where

depend

in our

consistency

with

al~@wed

We d e s c r i b e

individuals)

notation

Let

sets.

to the

it c o n f l i c t s

M U {a

we w i l l

is i n f i n i t e ,

and

>

is a n a l o g o u s

(with

set of d i s t i n c t

containing

Prim-closure ma 1.2

The

be an u n c o u n t a b l e

: { < n , 1n,"

individualSi~are

an a n a l o g

corresponds

logically

~• u-indexed a

has

now prove

section;

when

in a d m i s s i b l e

introduction.

not

proof

proof

i.i that

does ~

and

of this we w i l l

for this

Let

a simpler

LEMMA.

A

i-i

Symmetry.

onto

extending

g: u - - + v

determines an

g: T u - - + T v

ia,n~--~ig(a),n.

This also determines an extended map to those of

(-isomorphism

Tv.

For this map,

g

of the relations

of

Tu

171

b

z i u,

For

2)

Consider z c

~

is

y ( ~, such

ysz:

E-isomorphism

I-i

~ z

and

: 0.

These

U {y})

n ( w.

h

a,n

onto map

h

of i n d i v i d u a l s

of all the sets b u i l t on

h(x)

= h(~x

for

x (dom(h)

in

dom(h).

It can be s h o w n that

h(F(~))

functions;

but all

determines

of inner m o d e l s

of

and by a p p l y i n g

the a b s o l u t e n e s s

formula

F:

that

individuals

of

schemes

h: U - - + W

individuals, property

i.e., [9;

is an

<W,E>I = @ ( h ~ , h F ( ~ ) )

implies

I = ~ ( h ~ , h F ( ~ ) )

classes,

implies implies

: F(h~).)

(i), we show x E M,

y = F(a a ,...,a m ,x). l

re-

(4)] of the d e f i n i n g

I: ~ ( ~ , F ( ~ ) )

a l , . . . , a n E u,

recursive

of P r i m - c l o s e d

2.3

are

E-isomorphism

implies

For

TC(x)

for p r i m i t i v e

i: @ ( x , F ( ~ ) )

hF(~)

an

for every p r i m i t i v e

on the d e f i n i n g

with

an

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

= F(h(~))

or by k n o w i n g ZF

determine

E:

i (dom(h)

(by i n d u c t i o n

and

extending

a ( z,

k-+i

i-i

of

z n rng(s)

for

a,n

identity,

s: m--+~,

sequence

i

for

~

is

n ( ~o

= h(i)

cursive

g

r ( m,

by i n d u c t i o n

F

finite

T(z U r n g ( s ) ) - - + T ( z

h(i)

function

If

for

~-isomorphism individuals,

g ~ Tz.

is(r),nF--~ iy,mn+r

A

PROOF:

y

is

identity.

nonempty

that

b @ (g~) 9

iff

(g ~ z): Tz--+ Tg~z

g: Tu--~ Tv

then

~ (~)

n

gCCTu c Tv.

For e v e r y

and p r i m i t i v e Then

y E Tu,

recursive

there are

function

F

such that

172

gF(a a

i

,...,a

n

,x) : F ( g a

By c o n s i d e r i n g For

g-i

,...,ga

n

,gx)

we also have

(2), we show

putting

i

T = ysz.

ysz~T(z

For each

: F(a

g(ai) ,

...

,ag(~n),X)

( Tv.

g~Tu n Tv.

U rng(s))

r ( m,

c T(z U {y}).

define

primitive

Abbreviate recursive

by

Gr

so that

Gr(ay)

Consider

any

recursive

TF(a

=

{

:

{ I n(~}

{ 1 new} =

Tas(r)"

~ z

and any

x ~ M.

Consider

with

n + m+ i

arguments.

any p r i m i t i v e

Then

,as(0),...,as(m_l),X) n

=

F(~a a , . . . , T a

,Tas(0),...,Tas(m_l),Tx)

i

=

F(a

n

,...,a I

For

(2), we show

tive r e c u r s i v e new}.

=

Then

T-iF(aai

7-1~T(z

function

for

,G0(ay),...,Gm_l(ay),X)

( T(z U {y}).

n

G

al,...,~n

0 {y}) c T(z U rng(s)).

so that

r

~ z,

Define

primi-

G (a ) = {<mn+r (n)> r s(r) 'as(r)

x ( M,

and p r i m i t i v e

reeursive

I F:

n,ax, x)

F(T-Ia

... T-la

,T-lax,T-Ix) n

F(a

,...,a i

2.2

LEMMA.

, O {G0(as(0)),...,Gm_l(as(m_l))},x)

For every finite

20-reflection principle. PROOF:

By the

inductively

z ~ ~,

transitive

Also for finite

last

lemma,

z Define

( T(z U rng(s)).

n

:

z' a_~,

Tz

satisfies the

Tz n Tz'

we can assume

{al,...,an}

the inner m o d e l



_c ~. of

M

so that

= Tz n z'.

173

x

( Nz




It w i l l

be

shown

s-isomorphic The recursive proof

represent tion to

Nz

Nz

relations;

thus,

of the

lemma,

and

x

(and

x,y

g0-reflection

are r e s t r i c t i o n s

Nz

E0

and

E

since

are

to

M-A I.

is w r i t t e n

formula.

