Set Theory and Model Theory

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

872 Set Theory and Model Theory Proceedings of an Informal Symposium Held at Bonn, June 1-3, 1979

Edited by R. B. Jensen and A. Prestel

Springer-Verlag Berlin Heidelberg New York 1981

Editors

Ronald Bj6rn Jensen All Souls College Oxford OXl 4AL, England Alexander Prestel Fakult~,t fL~r Mathematik, Universit~t Konstanz Postfach 5560, 7750 Konst&nz, Federal Republic of Germany

AMS Subject Classifications (1980): 03Cxx, 03 Exx

ISBN 3-540-10849-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10849-1 Springer-Verlag New York Heidelberg 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 § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

FOREWORD O n the o c c a s i o n informal from

symposium

I.-3.

of J ~ n e

all d e d i c a t e d some

Each traces

Scholz.

concept book

The

directly

ontological too polite

that

o n them. prove

faith.

study

of

on them.

with more of P a r i s We

hope

set

fear,

Heinrich

have

a

today,

'real

rather

set

faith

content' than

its to

to

stronger

non-standard Some

of u s

immediate

the

some

models

will

and

but

provided

among

of

he not

us of

'taking and

be

to s h e d

light

of

able of

logic will

the g r e a t

perhaps For

this

encouraged

however,

a first

were

numbers

even

some

unsolved point

reason those

all e v e n t u a l l y

Recently,

volume

natural

tools

theory).

tried,

rewards.

and Harrington

the

he h a s m a d e around

turned

to

the b e a u t i f u l

vindication

of h i s

follow.

are grateful

to P r o f e s s o r his

association

the

should

of

to this

importance

Hasenjaeger,

(and in t h e p r o c e s s

in s o m e

on the

theorists of

that

undecidability

theory

contributions

that

direction

he h a s

that more

the

- his humor, our

deep

is an e a g e r

- a conviction

insistence

the primacy

We

made

contri-

from Hasenjaeger

in l o g i c

however,

qualities which

bears

platonists

content,

learned

the

of P r o f e s s o r

teacher,

must

its

theoretical

accuse

mathematical

of n u m b e r

to work

theorem

We

so, w o u l d

to their s o l u t i o n

fields

of

the e l e m e n t a r y

a lifelong him

as c e n t r a l

He recognizes

a major

of us

to H a s e n j a e g e r ' s

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

questions way

to d o

of

are

place

assistant.

call ourselves

by

took

witness.

questions.

road'.

All

himself

by his

an

at B o n n

proceedings

to m a t h e m a t i c s ,

judged All

held

co-workers his

Hasenjaeger

advocated

be

of m o d e l

bears

or

that mathematics

value.

preponderance

is r e l a t e d

view

should

or esthetic

the

approach

birthday

the meeting

extended.

or a n o t h e r

influence.

sense

a problem

this

easy

own

and

was

in t h e s e

Since

students

time

a l l of u s w o u l d

a profound

that

regard

day

in h i s

not

revised

former

60-th

theory

published

Hasenjaeger.

at o n e

of t h e p l a t o n i c

difficulty

feels

of us,

Though

we retain

which

are

Each was

Hasenjaeger's and model

The papers

been

of H a s e n j a e g e r ' s

proponent

- and

have

to t h i s v o l u m e

Hasenjaeger.

the

1979.

theory

to P r o f e s s o r

of t h e p a p e r s

butors

of G i s b e r t

on set

Hasenjaeger

tolerance, with

him

his

for

patient

the many

human

attentiveness

so p l e a s a n t . R. B. A.

Jensen

Prestel

-

TABLE

KEITH

CONTENTS

J. D E V L I N Morass-like

HANS-DIETER

DONDER, Some

SABINE

constructions

of

~2-trees

in

L

DONDER

Coarse

H.D.

OF

morasses

R.B.

in

JENSEN,

applications

37

L

B.

KOPPELBERG

of the c o r e

55

model

KOPPELBERG A lattice boolean

ALEXANDER

o n the

isomorphism

types

of

complete 98

PRESTEL

Pseudo

TASSILO

structure

algebras

real

closed

127

fields

VON DER TWER

Some

remarks

arithmetic

on the m a t h e m a t i c a l

found

by Paris

incompleteness

and Harrington

of P e a n o ' s 157

Morass-Like

Constructions

of ~ 2 - T r e e s

in L

by

Keith

J. D e y l i n

(Lancaster,

U.K.)

Abstract

Using the

a simplified

construction

Kurepa

~2-trees

techniques lin

~2-tree

principal, quent not

were

version

of a m o r a s s ,

expository

we

fine

originally

developed

uniform

that

the

filter

R. L a v e r

result

in nature.

reduced

on ~,

Rather

theory

to c o n s t r u c t

systems

power

tree

a much the

required Souslin

of c o u n t a b l e

by us in o r d e r

is a K u r e p a

obtained

here.

structure

show how

of d i r e c t e d

T such

this

the

as limits

to o u r work,

include

of

T°/D,

~2-tree. simpler

present

where

account

These

a Sous-

D is a non-

However, proof,

and

trees.

to o b t a i n

for

subse-

so we

do

is l a r g e l y

§

We w o r k ventions. nals IXI

are

initials

Most

the

refer

require

Our of

the

here

von Neumann

cardinality

of the

set X.

results

I] is n o t

require

importance

[De

and

concardi-

ordinals;

It t u r n s

Section out

(in p a r t i c u l a r ,

2 provides

that we morass

to k n o w

V : L,

do n o t

axiom

something

(M7)

of the

(in L). trees

o f the v a r i o u s of w h a t

is a p o s e t

ordinals

and

of constructibility,

b u t we d o n e e d

concerning

resum6

need.

notation

to d e n o t e

I] for d e t a i l s .

o f the m o r a s s

of the m o r a s s

etc.

the a x i o m

theory we

required),

terminology

A tree

are

the u s u a l

e,B,y,

full power

a quick

adopt

We use

to o u r m o n o g r a p h

construction

and

ordinals

o f the m o r a s s

the

[De

theory

ordinals.

of o u r

an o u t l i n e

from

in ZFC s e t

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

denotes

and we

I. P r e l i m i n a r i e s

is

fairly

definitions

standard,

to this

but

in v i e w

paper we

present

w e need.

~ = s u c h

that

the s e t ~ = {y 6 T I y

be

~ edom(f).

Hence

f c~xd

B = v or else

if p ~ > m,

(~ × m ) 6 J n ~ J s .

into

Case

~ = ~(~)'

is c o f i n a l

Z n ( J 8) m a p

map

let 7(m)

A = A(~),

of J 7

7 = 7(~)-

is J - d e f i n a b l e 7

from

U {q}. 2.2

in JB

there

is a p 6 J

from parameters

is a r u d i m e n t a r y

function

such

that

every

element

in J~ U {p}.

Since

J~ = rud(Jy)

f and

an e l e m e n t

q of J

such

of JB is

that

Y p : f(J

,q). We s h o w Y Let x 6 Jy. Then

unique

x in J~ s u c h

JB = m~j

implies

the

q = o-l(q),

D.