§x,

( y-+

proof

E

I = ~(~)

arguments

"Vx(<x,l>

of

(and

(M)

and

(Nz).

is

Tz.

an a r b i t r a r y

The

is a m o d e l

relations

of this

x c Nz x {i}

( y

that

to

or

Then

M

of p r i m i t i v e

For the r e s t

I= ~(~).

I= ~(~)

the q u a n t i f i e r

Also,

is a

~

M-E 0

"Vv s y"

of the will

rela-

translates

)."

that

Nz

is a m o d e l

of the

axioms

of a d m i s s i b i l i t y

is

->

now

indicated.

Pairs:

Here

x, x, y

{<x,l>,}

Unions:

{

I 3z(

{

Zo-reflection:

Assume

in

Nz.

(x

&

[ ( x &

I= Vv 0 ( x

(z)}

( Nz.

I: ~ ( y , ~ ) }

( Nz.

3Vl~(~,Vo,Vl);

we

I= V v 0 ( x

x ~ Nz • {i}.

Then

Vx 0 ( d o m ( x ) 3 x I ( Nz(l= ~ ( ~ , x 0 , x l ) ) .

gl-reflection

for

M

for

"3Xl";

x 2 = (w n Nz) • {i}

M,

3v I ( x 2 ~ ( ~ , v 0 , v l ) .

show

that

in

such

assumed

( Nz.

E0-separation:

x 2 ( Nz

are

since

i.e.,

Nz

is

M-A1,

We can

there

there

assume By

Vx 0 ( d o m ( x ) 3 x I ( w(l= ~ ( x , x O , x l ) ) .

( Nz.

This

is the r e q u i r e d

inductively

is

that

is a r e s t r i c t i o n

w

Put

x 2.

v

Define

that

v

x = {

I y(x}.

Then

x

( Nz

for any v

x ( M. to be

Define

~x,y}

~x},~x,y}},

to be the the

{<x,l>,}

'tordered pair"

of

above.

Nz.

For

~

Define ( z,

v v

b

to be

{<m,>

Define

I m(~}.

the p r i m i t i v e

Since

recursive

w

( M,

function

b F

E Nz. so that

<x,y>

define

174

a

(m)

if

x =

and

m E w

(m)

if

x =

and

m E w

i

F(a

,...,a

,x)

i

: a

n

n

{F(aal , . . . , a a n , Y )

Considering

a

,...a ~i

onto

F~Nz.

as p a r a m e t e r s ,

x E M,

F(x)

F~Nz ~ M U {aa1'

n}

Now

an i s o m o r p h i s m )

F,

under

satisfies Thus,

up

from

elements

= Fl(X)

E Tz I.

2.3

LEMMA.

The

PROOF:

Since

transitivity).

Note

that

We

show

from

the

has by

ness

property

that

finite

of

Nz

x,x

finitely

of

EO

the

z c u

such

) = a a.

and

and

FI:

Thus

(since thus

Tz

F

is

F~Nz

is

is c l o s e d

Nzl--+Tz I

for

n(w}

x E Nz: iff

and

~

lemma,

be the F(x)

x E Nz I

is

and

we

such

is a

E0

we

x,x

E Tz.

F

[9;

of the

Zo-reflection u

is not

a primitive Tu

principle that

z

last

2.6].

show

(Thus,

finite.

recursive

is t r a n s i t i v e . ) for

Tu

follows

is finite.

formula.

can w r i t e Suppose

the

can a s s u m e

Zo-reflection

~

to

since

Tz

= o(M).

of

need

arguments.

E Tv.

that

result

z ~ u}

many

the

is by o b s e r v i n g

we only

last

o(Tu)

and

general

formulas,

provided

let

I aEZl,

for each

E Tu

F(b

= Tz.

z I ~ z,

I finite

that

F'Nz

This

from

Z0-reflection

I: ~(~), Find

only

M c Tz

is a d m i s s i b l e

By the

= U {Tz

a E z,

By i n d u c t i o n ,

= o(M):

symmetry

Suppose

Thus,

is P r i m - c l o s e d ,

(and

function

E-isomorphism

z I = z N z'.

Tu

set

follows

Tu

Tu

otherwise.

of a d m i s s i b i l i t y ;

Since

{ia, n

Put

o(Tu)

or this

is an

is t r a n s i t i v e

axioms

For

of

For

F~Nz

e-isomorphism.

F(x)

lemma;

= x.

Tz c F~Nz.

= Tz n z':

corresponding built

the

FCCNz c FCCM c Tz. n Tz'

Tz

F

E x}

~n

For

Prim-closed.

I

By the a b s o l u t e -

I= ~(y)

for any

Vx 0 E x3x I E Tu(l: ~ ( ~ , X o , X l ) ) . Find

y

E u-z.

We w i l l

show

175

Vx 0 By

Z0-reflection

exists

w

E T(z

E x3x I ( T(z

for

T(z

U {y})

Assume z' c_ u by

x 0 E x.

such

that

a nonempty

ysz.

Then

i.i,

we

can

finite that

Tv.

z

is

b

finite. A

of

A

follow

that

there

I: e ( } , X o , X l ) .

assume s.

is

Tx 0 : x0;

(2),

~0

T

theory

Then

Tu

used

§

x

E Tu

find

i-i

Given

z' - z

take

T =

is a n

U {y})(l: ~ ( ~ , x 0 , x l ) ) .

u ~ v ~ ~,

of m o d e l

List

symmetry and

finite

I: ~ ( ~ , X o , T X l ) ;

Vx 0 ( x3x I E T(z

x E Tz.

z' - z # 0.

By

~

Find

is a n in t h e

and

elementary proof

formula

onto

QED

g:

of

sub-

Theorem

~( }) ,

u--+v

find

such

Then

I: e(~)

uncountable

subset

the

next

theorem

set

TA

o(M)

= o(TA).

of

holds

iff

~

I= e ( ~ ) .

such

for

any

that

X - A

uncountable

is

in-

subset

~.)

Also,

M c TA

is

can

since

= x;

iff

(Actually,

THEOREM.

could

2.5.

We

infinite

identity.

be a n

then

that

sequence

fact

that

it w i l l

x I E Tu,

u ~ w ~ v.

e(~)

2.4.

we

By t h e

such

Let

For

assume

g ~ z

Tx

Thus,

Remark.

of

some

finite

U {y}).

Incidental model

For

Then

~(~,x0,xl)).

( x3x I E w(b~(~,x0,xl)).

I= ~ ( ~ x , ~ x 0 , T X l ) ,

E-isomorphism. TX 1 E T ( z

such

x I E Tz'.

i-i

{y})(b

U {y]),

c Tu

Vx 0

U

have

The and

is

= L(a),

put

M

PROOF:

The

main

LEMMA.

Every

an

so

point

For that

is t h e

TA-a.i.d.

admissible

set

any

set

admissible

o(TA)

fails

ordinal

WFT. a > w,

= a.

following

is

which

a subset

lemma.

Tz

of some

where

z

finite.

PROOF:

Let

~(x)

be

a

TA-a.i.

definition

of

R.

Then

x

is

176

contained

in some

Tz

where

finite

For

z c A.

some

S,

< T A , ( , R , S > I : ~(~).

Suppose y

R d Tz.

( Tz'

Find

identity,

and

There i-i

is

onto

gCC(z'-z)

y

( R-Tz.

g: A--~ B

c ~ - A.

For such

Take

some that

the

g

finite

z' Z z,

A ~ B ~ ~, of

symmetry

g ~ z (i).

is Then

4: @(g~) ;

transitive gy

TB n TA~

( g R n gTz'

tradicts

placing

c_ R n gTz'

gy ~ gTz

F o r the HI

and

= Tz.

theorem, by

TA.

let

g~ = x.

By p e r s i s t e n c e ,

c T A n T g ~ z ' c_ T ( A QED

F

gR

= R.

n gCCz') c Tz.

But This

con-

Lemma

be as

in t h e

example

HI

of

0.7,

re-

177

SECTION INNER BOOLEAN

We assume algebraic

familiarity

models

here c o n s i d e r

a complete .

complement

of

b

bound

AX

(or meet)

least

upper

unit

( B, is

~.

VX

X

and

We assume

denote members

@ # ~.

identify

B

subalgebra

= V{b,c}. We write

with

BI

to

of

from

zero is

In this

ordered

~.

set

The

lower

of "complete."

for the

We

kind of par-

iA{Ib IbEX}.

2

B

[23]).

the greatest

The

.

and Boolean

and

definable

by d e f i n i t i o n

of any given p a r t i a l l y

A complete

are

X 5 B,

is equal

bvc

[22],

to be a special

For

exists,

algebras

[25],

operations

lb.

SETS

Boolean

e.g.,

algebra

(or join)

= A{b,c}

We n o t a t i o n a l l y

(see,

The other

of

FOR ADMISSIBLE

complete

Boolean

is w r i t t e n

bound

bAc

with

for set theory

tial o r d e r i n g

b,c

MODELS

3

The

For

@

and the

subalgebra

section

[0,~}.

p, q, r, s

~.

is a s u b s t r u c t u r e

of

B

such

that

Vb

(Bl((~b)

( B I)

and

VX 2 B I ( A X ~B 1 Then

B I

is a complete

by those

3.1.

of

algebras

infinite make

between

admissible [25].

complete

this

algebra

with

complement

and meet

given

B.

A countable

Boolean

Boolean

dense

sets,

is a c o n n e c t i o n A subset

However,

Boolean

fact useful,

set does

it does

algebras

we now

complete

list

some known

Boolean

dense

P

is called

called

dense

if

and

weak

complete

subsets ~).

elementary

algebras,

is commonly

B

contain

infinite

(when it contains

with what of

not contain

of some

In order

to

relationships

isomorphisms. forcing.

There

178

and Vb

(Trivial

example:

A partial ean a l g e b r a [25;

12B]

then

tion

Vpq(p to

~ @, the

in

< b).

B.)

is a d e n s e

substructure

~ q - - + 3 r ~ p Vs ~ r(s If

P

is d e n s e

is some

below.) up to an

rest

of this

Note

the

complete

B.

for b e i n g

a dense

The

~ q)).

in some

r ~ pATq; (minimal)

(For B,

the

and

P

section,

denotes

this

and

p ~ q,

is p r o v e n

of the

is dense

Bool-

fact,

if

converse

completion

isomorphism,

of some

by

Boolean

in the

comple-

P

contrapositive

a dense

of an e x t e n s i o n

subset

of the

of a

condition

set:

VpVb

pp.

~ @--+p

35.2].

For the

The



[5].