C and

which

That

Set

B and

claims

= ,

. Set s = . Let % be a E o - f O r m u l a such that: P (i)

<J

(ii)

<J

(iii)

( V y 6 J ) [y = o ( x ) P

P P

,A> b Vz 3y Vy'

[y' = y ÷÷ 3 u % ( u , y ' , z , q ) ] ;

,A> b Vz V y [ 3 u # ( u , y , z , q ) ÷÷

<J

P

+

(~)

,A> ~

(z = )];

Hu%(u,y,s,q)].

Let

b ~(Uo,~(x),t,q).

is E l - d e f i n a b l e

a(x),q 6 ran(d)

~ i < J p ' A > ' t ' U o 6 ran(o).

t = for some ~ 6 a. By (Vy 6 Jp) Applying

-i,

The m i n i m a l i t y

So as

t ~ j s, t 6 Je.

(i) above,

~ = o-l(t),

[y = x ÷+ <J~,A>

x is E l - d e f i n a b l e

in <Jp,A>. Since

[y = d(x\) ÷÷ <Jp,A>

and s e t t i n g

(Vy~J~) Hence

from o ( x ) , q

we have = 3u¢(u,y,[,q)].

from p a r a m e t e r s

of q is p r o v e d

b 3u¢(u,y,t,q)].

in [ U {q} in <J~,A>.

just as in C l a i m C.

[]

Thus

12

Claim

]

J:

Proof:

By

is n o t Claim

J-

In-regular

"(w × (J- x { q } ) ) .

P,A

In particular,

there

Since

~ = p~-l,

Claim

K:

is

{ = B(]), claims

n = n(~),

By

lim(B)

+

lim(~).)

[]

L:

q = q(~)

and

Proof:

By

a II(<J~,A>)

~ : A ~n - i , t h i s

Proof:

Claim

H and

claims

J.

so o ( p ( ~ ) )

o(~)

:~.

then

That

=

completes

Lemma

Let

p(~).

5 6 S[,

such

Let

~'

that

Case

the

[ onto

~.

~ : p(~), notice

A : A(]). that

~:

JB

~i JB'

so

: p(~). If

: p(v)

~ : ~ and

proof

of

v < p,

by

p(~)

lemma

p(C),

and

suppose

<Jp(~) ,A([)> ~i

~[[].

then

claims

v < p and we

G and

= ,

the so

fact

again

have that we

get

2.6.

] is

a limit

point

of S-.e L e t

<Jp(~),A(v)>

: p(u).

Then

v'e

<Jp(~) ,A(~)>

c~J-cd' v -

Set

a < a,

o(p(~))

= sup

that

Proof: =

of

[]

~ 6 S,

o':

such

a subset

2.7

o:

be

map

q : q(~).

= ,

from

is I n (S~).

~ 6 R,

o(p(~))

I a n d K,

If u = p,

map

(For ~ 6 R,

p(5)

o(p(~))

J~.

I,

: h-

P

over

S

~i

o'(p(~))

and

and

there

is

an e m b e d d i n g

<Jp(~') ,A(~')>

: p(m').

B : B(~) , n = n(v) , p = p(v) , A = A(m) , q = q(~) , p : p(~) , ~

= A(~),

I. ~ 6 P.

Set

~

:

q(]), ~ : p(]).

y = y(v) .

13

For each m £ ~, set X m = {x 6 J

Y

I x is E m + l - d e f i n a b l e

from parameters

in ~ U {q} in

J } Y Thus X m ~ m

Jy.

S i n c e @ is the l a r g e s t

X m @ ~ is t r a n s i t i v e , a Jy-definable

is E l - d e f i n a b l e

t_] X = J¥, m ~ O So,

as o is c o f i n a l

<J

,A>.

in q,

Set

X = hn, ~ "(~ x <Ja × {q}}).

Let

~: <Jy,B> ~ <X,~A X>.

Thus

~: < J y , B > 4 1 <Jn,A>.

%

C l a i m A: ran(u) c X . Proof:

L e t x 6 ran(6).

T h e n x is E l - d e f i n a b l e

from parameters

U {q} in <J

in

,A>. Let x = o-l(x). A n a r g u m e n t as in c l a i m I of P l e m m a 2.6 shows t h a t x is E l - d e f i n a b l e f r o m p a r a m e t e r s in ~ U {q} in <J~,A>.

So for some i 6 w, z 6 J~, x = h ~ , ~ ( i , < z , q > ) .

Applying

14

e:

<J~,A>

x

= h

a n d × {q'})).

= 6(~'),

n : n(v'),

C and

D.

Hence

as

.

Hence

= o(~')

it

suffices

to

show

J6'"

over

41 < J

: v'.

= q(v').

In-regular

: v' n h q , ~

p =

-l(p)

2.6.

~'

(v)

= 7-

that

that

-i

Hence

v' < y.

[]

"(~ × ( J

~' c X Thus

x {q})).

= ran(~), there

n-I 871 y = PS' ' B = A , so this

we

is

map

have

a II(<Jy,B>) is E n

~' 6 R, y : p ( ~ ' ) ,

(J6

,).

map

of

[]

B = A(~').

[]

= q(~').

"(~ x ( J

parameters

lemma

B,

lemma

"(~ × ( J

claims

q'

B,

B,

to s h o w

i.e.

claim

H in

By definition,

= h¥, B

from

The

,B>

= ~' n hy, B

onto

Jy

y

J6

6'

p =.

p = ,

claim

in Claim

By

by

e>,

Moreover,

is I n _ l - r e g u l a r

Proof: as

using

in o r d e r

as

~'

But

[ : ~ and

= q(Y'),

~'

)>.

= .

And,

above,

that

, -i (~

p'

~ = p.

p'

Proof:

(v)

a unique

~ -i ( p ) .

=

Suppose v < @. T h e n ~ < @ a n d p = < q , ~ p'

is

X = h

× {q'})).

in a U [q'}

completes

is p r o v e d .

the

[]

,~ "(~ × (J~ × { q } ) ) . Hence

every

in < J y , B > .

proof,

u

member

And

an

So, of J y

argument

applying

~

-i

,

is E l - d e f i n a b l e as

in C l a i m

C in

18

Suppose iff for

now

that

v is a limit

all T o - f O r m u l a s

¢(Vo,Vl)

ordinal

and X c J

of set theory,

. We w r i t e

with

X 4Q J

parameters

from

X,

X ~

(¥~)

Clearly,

(B8 > ~)

if X ~ = p(T)

~Q

most

one ~ y

bedding J

P(~

~:

= h

ran(o) mined

It s u f f i c e s

iff e

1 <J

such

,A(T)> (o~J):

J

~

p(~),A(T)

< ~

and

T

there

is

that

4Q J T

partial

to show

that

= ~. Let

<Jp(v) ,A(v)>

0(Y) ,A(~)

= h by

with

and X N v is

J~.

~ on S by ~ T

and

~ is a w e l l - f o u n d e d

is a tree.

$(8,J8).

if X

"(~ × ( J - × {p(~)})),

p(~)

2.5,

the m a p

show

and [ < e,

that

there

= [. T h e n

there

as above.

Now,

is at

is a E l - e m -

so

"(w × (J- × {p(~)})).

T a n d [. H e n c e

on S. We

Thus

is c o m p l e t e l y

ran(0)

is e n t i r e l y

determined

by

deter-

~ and ~. H e n c e

so is ~.

By

lemma

if v ~ T ,

so we m a y

denote

{0 T I ~ T }

and

Lemma

if by ~ T

{~ T I v ~ T }

o which

. Set are

~ ~r :

testifies

this

fact

(a ~T Iv) U {}.

is u n i q u e ,

The

systems

commutative.

2.8

Let ~ T .

Then

maps

S

~ (v+l)

into

S

N(T+I)

in an o r d e r - p r e s e r v -

T ing f a s h i o n (i)

such

that:

if y = m i n ( S

) , then

z T(y)

= min(Se

); T

(ii)

if y i m m e d i a t e l y

succeeds

6 in S

n (~+i) , then V

immediately

succeeds

~

(6)

in S

N (T+l); T

~

~T

(y)

17 (iii)

if y is a l i m i t

point

of S

point

of S

A (m+l),

then

z

(~) is a limit

~ (~+1). T

Proof:

This

following We wish

follows

case.

trivially

Suppose

to s h o w

that

from

lemma

~ is a limit

T : z

(~)

2.1

point

is a limit

(vi),

except

of S

and

point

of S

in the

that

~ = p(~).

. This

follows

T

easily

from t h e

fact

that

where

the Q - e m b e d d i n g

Lemma

2.9

T ~T,

Let

~ 6 S~_~

(g

~J ) : J

condition

4Q J ~ .

This is

is r e q u i r e d .

T, ~ : ~--T~(~). T h e n

the only point

[]

~ ~,

~-

I~ = ~-~T I~' and

g-

(A(~))

T

By

Proof:

lemma

2.4,

-

(p(]))

= p(y)

UTT

~

(p(~))

Lemma

= p(~).

The

~

lemma

follows

= A(Y)

and

TT

'

immediately.

2.10

If T 6 S is a limit

point

of 4, then

T =

k_]

~[~]

and p

J

=

p(T)

Proof: and

t_]

o

v~T We

<X

Iv ~ T >

~:

lemma

o(p(~)) since

X

p(m)

commence

<Jp(T),A(T)>.

2.6

I~ ~ T }

•~

that

Set X = U{X

<J~,A>

there

in < J p ( T ) , A ( T ) > . T)

Iv ~ < } .

~ = sup

for

all

Then Pick

~ such {~

V~T

let X

Suppose = ran(o

Z 1 submodels

not,

T) •

of

X K1 <Jp(T) 'A(T)>"

v-~T,

some

we

we h a v e

Let

with

~i <Jp(z) 'A(T)>'

~6 e

clearly

A = A(~), have

X A JT ~ Q JT"

contrary ~ 6 ~T,

~ = p(~),

that

I~)-~T],

5 in ~ ,

for

~ ~T

of

Thus

[~\~I~ ~ T ) .

X>.

is a u n i q u e

T succeeds

let x 6 Jp(T).

chain

= <X,A(T)N

Since

a T : sup

. For e a c h

is an i n c r e a s i n g

N JT T.

place

point

Given

ourselves

in

of

~ follows

at

~.

8 < e we

show

that

Jl"

X be

the

Let

e l e m e n t a r y submodel o f < J p ( T ) , A ( T ) > c o n t a i n i n g

smallest such

~

2.1 1

Proof: is

~~

o

Lemma If

=

p(T)

that

X n a is

return

to

the

transitive. real

Since

world. Let

a = ~i'

q:

<J

,A>

X n ~ 6 ~.

p(T)

Let

= <X,A(T)

there

and 8,

~ = X N ~.

n X>.

By

lemma

2.6

P there q

is

= o.

a unique

~ 6 S-

Since

= ~ > 8, w e

~

such

that are

p =

p(~),

done.

A

= A(~).

of

S

. Let

~

[~]

Then

~ ~T

and

[]

~T

Lemma Let

2.1 2

~ ,

Then

and

]~'

Proof:

and

Clearly,

Hence

(q~IJ]):

now.

[]

suppose a~,

~ is

limit

point

~'

= sup(~[-~]).

~J~ = o ] ~ I J ~ .

(o~IJ~): J~

a

.

<Jp(~) ,A(v)> ~ % ( u , v , x , p ( v ) )

for u , v 6 J p ( ] )



and T ~ T ,

T ~ which

and e l e m e n t s

u and IT ~)] = T m

N ran(~5~),

we

'

<Jp(~) ,A(m)> ~ ~ ( u , v , x , p ( m ) ) .

by

iff

<JD(~),A(~)>

s e e n t h a t B is a b r a n c h

b %(u,y,p(5)) . through

T 5 and that ~L~[B] _cB.

29

Now,

the b r a n c h

there

is

shall

ensure

B just

defined

(by c o n d i t i o n that

~-

(iii)

is E l - d e f i n a b l e

above)

(x) e x t e n d s

a point

B on T T.

over

<Jp(5),A([)>,

x 6 T ?- e x t e n d i n g (This

is w h e r e

so

B. We

we

lose

the

TT

full

normality

with

El-definitions

tions

as if they

not be

As

the

Case

~

T~T

rise

to

not w i t h

and we m u s t

"different"

branches handle

so m u c h

different

branches,

though

as defini-

this m a y

of course.)

construction

a limit

T T = _kJ

We d e a l

of b r a n c h e s ,

3, the

I. T is

trees.

gave

case,

in s e c t i o n

Set

of our

point

[TT].

prpceeds

by

cases.

in ~ .

Conditions

(i)

and

(ii)

are v e r i f i e d

as in

TT

section

3. W e

Suppose

then

cheek that

(iii)

and

Y = s(m),

(iv)

where

simultaneously. ~ is a limit

point

of S

, and that T

T ~) i s

71-definable

which

is E l - d e f i n a b l e

qb,t) o f

over

<Jp(x)),A(~))>.

over ÷

o~(A) a n d e l e m e n t s

Let

<Jp(~),A(~)>. ÷

x , y 6 cry s u c h

B be a branch Then

that

for

there

all

are

through

El-formulas !

u,VEJp(,~)

u

~ %(u,v,x,p(~)),

u 6 B

iff

<Jp(~),A(~)>

b ~(,m,y,p(~)).

T ~)

÷

Pick

~o ~ T

with

x÷ , y÷ 6 a~

. Let

~o = S(~o) " S u p p o s e

To ~

~T,

and

let

O

= S(5).

Define

Then,

u

<Jp(5),A(~)>

B 5 is a b r a n c h

z--V~. [Bs] B

clearly,

through

T~

•

= B~, n r a n ( z s ~ , ) . Also, ~~

[B ] ~

By "

condition

p (5) '

b %(u,v,X,p(5)).

~ ~(u,~,p(~)).

Moreover, ~sv[Bs] (iii)

for

~o

~T~TI~T,

= B n ran(~L~) , and

below

T, let x- e x t e n d M

B- on

30

T .f x

By

condition

= 7-YY (x ~

(iii).

Case There

T,

~TT'-- (X~)

any

T 'To 4T

~T.

Clearly

the

method

of

T Is m i n i m a l

are

Case

below

for

And

2.

(iv)

2.1

three

T is

in

proof

~

for

To ~ T

x extends

establishes

~T'

B on

~T.

Let

T T.This

proves

(iv).

.

subcases

initial

= X~,

to

consider.

in

S

no

checks

T

T T = ~.

Set

Case

2.2

There

T is

are

a limit

point

to

of

be

made.

S T

TT = U{T~I~ 6 S

Set

N T}.

There

is

nothing

to

check.

T Case

2.3

Y = s(~).