~

~ P(b

is d e n s e

so there

is u n i q u e

[25;

{@}

ordering

iff

considering algebra

B-

refers

pA1q

( ~3p

( ~(Vr

following

s pBs

is a slight

s r(s

~ b)--~p

modification

~ b).

of what

is in

[5]

or

[8;

is a

com-

100-103]. Define:

7S = {p

I Vq ~ p(q

= {S

Then

~

plete

Boolean

partial

is o r d e r e d

and

by i n c l u s i o n

algebra

ordering).

P~-~{q

I S ~P

(this

~ S)}

c.

for

S s P;

S = 77S}.

For this

is p r o v a b l e

from

ordering,

the

fact

I q ~ P}

~-embedding

of an e l e m e n t

S

that



is a

Further,

(: 77{q

I q ~ P]

~ 6,

by the

for b e i n g

is a

~

~: P - - ~ @ of

~

onto

is the

a dense 7S

just

subset

of

defined.

condition

a dense

@.

The

For each

set)

complement X ~ ~,

179

the meet

AX

Then

is

~

extending

RX,

and the join

is i s o m o r p h i c

U: P - - ~ B

with

is given

p(b)

VX

~.

is

~UX.

The unique

isomorphism

U: B - - + @

by

: {p I P -< b};

and -i

The f o l l o w i n g

uniquely

Vp(p This

(S) = VS

can be p r o v e d

for

determine

S (~.

the o p e r a t i o n s

Vp(p

_< ~b +-+ Vq _< p(q { b))

Vp(p

~ AX~+Vb

of

~:

( X(p ~ b))

5 VX+-+ Vq ! pBr ~ q3b directly

~ , A, V

or by using

7b : V{p

( X(r ! b)). p

above.

For example,

I P -< ~b}

and p _< ib~-+p Let ~i

~i

and

there

( u(Tb)~-+p and

~2'

Consider

extensions

which

~: ~ i - - + ~ 2

p2xpll:

p2TUI-i : p ~ l - - +

be a that

PI: ~i---+~i '

~2:~2--+~2

Then

of c o m p l e t e

T: ~ I - - + P 2

the e m b e d d i n g s

etc.).

extends

subsets

let

isomorphism

PI: BI---+~I'

b ( ~i'

for

be dense

respectively;

is a unique

33.4].

P2

( ~p(b)~-~ Vq _< p(q { p(b))~-+ Vq _< p(q ~ b).

(where ~I--+@2

p2c~P2 .

Note

Boolean

algebras

~-isomorphism.

extends

T

[25;

P2:P2--+~2

that

for

12.5,

and t h e i r

~i (b) = {P ( P I is the unique

Then

I P s b}

isomorphism

p ( ~2

for

S ( ~i' -l($)+_+p p ( ~2TOI Thus,

~2T~I(s)

parameter

< ~p~i($)+_+ _

: T~CS.

T: ~ I - - + P 2 .

Then

-i

p ! P

~2T~ I

~l(s

) ~-+ T

-I

is p r i m i t i v e

p ( S +-+ p ( T~CS. recursive

with

180

Suppose of

B.

101 ~ P

Then

for

onto

for

S ( ~i,

(~

I)-I(s)

Thus,

~i I

ters

:

subalgebras

~.

plete

Suppose

~(VS)

:

{P I Vq _< pgr _< q3s

~i (vS)

:

:

BI

isomorphism

{~

of

~.

of

Note

{P I P ~ VS} ( S(r _< s)}

{P ~ 101 i p -< vs}

are both primitive

for each isomorphism B,

that

I~

an isomorphism c 10

~i' B2

=

T T*

101' I~ of

with extension

T*: ~i--+~2 T*

and

(respectively)

is an isomorphism

Then we define

:

101 n s.

recursive

with parame-

of dense

subsets

of complete are dense lB.

of com-

subalgebras

subsets

Suppose

m: BI---+B2,

of com-

that as above.

by ( ~ 2 i ) ( ~ 2 ~ i i ) (~oil) -i ,

,

By the above, and



namely

or w i t h

of

U--+ U'

of some

entity

of

of B o o l e a n T

and

9

structures

such that

of the same s i m i l a r -

(letting

'

denote

the

U'):

~: B - ~ B '

is a

T: U--+U'

is an

s-isomorphism

i-i

onto

function;

and T(Ri(Xl,...,Xn.))

= R l ( ! ( x l),...,~(x_ n.))

i

for all

i S m,

1

all

Xl'''''Xn.

( U.

i

For the a b o v e isomorphism

T I~l,

Consider R0,...,Rm,_

T, ~,

the

where

~ IUI

--IR" is

Boolean

value

formula [~(~)~

U I ~ U,

there

is the B o o l e a n

of the s u b s t r u c t u r e s

(finitary)

ed an i n t e r p r e t a t i o n For e a c h

and

language

ni-ary

of this ~(~) of the

for

UI

and

w i t h the r e l a t i o n i s m.