There

are

three

Case

2.3.1

subcases

~ is

to

initial

consider.

in S T

Set

T Y = T -m

Case Use of

2.3.2 the

~ =

member

2.3.3

~

finish.

s(D).

ordinals

each

Case

and

of of

is

T -mto

T ~. n

provide

There

a limit

is

point

infinitely

nothing

of

to

many

extensions

on

TT

check.

S T

For use

x 6 T~,

each the

ordinals

branches This

let

In

Zl-definable

over

extensions branches

be

and

this

possibily only

case

distinct

definitions

may, have

more

than

through

to p r o v i d e

arises

we

other

when

extend

branches,

one

than

of

thereby

actual

We s h a l l

of

these

many)

through

distinct

and

x.

Zl-definable

branch

extension.)

each

(countably

T ~ is

each

treating

rather

T~ containing

extensions

some

<Jp(~),A(~)>,

produced with

a branch

T - v

possibility

<Jp(~),A(~)>.

they

x

from

b x on T T

latter

though

b

over

T ~ which

Zl-definitions

is as

associating

branches.

There

branches.

are

(Hence, no

checks

some to

31

be

made.

Case As

3.

in

Case

T immediately

Case

3.1

2 there

T is

succeeds

~ in

three

subcases.

are

initial

in

~.

S T

Set

TT = $

Case

3.2

and

T is

be

done.

a limit

point

of

S T

Set

TT =

U{T~I~ 6 S

N T}.

The

only

non-trivial

check

to

be

made

T

concerns

condition

(ii).

Case

3.3

T = s(v).

Then

~ = s(~),

where

This

z-

([)

is

handled

= v.

There

as

are

in

section

three

3.

subcases

to

consider.

TT

Case

3.3.1

~ is

initial

in

S T

Let

T T = T -m.

Case

3.3.2

Then

~ = s(~)

There

are

q

, y 6 T_, ~

ordinals has

where

checks

to

be

made.

~-(~)

in

3.3.3

= n.

For

each

pair

x,y 6 T ~ such

that

T%

and

T -

infinitely

Case

non-trivial

~ : s(q).

'

x 6 T

no

x

~- ~ ( u , y , p ( ~ ) ) .

through iff

%(u,v,x,p(m))

we h a v e

iff

that

u 6B

Then

% of ~(A)

for u , v 6 Jp(v) ,

U

BC_Jp(~)

is d e f i n e d

b ~(u,y,p(~)).

by

such

that

33

We k n o w with

the a b o v e

being

We

t h a t B has

the

not

definition.

one n o w

complete

an e x t e n s i o n ,

to be

above,

let z- (x) e x t e n d TT

in case

ensure

usual,

that branch

T m is E l - d e f i n a b l e

over

we regard

the

above

over

distinct

<J p(~) ,A(~)>

definitions

(this

extension

definition

of B). T x 6 T_

T ~ containing

y' (x)

each

the r e m a i n i n g

an e x t e n s i o n

<Jp(~),A(~)>,

associated

point

For

using

of T ~ has

on T_,

B on T T

through

on T T. Now,

point

is E l - d e f i n a b l e

with

a branch

each

T ~ which

it w,

of T T as f o l l o w s .

choose

that

call

~- (w) e x t e n d TT

associated

the d e f i n i t i o n

considered

in T - V,

Let

let us

every

extends

as if they

ordinals

o n T T and

branch onto

defined

and

that

through

T ~' T where,

as

distinct

branches.

There

The

are no n o n - t r i v i a l

construction

checks

is c o m p l e t e .

to be made.

Clearly,

T =

[J T6S

T T is an a l m o s t

nor-

e1 mal

tree

of h e i g h t

~2

Suppose

not,

branches. branches

case

~2-branch.

one,

this

Define

of the Since

will

exhaust

submodels

NI~

show

transitive.

Set

L e t N I + I < J~3 transitive.

'

argument

sequence

T has ~3 m a n y

as f o l l o w s , such

all

the

~2~2-

that

T has

not need

at least to be o n e -

that

for

I < m2"

el U {T,B}c_ No.

Clearly

N o A ~2

is

mo = No A ~2"

be s m a l l e s t

Set ~ l + l

let N I =

~I = s u p n < l

shows

< b ~ l ~ < ~2 > does

such

t h a t N 1 U {N I} ~ N I + I. T h e n N X + 1 n e2 is

= N I + I A ~2"

n-

attempt

~ < w2"

= idI~ 1 and

is a limit

= is d e f i n e d ,

~ < e2"

then

q 6 J B ( v ) '

(Recall

that

so <m' n

~(I) < 8(~)-)

as Nq

Set vq' = Nq' N v for each

,q < w 2 ,

q < I. Then,

lq < l> 6 J s ( v ) " But we may r e p l a c e

definition

of Nq for q < I. So,

shows

that

(pl IN n) : Nq ~ N q ', and hence

Hence

<mqlv < I> 6 JB(v)'

J~3 by N 1 in the o r i g i n a l

as Pl: N1 = J~(1)'

and we are done.

that

a simple_ i n d u c t i o n

v q = v'q

, for all q < I.

38

References

[Del].

K.J. Notes

Devlin, Aspects of C o n s t r u c t i b i l i t y . 354

Springer Lecture

(1973).

[De2].

........... , C o n s t r u c t i b i l i t y .

[Je].

T.J.Jech.

Trees.

Springer,

to appear.

Journal of S y m b o l i c Logic 36,

(1971),

1-14.

[La].

R. Laver & S. Shelah.

[Mi].

W.J. Mitchell, A r o n s z a j n Trees and the Independence of the Transfer Property.

[Si].

J.H.

Silver,

The

~2-Souslin Hypothesis.

Annals of Math. Logic 5,

(1972), 21-46.

The Independence of Kurepa's C o n j e c t u r e

Two-Cardinal Conjectures

in Model Theory.

Pure Maths. XIII, Part I, 383-390.

AMS Proc.

and Symp.

i~

COARSE MORASSES I N Hans-Dieter

Donder

Mathematisches Universitat

Introduction several retic

Higher-gap

years

transfer

definition morasses Jensen

ago

(see

in

of h i g h e r - g a p

exist in

noticed

L

morasses

that w e a k e r

his notes,

J e n s e n gave

Our a p p r o a c h

son for

treatment

the simple

the

can be f o u n d uses

questions

morass

structure

can be o b t a i n e d

L

This

treatment We

restriction

In addition,

w h i c h do not

the n a t u r a l

rial

questions

some

direct

in

in § 2 that u s i n g properties. gap-1 torial using

L

which

application

the morass

result w h i c h an a p p r o p r i a t e

can b e more

O-principle. one

We also give some

coarse m o r a s s e s

other

to be new.

easily v i s u a l i z e d .

coarse

can get K u r e p a

~-principle

In

just define

the

One reaout

the

is not n e c e s s a r y some proper-

from the axioms

methods

b u t not

direction,

we show

trees with a d d i t i o n a l

prove

result

but we think

that

combinato-

In § 1 we only deal

things This

the r e a d e r

to a n s w e r

In this

applications.

and among

seems

tool

can be p r o v e d b y

of a

L.

of these struc-

seem to f o l l o w

is to try to c o n v i n c e

are

But

for fine m o r a s s e s .

coarse m o r a s s e s

L

that

easily

in

we use

The m a i n aim of this p a p e r in

but a

L.

rather

arguments

different.

of

that J e n s e n has only w o r k e d

gaps.

false

model-theo-

and use its p r o p e r t i e s .

we deal with.

are a c t u a l l y

strong

in [6]. The p r o o f

the fine

axiomatic

in

is the fact

for small

ties of the natural a n d some w h i c h

to prove

is still u n p u b l i s h e d

in § 2 is s l i g h t l y morass

by Jensen

them

and c o n d e n s a t i o n

a thorough

coarse

this a p p r o a c h

axiomatic for

global

introduced

work

structures

coarse d e f i n a b i l i t y

"natural"

This

essentially

BRD

have b e e n

He u s e d L.

just u s i n g

tures.