Then

U

structure

~ U I.

symbols can be c o n s i d e r -

language. of the

l a n guage,

U-sentence

and for

is d e f i n e d

x ( U,

the

inductively:

182

E R--i .(xI,...,x

An

U-sentence

EVvm(v)]]

:

A{lI{p(x)]] I x ( U}.

e

is B o o l e a n

of

is

Boolean

inference valid

model

valid.

axiom

of p r e d i c a t e

that

the

logic

If some equality

3.3

list

recursively

the

~ y~

~

takes

axioms

are

T, T

of

<E~(~)]]

:

Boolean

Boolean

is the

valid

equality

Boolean

Boolean

this

the m a p

some

definitions

:

V u(dom(y)

:

(

Ex

A

Boolean that

the

[~

= I.

model every

premises symbol,

U

is rule

of

to a B o o l e a n

then we will

valid. one

proves

by

E~('P~)]].

and

Boolean

is a p a r t i a l

facts

x:

Boolean

pair

One d e f i n e s

:

V (~)

>B

s y~)

A (

A

ly(v) V E v

s x~).

v(dom(y)

sets

:

[22].

e u~ A y(u)

structure

takes

from

structure

function

Ix(u) v [ x

~< Define

iff

iff e a c h m e m b e r

valid;

structures,

u(dom(y) Here,

sentences

of a B o o l e a n

is

U

structure

of

point

logic

R. --]

(improper)

x ( V (B) + ~ x

Ix

i < m

that

We n o w

E x s y~

for the

of a set

The m a i n

For an i s o m o r p h i s m

induction

valid

of p r e d i c a t e

conclusion.

require

for

l

I E~]]

is a B o o l e a n

every

Ri(Xl,...,Xn.)

=

C

that:

=

I]-7(I)]]

Also, ~

)]]

n. I

will

be c a l l e d

M(B).

x

to t h e i r

Boolean

{

I Y E x}

( v (~)

and

ordered

pair:

Define

analog,

recursively

183

~x,y}

:

{<x,l>,}

:

iix},i•

v

<x,y> For n o n e m p t y

N c V (B) ,

N

Vxy(x ( d o m ( y ) A dom-transitive Then

N

found

in

and

equality

[22]

for

For a c o m p l e t e structure

of

be c a l l e d

dom-transitive

iff

& y ( N--+ x ( N ) .

determines

is an i n t e r p r e t a t i o n

extensionality ment

N

will

a Boolean

of the are

substructure

language

Boolean

valid

Ls. for

N

The N,

of

M(~).

axioms by the

of argu-

V (B)

subalgebra

s

of

B,

M(C)

is a B o o l e a n

sub-

~[(B):

v (r

c

v (~).

Ix

e y](C)

:

Ix

E y~

for

x,y

E V (c)"

Ix

--- y~(r

:

Fx

= y~

for

x,y

E V (c)

and

(The

superscript A Boolean

({)

isomorphism

T: V ( ~ ) ---+V (B 2)

if

function T

s o is

is a T.

T

the

value

T: B I - - ~ B 2

for the

structure

determines

a

i-i

M(C).) onto

map

by

! (x) Then

denotes

:

{ < ! ( Y ) , T ( x ( Y ))>

is p r i m i t i v e

(partial) Together,

recursive

primitive T

and

I Y

in the

recursive

T

form

( dom(x)}.

an

function

function(with isomorphism

T;

that

parameters),

of B o o l e a n

structures:

For a n o t h e r

TEx

E yl

=

E~_x e T_y],

T[x

~ y]

:

[Ix

isomorphism

o: B 2 - - + ~ 3 ,

~ ~y].

oT = o ~.

If

T

is i d e n t i t y ,

is,

184

so is

~.

For c o m p l e t e

generally

3.4.

write

Let

a dense

above.

Boolean

is the

relations

7

Pu

E X(Pu

~u

Then

Then

_< b) ~

for each

Pu _< AX.

P

188

Bu

is closed

We show

Pu ~ ib

Consider

LEMMA. A

i-i

Assume

by showing

i b).

contradiction;

i)

7:

P2 = PI u U P(~-u)

VP2 ~ P(P2

4.2.

under

Thus,

thus,

b ~ Bu.

Then assume

VP I ~ Pu(P I ~ b). ~ Pu U P(~-u)

P2 ~ b.

If

Assume

= P.

PI ~ b,

Since then

P ~ lb.

PI ~ Pu. P ~ ib,

b ~ PI u ~ P2'

PI i b.

Symmetry.

onto map

determines an isomorphism

g: u--+v

g]p: ~ u - - + P v

by

I

),

namely

g~({,...,})

For the unique isomorphism

:

= gB ~ w

for

(glg)B

= glBg]B

for

g

is identity,

then

2)

A finite partial map

gB

extending

gp:

w ~ u;

onto

l-i

gl:

v--~w;

is identity.

p: ~x m--+ 2

determines an isomorphism

by



I

~

for



(dom(p)



I

~

for



(dom(p)



I ~

for



~ dom(p).

For the unique isomorphism

Pu: ]Bu--+]Bu

Pw = Pu ~IBw

if

{,...,}.

gB: B u - - + ~ v

(g ~ w)B

if

Pu: P u - - + ~ u

a

p~ux0~ c_ {I},

then

Pu

for

extending w c_ U;

is the identity.

Pu: Pu--+lnu:

189

3) i-i

Consider onto

7 ( ~,

nonempty

h: ~ - r n g ( s ) - - + ~ - { 7 ) .

ysh: P w - - + P ~

for

r ( m

~



for

a (dom(h).

z ~ dom(h)

and

The above

u

ysh

h ~z

V (~u),

The p r o o f

P~

~: B ~ - ~ operation

defined

on

Bz

~*z

b (~

z

of the respective

P, ~,

g, p,

and

u.