Institut

Bonn,

morasses

[5]).

theorems

L

a simple

with

combina-

could also be p r o v e d

that

the m o r a s s

proof

38

I.

Coarse Gap-1

for

gap-1

morasses

the m o m e n t

define

coarse

axioms V=L,

have b e e n d e s c r i b e d b y D e v l i n

that the r e a d e r has gap-1

(M0) -

(see

a copy of that b o o k

as a s t r u c t u r e

[I], p.149).

such a structure

in his book.

which

We first

available

satisfies

show

Assuming we can

the m o r a s s

that a s s u m i n g

can be o b t a i n e d very

easily.

There-

we give some a p p l i c a t i o n s .

Assume

V=L.

Let

L

is a model of Y In addition, set S+ =

For

morasses

(M5)

of course,

after,

morasses

ZF-.

Iv~ S I L v ~ v~ S +

~

be

the

Let

S

"there

class be

of all o r d i n a l s

T

such

the class of l i m i t points

is a l a r g e s t

uncountable

that

of

S.

cardinal"~

set

av =

the l a r g e s t

Lv-cardinal

a n d let

S ~v:=C~ ~

=

Now let

~

S+

such

v* = the least

that

v~ S

and for some

~v

in

aVUI,~} ~

has a d d i t i o n a l nice p r o p e r t i e s .

not good enough,

for there are p o i n t s

i m m e d i a t e successors. K u r e p a family. For of

B

(i.e.

L e m m a 4s Proof:

let

gv I m v

v,~

w h i c h have

$

,v~.

~ , 2

F~a

is

Row let T

6~ ~ ) .

Set

Let

~=a~. T

I:~Ig~

T1

shows that

b r a n c h of

~

=

~ =

Ta~L ~ So

is an

T U

is an T

~-tree.

Choose

is a s u c c e s s o r in g ~-tree

for

and and

(since g~ L ~

.

~

a.

We h a v e

v~ 8 1 s.t. HVv(~ HVv

So there is some

is u n b o u n d e d in

v = m a x Sa).

IBvlv~ S 1

countable.

Since

ly have

Bv~B~

F =

we h a v e

has an u n c o u n t a b l e branch.

v ~ v s.t.

is a

tree w h i c h c o n t a i n s no A r o n s z a j n subtree.

a_k,

then

K=k ÷

we can s t r e n g t h e n

(*) to~

... +

Actually,

Jensen has shown a long time ago that for

strengthen V~

~-5

morass..

(*) to:

if

~v,~g A ( f v ( ~ ) = 0

A C ~ + , IAl=k, and

there is some

fv(~)=1).

~z=-k 8

f(~(~,i))

= ~

v, v.

if Then

= ~(~f(i))

f*

is u n i q u e l y

So, by slight abuse of notation,

V,v.

we do not d i s t i n g u i s h

Note that

So

~v.

f

and

We also set

We clearly have

f(k~) = k v for

f*.

i(_k~v .

and

given

f(~*)=v*.

48 We now introduce another notation. Let

f : ~ ~

~

and

f(~)s ~(~) --~ ~(~)

~

~+ s.t. ~(~) ~ ~(~).

by

f(~

= f~(~).

Set

~ = f(~).

Define

We have

(E 2) f(¥) , ¥ = = ~ Proof:

Since

~'¢v*

we get

~(~) & ~(~), f(

Let

v.

L =L~. , where

We set

f(a,x,v)

:= f"

49

Obviously, (E 4)

the characteristic

Let

f(a,x~)

: ~ ~

Then there is some we

property ~

and

of

f(a,x,v)

g : x ~

is

~ s.t. a u l X l ~ rng(g).

h : ~ ~-~ ~ s.t. f(a,x,v) = g-h.

In addition,

IVI~I~I+~.

have

We now come to the main p r o p e r t y ion. Let

4~(Ir)JT(p>

iff there is some (E 5)

Let

where

lim(p)

By

Let

We say that

for all





conver~es

l'(P. be a convergent

R(T) = f ( ~ ( T ) ) ( Y < P ) .

Then

-~-chain

(~(T)IY

is

-(-chain.

(E 2)



is o b v i o u s .

and ~yy, (y) = y'. N o w d e f i n e

~ y

y 6 ~yT, (X)

=

is n o r m a l

and < ~ y Normality

~ y = id

yy'

X 6 U7 (4) Uy

f~ ~,,

, Uy >

is a m e n a b l e

To p r o v e

y < y'

.

amenability

set U n = U

•

T

It s u f f i c e s

to s h o w t h a t U n 6 ~ But this T Y" O Z - d e f i n a b l e from ~yy, and U nY 6 0 ~ y ÷ .

Since

that z yy'

~

7,

: < ~

N O W set

< < o Z y , Uy >

Y

is i m m e d i a t e

since U n is Y

n = U ny, of U n is u n i f o r m we get ~yy, (Uy) Y is c o f i n a l we have:

the d e f i n i t i o n

So o b s e r v i n g

(5)

N ~n Y

, Uy>

,

T* = sup I ,

, < ~yy.

, ,

U

, >

I* = I U {y*}. C l e a r l y , O~y, ~ O ~

[ y £ I >

I Y 6 I > , < Zyy,

since ef(IIl)

~ and:

62

X 6Uy~

Our

next

.

aim

~ X 6 ~y

is to s h o w

(7) ? (y) D

and

that

M Y { ~n

3 y< y*

I -yc_X

(gZy 6 M Y

for

. We n e e d

first:

y 6 I ,n

for (~.

NY .

the p r o o f

such

~yy, (~y)

= ~y, .

fl j U y ~ J~Y ~+i Hence

Set

there

methods

is a

Z l(NY)

since

it is

'

N Y is a m o u s e ,

be a r e c u r s i v e

By s t a n d a r d

that

of L e m m a 1.3. U O~y 6 J~+l ¥ "

enumeration (cf.

the

of the proof

of

first

63

Lemma and C~y

1.2),

if ~

By

one

v 6X i Y

, ~

~i(~'~)

the

for

Now

I'

set

f o r C Tt,

~

=

' (]rAy b

< ~>

X i 6 N Y such Y

that

Xi £ U Y Y

, < ~ >

6 [ X i - ~ in Y

, then

~ i ( ~' T%) "

choice

i i~< ~ X Y*

is

O~y~

we

"

have

Then

~

OZ.

a mouse.

So

Before is

and

choose

Zyy,(X$)

=

X y, i

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

y < y'

since

N Y~

canonically

< v

canonical

y 6 x y, i

since

may

But

I'

and was

I' 6 K is

proceeding

a natural

I ~I'

further

prewellordering

is

defined

the

let

of

I'

the

set

us

good

for

OZy~

from

N Y*

and

we

were

remind

mice

is

hence

N Y*

£ K,

looking

for.

reader

that

the

which

,

defined

there

as

follows

M

It

is

~ N

shown

iff

in

M 8 6 N8

[4

] that

where

~

@ >M,N

restricted

is

to

regular.

the

core

mice

is

a well-

ordering. We

often

use

Let ~

(¥)

M

D

Our = K.

(cf.

the

HM

= KM


~,T

is

1

Then be

regular.

of

if an

which

It

M

is

is

satisfy

easily

seen

critical,

iterative

of

M

then at

M,

. Hence

model

models

H M< 6 M .

that,

if M. '

W

this

iff

M.

HM < H i <

Let

all

[ 4])

w.

a structure

w1-Erd~s

that

and

application

conjecture

show

immediate.

following

there

proved

is

< is



Then

a model

r be

in w h i c h

proposed

Let

where

1

1

hand

and

~ = w2, ~'

= ~.

starting Kunen

from

showed

existence

of O #.

We

strengthen

in S i l v e r ' s

result

cannot

be

weakened.

Theorem

1.7:

Assume

~-Erd~s

in K.

Proof:

Let

A6~(~)

a good

set

of

fice,

Chang's

N K

and

conjecture.

set

indiscernibles

since

there

will

then

.

K < K ~+" P first By

two

un-

Chang's

P e n~'

Then

is c o u n t a b l e

and

con-

6B

6 ~'.

L e t b':

~-~

~

where ~

a n d b' (~) = ~ .

Our

choice

b' (e) = i s

U is

rug T ~ and g ~

a regularity

(y,~)-regular.

(~) < gT,(~)}.

sequence

Contradiction!

for

U and ~

< ~

for

~ 6 Y.

70

We model T s

return

to

of

+ V = K and

ZF-

: KT s

Case

the

main

Hence ~

I:

is

the

T 6 E

such

of

our

notion

KT c K S T

~ K +"

There

line

proof. of

We

mouse

note

is

that

absolute

KT

is

a

in

KT t

since

s

that

{s 6 C

IK T

J KT

} 6 U.

T Let

~

Set

W

be

be

tion

L e m m a

s Y i -> T

such

KT S

{aE

7s

and

Let

T 6 E,

. Then

~(~)

let

:+ 7~ < ~

2.4:

> T

Suppose -

M.. 1 that

KT

M

= M ~,

2.5: WN C

for

L,

a

since

are

IT 6 r n g

s

since

mouse

KT N s

7~ < ~

There T

K ~

AM, cy.

at

C

set

since

iterate

WN

and

otherwise

ME K { , ~

, s6

all

N

since

for

for

have

a6

the

Hence

and

Bi

7

S

T £ rng

KT ~

at

a

Let

M s with

if

T > T.

a c-~ ,

itera-

M is

= 7~}

, such ~

T: S

KT ~

6 >- ~ , y .

Let

N =

the

large

Let

~ < T

Since

since

contradicts

S

~T

a N f M i . Hence

and

T

of

.

= y s.

M 6K T

arbitrarily

~T S

and

~ < ~

a 6~(~)

- This

iterate

M exists.

N6

K

i-th

:+ i < s

~ l

not

We

M~ be 1

n K TmM ~ - -

then

Lemma

at

Then

N ~ M

~-th

that

7~-

AM, 6 Nc But

}. T h e n for each ~ £ W there is a core mouse s the ~-th iterate of M s is a mouse at a 7 < T . Let s

point

follows Then

such.

J K~

a mouse

Proof: a6

least

{~IK [ ~

Ms ~ K~ Ms

the

M ~ N.

KT

is

a

unique our

T 6 E

But

choice

such

of

K < +t

core

it

(N).

then

model

core

that

mouse

of

ZF-. with

an

M.

that

E U.

+ Proof: fine

Let T ~ 6 E,

~ < K f

. We

6 sT

show ~+

by

that

there

induction

is on

T > ~ ~ < K

with as

-

T~

=

the

least

{a6 WN CTI

T

such

that

T > T,~,

f~ (s) < T s } 6 U;

SUP ~ sup

{T

least

I~ < { a n d

i such

~6 C

that

. T

Let

T = sup

T ~ and

= {e6

C I

sup

T h e n C is c l o s e d Z

= {~6

T~ < f f

unbounded

= sum

{~16

(~) h a s

the

On>

NOW

is

Lemma

Set

2.6: that

Let

we

Z 6 U such

(a) T

~ E Z set

the y~

are

X6~(y~) This by

Lemma

We may

Set

Now

;f

let

This

some

shows

T = sum ~
_ ~a} 6 U. set

Xn

:

Then

Z =

{~ 6 Y N

and

,

fn* = ~ ( f

) . Then

~6

Xn

X* = z n

*

=

hence

pick

~

W-TIT 6 rna~ ~

{~If ~ (~) 6 f ~ ( v ) } , n+1 n

N ~ ~

where

N6

K=+.

T ~ T such

--

and

~(Xn )

T~_> @~} 6 U.

Let

and

*

*

{~I fn+1 (~) 6 f n ( V ) } ,

But

.

~

{~[T

that

~:

hence

*

fn+1(~)6

fn(~)

Contradiction!

Case In

2.2:

this

case

we

theorem

of

Un

normal

K

is

+

(K+) K =

K

actually

[ 3 ] it in

show

suffices K.

This

that

to is

U N L[U]

show

that

equivalent

is

is a m e n a b l e K the following

a

and

+ Claim:

There

are

arbitrarily

large

T <
.

+ Let For W

T

y < K

the =

moment,

{~ 6 C vT :

We

. We

first

~

shall let

PK T = K } 6 U. ~ T~

{X6~(a)nK

see

that

T 6 E be

T

there

arbitrary.

For

~ 6 W

is

T > y

Recall

satisfying

the

claim.

that

define:

I~(X)}.

show

WT = { ~ 6 WTIVT6 6~K} To

show

this

pick

T 6 E,

U. T ~ T.

Then

Y =

{6 6 W - I T 6 r n g

T}

6 U.

Let

a6

T

Let A

A

= o ~ U u0 ~M U u o

= ~

0 = ~ ~

z ~eq~

:]~s

= l~

~ON

£IIeU~ 7 ~M ~ M =

~

z0

pu~

:%~s

'eAT~ISUel]

ST

i0

alaq~ ,D9 ~

"~IOa

IeTOnXo

~q~

£eId

~

s~an~onx~s

~q~

'/ooxd

l~ ~

:1~

1o

'Z ~ {0} 9 i ao 7

~u~s~xd

~q%

uI

o~ 9 x

"g% 3 ]eq~

qons

co> I

sT

~a~N%

pu~

o~9

u~qm

~ ~sooq0"

ST

qons

~)q

pu~

x ~q~ 96A

1

n3 ql~>

A ~]!uT7 ~)

dns

1

( nuo)dns

e

~eq~

n3

X)

o>

" {~n 3 (~) ~0

•~ 9 { ~ =

I D3

~]

{g ~ [g > @ ( i ~ ) q } l -~no~



e UT

~IqeuTl~p-

= ~ u~q& = X 6"g

(g~)q}

ox~q~ %~S

'~>

oS

M

x u~q~

l~ U

~eq~

"~ uI

"s~Inu~xo7

{(x) T~

70

~ou~ H

= x qons x ~q

II~qs sT

~ ~

~AIS

I<X'T>}

:q

H

~.eq

= g~

:7oo~d

:OL'~

,D

~M

0 uaq&

uoI~ex~mnu~

[V]@q3x

0 ~ eq

H9

~oqs

papunoqun

~q;

~0 p u e O

~Iq~uIT~P-

]eq~

• ~D0

0 s!