(g ~ w)~

in

B~,

@(~)).

and

we d e f i n e d and

I VP

gB ~ B w

: g~ ~Pw.

For each

: {b ~ ~

For example,

The

z c ~, and

the r e l a t i o n

z c ~,

are each the

(here,

this

( b)},

V (~*z)

=

in 3.1 an i d e n t i f i c a t i o n

b = ~Ib}

( b(Pz

(g ~ w ) ~

QED

~

l

is that

identifies

since

( ~z*-+ VP(P ~ ~-ib--+ Pz ~ ~-ib) +-~ VP(P

( b--~ Pz

is p r i m i t i v e

~ b).

reeursive

in

P~:

z+-~ b ( ~

x ( V(~*z)*-+x

& ~P

( V (~)

Let a d m i s s i b l e We now b e g i n

(g ~ w)B

= {b I b ~ P ~

= ~C~z

b ( ~ B z + - ~ ~-ib

parameters

is identity,

We call these isomorphisms

is left to the reader.

is d e n s e ~

For each

isomorphisms

We omit the subscripts

extending

where

with

ysh I P z

the identity.

f r o m the fact that

isomorphism

Since

etc.

then

(2) is to be s e l e c t e d to fit the context.

follows

unique

is also

determine

respectively.

PROOF:

is identity,

ysh ~Bz

(1,2,3)

of symmetry

g~ [ B w

and

These determine an i s o m o r p h i s m

~

Boolean structures and

s: m--*~,

I

and its extension

4)

finite sequence

by

<s(r),n,k>

If

i-i

set

to d e f i n e

( b(Pz

(b)

& Vy ( d o m ( x ) ( y

M Nu

be g i v e n

( V (~*z) & x(y)

such that

(which d e p e n d s

on

~ ( M. M)

(~*z).

Then

for each

~

( M.

u c ~.

190

The

Mz

4.3.

of Lemma

Definition.

Nz = D-l(CN*z. in

M,

of

V (~*z)

x(y)

N*z

be

For each

Since is

Note

Nw U z

finite

V (~*z)

M-A I.

yields

E ~*z).

Also,

"reduced"

z c w,

is p r i m i t i v e

The r e s t r i c t i o n

x E N*z+~x that

put

z ~ zI

M

& Vy

implies

N*z

recursive

to

E M~w)

to a t r a n s i t i v e = V (~*z)

set.

n M

and

with p a r a m e t e r s

of the above E dom(x)(y

N*z ~ N*z I

definition

E N*z

and

&

Nz ~ Nz I.

Nz c V (~z).

4.4. i)

1.2 will

LEMMA.

For

z,z I ~ w ,

finite

z I _c dom(gl),

g" gl

maps

of s y m m e t r y

(i),

z ! dom(g),

g~Cz : glZl :~c

and

~C

g~CNz = glNZl . 2)

For

finite

z c w,

g

pg~Nz For

finite

PROOF:

there and

of

z c w

(i) M.

becomes Pw.

and

X,

s,

--

Since

Put

finite

h

of symmetry

= N({y}

h : gllg ~ z

Further,

h*: ~B*z--+~*z I

]~

etc.)

Thus,

h*~CN*z c N*z I N*z I.

and

of

here, h*:

V (~*z)

Similarly,

~

h = D-lh*~,

(3):

U h~tz). E M.

Now

was d e f i n e d

and it was p r i m i t i v e

) V (~ * zl )

maps

h-l*CCN*Zl c_ N*z h(CNz = Nz I.

M .

Thus

into

h: l~z--+IVz I

Consider

the finite :

Pl(a'n)

Pl

(where

recursive

in

M.

Combining,

IP(g(a)'n)

h*~CN *

~undefined

(2) d e f i n e d

if d e f i n e d otherwise.

2B h

Thus Z

-ig g~CNz = (glg I ~ z)r

for s y m m e t r y

is a

in 3.1

glh~CNz = glccNZl . (2)

p

= g~Nz.

ysh~(N(z U rng(s))

member

z c dom(g),

(2):

symmetry

3)

(i),

of symmetry

by

--

=

191

Then

g - { p g ~l~

is the same as

PI: l~

Then

(Pl ~l~

V(~*z)---+V(~*z)

maps

verse;

(Pl ~l~

= N*z.

pgCCNz

(3)

Consider

morphism M.

onto

Thus,

Thus,

(ysh ~ z

(ys h ~ P z

Nu

=

h: u - + u ,

Nu = h(~Nu.

Thus,

O rng(s) H,

is an iso-

is a m e m b e r

as does

finite

when

and

U rng(s)

= N*({y}

For any

( M.

gCCNz.

into

U r n g ( s ))

Pl ~ P z

as does its in-

= Nz

ysh ~ P z

M

: Ng(Cz c Nu

M,

=

ysh I ~ z

maps

u c r

g~Nz

PlNZ

gpl~(Nz

Further,

U rng(s))*~(N*(z

any f i n i t e

into (C

Then

U rng(s))*

g: z--~g~z,

identity

gg- i p g ~ N z

U hCCz.

M

Therefore,

ysh: P r 1 6 2

P{y}

Consider onto

:

Further,

its inverse.

U h~z).

gCCz c_ u.