" ~1 D ~-X

"X3 ~ ~q

eurmaq £ q

= 0 ~S ~ ~o7

"uo!%o~fIq

l

"F/3

'~umi~q

ZL

78

Lemma

2.11:

{sIT

Let

W 6 U,

f 6 ~6

cf(~) +.

Then

there

is T 6 E s u c h

that

> - f(e) } 6 U .

Proof:

Suppose

set

= {~ 6 W f l C ' I T ~ ~ < f ( ~ ) }

YT

not.

(b) (c)

above,

the

K many

predecessors

For

gT

~ 6W

are

let h and

define

pairwise

(mod U)

g

inject

f(a)

into

gT 6 6 ~ T

distinct

mod

(6 < T~,T 6 and v Then

= v

We

now

prove

C

T



we

(?B~) < B) •

then

,

get:

6 [C] n such

that

~ .

Let

f. 6 W, 1

f• :

~ , B , O n R M such iteration

~

Let

some

n. For

= ~o,~+B

(f)

Now define

(x 6 H)

that

only and

an

.

iterable

L~[M]

premouse

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

set C = {Kili < ~}.

~ = ~M,a,B

n ~] , < ~ > and

f,

6 [C n 7] <w,

6 W such

1

that

f.

1

:

K

n

~
6 K such

that M,a,6

are

countable

in

is c o n s i s t e n t .

<M,~,B>

= b-l(e),

there

foregoing

is < M , e , 8 > 6 K + such

There

M , ~ , B 6 X. L e t b:

But M,~,8

of the

formula

be as in the

H~-~X

6 = b-l(6).

are

countable

is n o t h i n g

<M,~,8> 6 H L[Vo]= K0

where

lemma.

Let

H is t r a n s i t i v e .

Then

H = H~,~,~

in K +. We

claim

to be proved.

Otherwise

K

argument

. The

X ~ H,

same

~ = w in K + such

Set M = b - l ( M ) ,

and~,~,~ that

that

is c o n s i s t e n t .

< M , ~ , B > 6 K.

If K = K +,

K + = L [ V o] and shows

that M , e , 8

are

countable

K0

in K.

We

are n o w

and ~ = ~ M , ~ , 8 theorem part L~[a]

ready be as

to f i n i s h

in C o r o l l a r y

let 0~6 K be a m o d e l

of O i i s b A(a).

the p r o o f

transitive. It s u f f i c e s

Let

of ~ . a be

to show:

3.12. We m a y

of T h e o r e m By

the

assume

3.1.

Let M,~,8

Barwise compactness that

the ( ~ - i n t e r p r e t a t i o n

the w e l l

founded

of ~. T h e n

87

Claim:

L[a] ~ A(a)

Let ~' be the Z F - l a n g u a g e w i t h the c o n s t a n t ~ and ordinal c o n s t a n t s (~ £ On). Define a class S of ~ ' - s e n t e n c e s • Let 6 [C] n, fi 6 W, fl: Kn

~


6 [C N ~ ] n ~ < ~ > ~ < T > and L~[a] b ~(~(~)))

I n d i s c e r n i b i l i t y a r g u m e n t s show that this is a c o r r e c t d e f i n i t i o n and that (i) S is a consistent, (2)

r~(~)q 6

(3)

rBx ~(x) I 6 S

S

iff

d e d u c t i v e l y closed class of sentences

L~[a] b ~(~) iff

Bt £ T

where T is the class of (4)

rBx 6 On ~(x) I 6 S

Now l e t ~ b e

iff

for ~
)

of

algebra

(F(~) ,e,

otherwise,

have

show

T (b i

smallest

if

B has

to

~
to ( 2 a ° s + 1 )

< ~

with

, and

result

such

. Now by

choice

of

prin-

(P%0,z(~) l d >_ r =

. Thus,

below

all

~ ( x , y I .... ,yt ) 6 ~o set

By

existence

with

bounded M =

~

definable will

subsets

remain

can

now

be

so

that:

established:

2.3

Theorem:

(i)

For

all

finite

If

M ~

PA

(eiJi

sets

, M ~

[a,b]

e. 2 l


n

a < e 1• < b

< max(a,c)),

(I)

F

'

i+I

< max(a

¥x

, then

is

n 6 IN

there

is

a sequence

, satisfying

'

~0(x,y I . .• . . y t ) 6 M ~

there

c)

'

? , i°


n ~

thus

f(m)

its

~ g(m)) . We

S c

T

(the

so

graph

is

to

show

of

3.1),

< g(a)

that < ci(

c,

> g(m)) .

want

T

, a 6 IN , b = f(a)

total

~ g(c)

e i the

definition

of

ci, the

~ x)

i < k}

i K k, graph

"g(y)

of

g

, ~ x"

.

system

of

~ PA.

is

Now

~

function,

S U {c ~ c o } U { V x

the

~ T')

Vm

> n

a ~ d

arithmetical

M'

I'

dicting

total.

~ c O } U { Vx

a model

IN

B m

PA) :

g:IN

a recursive

6 IN

in

in

If

e o , . . . , e k 6 [a,b]

(IN , a , e ° .... ,e k)

abbreviating

:

B n £

be

Z1-definable g

(provability

< ci(~

this, If w e

M'

then had

~ g(c)

It-formula

g(c)

~ x) ]i 6 IN}

the

c

I' ~ d

by

= g(c) ^ d

and

M' l ~

< c

I ' ~Z

d

M'

consistent.

2.3,(ii),(3) , then

for

1

is

we

some

As

we

determine

had

(contra-

i 6 IN might

an

, since

have

added

o Th(IN ) t o 3.4

T'

, we

Corollary

obtain

(Paris,

Harrington)

: PA

U Th

(IN)

~

~

,

nI noting the

that

recursive

total

proof

there

is

2.3(ii) I ~

just

of

3.4:

M ~

for

, a 6 I < b

had

(Xl-formulas

I ~e

contradicts

M

Let b

£ M

models . If

~

of

were end

the minimality

of

f

(a+1) a

M ~

b'

(a+1) a a

there

is

with

. Hence

remark

any

~

extensions) b

the

IN . S i n c e

[a,b]

6 I

. By exceeds

, a 6 M ~

arithmetic)

to

I ~ Th 1 ( I N ) .

up

that

true

there

going

Th(IN)

so

of

totality

~b[O,b]

too.

PA

which

a ~

the

function,

a smallest (only

states

function

recursive

Second

we

a

I =

to

1.4

provably

N ~

a

. Thus

an

I c

[a,b']

M ~

[a,b']

I ~

~ a

~

, by

e ~

M

,

a (a+1) a,

(a+1) a

, and

by

'

167

Remark:

In

finable iff

Paris

function

there

graph

[2]

of

Y:MxM ~ M

is a n

I c M ~e

Y

a

countable

has M ~ PA

Y(a,b)

, then

indicator

M ~ PA

the

in PA,

and

for a l l

, which

which

sentence

the We

can

just

countable

be proved

MI,M 2 ~ PA

one has

MI ~

M3 ~e M2

. This

of

PA

system

formula

~

initial

segment

account,

in

in

by

of

a turns and D.