QED and

z c Also,

we can e x t e n d

of

i-i

for the

the d e f i n i t i o n

of

as follows:

4.5

Definition.

Nu

Then

:

For

u ~ ~,

Nu

U{g~CNz J f i n i t e

is d e f i n e d

z ~ ~

and

to be

i-i

g: z--~u}.

Nu c V (Bu) . If

u c v c ~,

If

u,v c ~

h~(Nu

then

and if

Nu c Nv. h: u - - * v

is

i-I

onto,

then

=

U{hg~Nz

J finite

z ~ ~

and

i-i

g: z - + u }

=

U{gl(cNz

I finite

z c_ ~

and

i-i

gl : z--+ v}

:

Nv.

In p a r t i c u l a r ,

for f i n i t e

z c ~

and

i-i

g: z--~g(Cz

gCCNz : Ng c~z

Thus,

Nu

:

U{Ng~Cz

=

U{Nz

J finite

J finite

z ~ ~

z ! u}.

and

i-i

g: z - + u }

192

Then

~-Ir162

show l a t e r that so that

N~

transitive

is N*~

U{N*z

of

For the p r e s e n t , M aP~);

it is

x ( N*~-+3z(finite

For

g: u - ~ v

of symmetry

Boolean s u b s t r u c t u r e s

2)

For

p

and

u

it

N*w.

of the a d m i s s i b i l i t y we note that

M-A 1

N*~

We will axioms,

is a dom-

since

z c ~ & x (N'z).

For

Y, s,

Nu

(I),

and

Nv

of symmetry

Boolean substructure

3)

Call

LEMMA.

4.6.

i)

z ~ ~}.

is a B o o l e a n m o d e l

is also. subset

I finite

h

g ~Nu of

(2),

is an i s o m o r p h i s m of the

V (B).

p ~ Nu

is an a u t o m o r p h i s m of the

Nu.

of symmetry

(3),

ysh ~ Nw

is an a u t o m o r p h i s m of

N~.

PROOF:

(i)

We h a v e

shown

g~rNu = Nv.

As m e n t i o n e d

in 3.2, this

suffices. (2)

Lemma

p~Nu

(3)

structure

(2) i m p l i e s

=

U{pg ~Nz

:

O{g ~Nz

:

Nu.

Lemma

Define

4.4

4.4

[~(~)~

I finite

I finite

= Nu:

z c ~

and

z ~ ~

and

ysh~Nw

= N~.

to be the B o o l e a n

value

(3) i m p l i e s

u

p~Nu



of

V (B)

Define

i-i

i-i

g: z - ~ u }

g: z - ~ u }

QED

for the B o o l e a n P j~u ~(~)

sub-

to m e a n

P s [ ~ (~)~ u . 4.7.

LEMMA.

if

x ( Nz

and

z c u,

then

E~(~)~

--

PROOF: Assume

It s u f f i c e s

P' ~ Pz.

to show that

For s y m m e t r y

(2),

(Bz. U

P I~u ~(~)

find

implies

p: dom(P)

Pz

I~(~).

n dom(P')--+ 2

193

such that

(pP) ~ d o m ( P ' )

if

P(a,n)

= P ' ( a,n)

p(a,n)

-- 0

if

P(a,n)

~ P ' ( a,n),

undefined

l~

and on

V (~z) .

VP' By the

(extended)

set,

Pz IFu ~(~).

4.8.

LEMMA.

E~(~)]

u

"~ ~ Nz,

~

v

(I), find

Then

g ~]Bz

4.9.

THEOREM.

by

defined both defined

extends

p~ = ~.

_~ P ' ( P "

Pz,

This

p

is the iden-

shows that

I}-utO(~)).

of the c o n d i t i o n

finite

z c u n v,

for b e i n g a dense

and

u ~

v,

then

~z.

i-i

onto

is i d e n t i t y ;

No~

is

dom-transitive,

PROOF:

P'

Thus,

By the p r e v i o u s

metry

Since

-~ Pz BP"

p

otherwise).

contrapositive

If

= E~(~)]

PROOF:

is

define

: i

P' _> P' U pP l~u~(p~).

tity on

It

(i.e.,

p(~,n)

p(a,n)

Then

c_ P'

lemma, g: u--+ v

E~(~)~

a Boolean as

b o t h values

is

such that

= g([~(~)~

u

model

the

g ~ z

]Bz.

is identity.

) = [9(g~)~v

axioms

For sym-

= E~(~)]] v 9

of admissibility.

Nu.

every

The d o m - t r a n s i t i v i t y

of

u

are in

f o l l o ws

directly

from the d e f i n i -

tions. We n o w p r o v e missibility. transitive N = N*~.

is

N*~

N o t e that

and

M-A I.

T h e n for

For the p u r p o s e The

that

is a B o o l e a n m o d e l

N*~

is an

genZn+ I

are in

ZO: { <x,]l>, }

of 3.4,

We use the n o t a t i o n formula

of this proof,

following

N

N*~

P

~,

there

i.e.,

x, y,

it is dom-

is

M - g e n Z n + I.

of

P~.

are in

N*~

a member x

of ad-

for the s t r u c t u r e

P ]F~(~)

denotes

when

of the axioms

and

194

{ {,}

satisfies

Let

Lemma

N'w,

of [22]

that

M

does also:

using

Z0-separation

see the p r o o f obvious

find

are

E dom(z))}

and the above

x, y, x, ~;

are,

values

considered

N*w

§x

x, y,

at

E dom(x)(y

I y E dom(x)].

is P r i m - c l o s e d

functions

that these

of pair,

s x & Q(~,y)]>

I 3z

E 0-

schema;

it is

"pair.")

g e n Z n - s e p a r a t i o n , then

Let

x,x

E N*w.

~ V V l ( V I c x2+-~ v I c x & Q ( ~ , V l ) ) ]

: ~.

We w i l l First,

-w

x I E dom(x) is

M-genE

w By

(or

n

=

M-A I

{<xI,P>

E M.

x2 Then

:

(*)

:

~x I e x & ~ ( ~ , X l ) ]

comprehension tance

of

& P I~ x I ~ x & ~ ( ~ , X l ) } .

s

w E M,

I x I E dom(x) M

& b : {P

is P r i m - c l o s e d .

{<Xl, I x I ~ x & ~ ( ~ , X l ) ] >

z c w

x 2 ~ N*z ~ N*~,

Put

since

Put

since

x2

Find f i n i t e

(or

{<Xl,b>

x 2 E M,

n : @).

I x I E dom(x)

genZn-Separation

dom(x) •

if

& P [ ~ - x I s x & 9 ( x , x I)

such that E ~*z

for all

The p r o o f that

schema

given

genAn-separation

x,x

E N*z.

I <xi,P>

Further,

1 x I ( dom(x)}. By L e m m a

x I E dom(x) x2

in [22],

is s l i g h t l y

equation

harder

4.7,

~ N*z.

is as d e s i r e d using

E w}].

Thus,

is the p r o o f of the (*).

to show.)

(The i n h e r i -

195

4.10.

LEMMA.

If

M

satisfies

the schema of

E -reflection,

then

n

N*~

In p a r t i c u l a r ,

does also.

N*~

satisfies

the schema of

~0-

reflection.

PROOF:

Let

~ ( V , V o , V 1) P I~ Vv 0

we will

show that

there

is

be

Z n.

Let

x,x ( N*~.

Assume

3v1~(~,v0,vl) ;

sx

x2

such that

P J~Vv 0 ~ x3v I ~ x 2 ~ ( ~ , v 0 , v l ) . Find finite

P (l~

z c ~

Find

such that

x,x

( N*z.

By Lemma

4.7, we can assume

y E ~-z.

For each

x 0 ( dom(x):

P I~ x 0 ~ x - - + 3 V l ~ ( ~ , x 0 , v l ) ; P I~ 3 V l ( X 0 e x - + ~ ( ~ , x 0 , v l ) ) ; VP0-

Find

then some

We

extensionality

The

Then

~ dom(P). s x~.

I

b 0 ( N{0}.

# ~,

is similar).

P U

E Pw

and

= a.

{ x~

P' ~ l~n

case

{ w:

z ~ e.

such

or

implies

true:

If

n U.

is true.

is a limit

w ! 6 < Y < ~

part

Pi

= {}

n



P ~

that

Ui<w

exists

1

n ((pg)-ipO) --

~ ({B}

F ((z-{B)) x w) ~ G B g

is i d e n t i t y

on

n dom((pg)-iP*))--+

2

x w)

=

pO ~ ({B} n

since

dom(p)

z-{B}.

For

such

that

x w)

=

pO. n

is d i s j o i n t

symmetry

(2),

203

p l ( ( ( p g ) - i P *) ~ ((u-z) • ~)) ~ G B.

Put

P : p l ( ( ( p g ) - i P *) ~ ((~ U z-{B}) x ~ ) ) gives p r o p e r t y Plqn

pl(pg)-ipgqn

:

= Mz' N~

U

we

z,

3P

iff

c

have

G(P

_< E ~ ( ' ~ ) ] ]

iff

3P

c G(P

< [~(~)~

--

--

iff

Thus,

Mz

is

Since is

N~

admissible. Mz

x

an

( Nr

n Mz' U z

P ~ G, such

that For

Lemma

is

a

It

follows

and

y

each

~ Then

( Nr

E w]. ( ~i'

U z'

each

such

Corollary Thus,

t

define ( M{~} .

of

t

Z

(used

4.7)

I

Mz'

the

admissibility

Xz

is a d m i s s i b l e .

( Mz

N Mz'.

that

H(x)

4.13,

there

= H(x) = H(b

as

O

I: ~ ( H ( ~ ) ) .

of

Suppose

) ~

Since

~b

Then

= t

~ N(r

( Mz

N z'.

b6~

( M}

:

b :

r

U

comes a

~

M0

are

For

w

where E

there

= H(y).

is

= H(w) )

axioms,

some

(z n

from

a 6.

o M:

Mz

o(Mz) o(M)

By

model that

N z':

a

submodel

Boolean

~ y~.

P ~ Ix

4 9 Ii. Hz

elementary

= Mz

P < Ex

<Mz',(>

)

~uz

--

= o(M):

~

o(Mz)

greater

than

o(M)

follows that

n_ H " N z

of

c o(Mz)

from x.

n_ H~'{~

I x

follows

observing

that

from the

M.

the rank

preceding; of

H(x)

is

no

z'))

205

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Berlin,

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Disser-

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12.

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Length,

North-

Logic

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206

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22.

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24.

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