3.5 C o r o l l a r y : segments

{X c ~

o u t to b e Jensen

There

a nice

asked

are models

(up to i s o m o r p h i s m )

Let

determined

by

language

Define

M 2'

M I' {c i li 6 ~ of P A

S S y ( M I) = SSy(M2) , h e n c e

if

this

MI

part

of t h e

Note

that

for

any in

[5]

3 b 6 M I a z

, cf.

of o n e

. Since

Y(x,y)

for:

T' b e as in the p r o o f

countable.

in a n y

satisfies

in M } , t h e n

S S y ( M I) = S S y ( M 2)

an i n d i c a t o r

is an i n d i c a t o r

JX =

M 2 . Furthermore,

if t h e

for unprovability.

shows

>

that

but

by a r e s u l t

[4] t h a t

M I , is

parameters

, then

Ehrenfeucht

the

of

with

MI,M 2 ~ PA

M I'

also

: Y(a,b)

. He shows

, then with

c a n be r e p l a c e d

showed

a,b 6 M

, Thz1 (M I) ~ Th(M2) , S S y ( M I) = S S y ( M 2 ) ,

standard

Proof:

stated

M ~ PA

M I ~ M2,

H. F r i e d m a n

all

requiring

this

P A to be a d e -

V z H y

(c+I) c} c

If

I c M --e

M ~

defines Vx

latter

exploited

easily

for

for

, a 6 I < b

reading:

indicator

that

= max{cl[a,b]~

indicator-property any

such

, I ~ PA

indicator-property.

function

an i n d i c a t o r

Z1-definition

true but unprovable of the

defines

are mutually that

C--e MI' 2

U T'

substructure of

ThzI(MI)

are m u t u a l l y

M

M I ~ a , M 2 ~ ~ s.

~ Th(~)

the r e d u c t s

initial

of

, M I'

M I' , M 2' = ThzI(M2)

initial

to and

segments,

168

and

by

M I' l= T'

As theorem

a second that

PA

i~

~

3.6

Lemma:

provable

The

PA

is n o t

All

sentences

provable,

Let

us

but

for

now

this for

these.

Cf.

one

be

so

Theorem that

a

even

V x3

y[x,y]

(~+I)[

remark the

to

v xV

Ryll-Nardzewski's First

we

n

~

(n+1) n

being

provided

give

to

, n 6

IN , a r e

one

given

in

Wilkie)

, even

axiomatizable.

Remark:

By

using

3.6

is

[2].

fact,

3.7

:

For

S U Th

all

(~)

restricted

the

"from

i~ a

provable,

do

far

is

Ramsey

outside"

on

.

theorem

by

finite

are

Infinite

will

which

by

Paris

and

than

the

simpler

a corollamyof

S ~

PA

there

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

2.3,(i)

is PA

n 6 is

not

n

truth-predicates,

induction

of

a proof In

El-sentences,

(Z)y

following

~I

finitely

~

induction

here

the

true

.

version

But

proved,

with

1.4

using

topic.

, being

y H z[x,y]

a definable

this

once

S I~ s

n

~

superscripts,

(Paris,

PA with

reprove

axiomatizable.

=

n

n

to:

can

e-incompleteness:

the

needs

for

game-theoretic

3.7

use

fixed [6]

can

an

H y[O,y]

that

2 . 3 , (i) Wilkie

we

.

mention

Theorem

of

sentences

are

s .

finitely

form

PA

~

application,

the

in

Proof:

, M2 b

the to

result

can

E -formulas

easily

for

be

fixed

extended r 6

r

does

not

Proof:

imply

Let

a finite

meters)

S

set

induction

all

be £

axioms in

such

i_~f (M, (ei) i E ~

the

a

n

a finite of

Eo-fOrmulas

contained a way ) b

subset

in

of

PA.

By

the

~(x,Yl,...,y S

(which

may

be

proof

of

t) m a t c h e s

2.3,(ii) with

assumed

without

(where

TF

the para-

that

PA U {c~

< ci+1 Hi 6 ~ }

U TF

is

,(3)

the

.

169

last

group

of

then

the

initial

Set

M b

Th(~

according [a,b]

~

to

% ~

the

. By

Remark:

the

As

Paris

model

models than

~

PA

(the

T

of

Proof: are

for

is

a model

Let

S

(eili

6 ~

of

so

(M,eo,...,ek) ~

the

in

number

in

F )

) models

M

S

for

Theorem:

There

not

even

with

such

are

.

S

, so

that

, hence (~)

nl [a,b']

, but

not

s

, n

~

(n+1) n n

would

means there

that are

whereas

not

3.8

segments

in models

of

initial

will

segments

imply

modelling

that

PA

in

, other

3.9, from

PA

nonexistence

paragraph

M ~ PRA

defined

, so in

model by

of

, as

same

~

ce M

argument

as

recursive

that

arithmetic)

(M' (ei)i encoded

of

an

PRA of

. By

( ~ , e o , . . . , e k)

recursive

a sequence

a set

T

of

2

(primitive

, ei 6 M

subset

S

on

~

has

b T

by

an

initial

seg-

Eo-indiscernibles.

3.2

and

S

. Hence,

. Using for

6 ~

PRA

~

~

b

a

there

for tro(~,x)

2 . 3 , (ii) ,(I) ,(2)

~ ~(x) yields

) as d e s i r e d .

3.9

,

with

, a ~ ei < b

U Th

I b

) contained

that ~

6 ~

TF

3.7

nonstandard

a finite

6 ~

element

S

6 ~

a natural

ei,i

initial

6 ~ )

6 ~

any

(eili

formulas

.

element,

extract

(eili

the

all.

(eili

which

be

6 ~}U

[2],

nonstandard

. Thus

Eo-fOrmulas

b

application

M

eo,...,e k

of

at

of

M ~ PRA,

< Ci+lii

any

we

3.1),

be

are

to

by

smallest

there

It-induction

any

a sequence

n

in g e n e r a l

models,

is

the

,

} satisfies i b' 6 I with

above

there

ment

6 M

~

induction

third

For

determined

in

present

Lemma:

M

restricted

pointSout

, are

the

3.1

< e

of

of

nonstandard

element

d

# ~

For

b

minimality

prefix-restricted which

of

2 P A U {c i

) ~

in

2.3, (i)

existence

contradict

T

, a 6 M ~

2.3, (i),

{d 6 Mi 3 i 6 ~

since

3.8

of

segment

), M

(n+1)~

(M' (ei)i 6 ~ I =

axioms

no

recursive

recursive addition.

nonstandard

models

of

PRA

,

170

Proof:

Let

M b PRA,

(M'((e)i)i I 6 ~

6~

, the

) b

T

proof

of

(*)

H x I Vx2...

Set

A =

of

Using

Hence PRA be

we

an

2.3,(ii)

G~del

take

initial ,(3)

numbers

one

e 6 M

segment

shows

a 6 M

as

the

but

on

n)

of

realizes

containing

induction

get

taken

the

3.8,

H x n ~(Xl,...,x

tr °

p(z) by

. By

so

I

that

that

given

for

any

by

(e) i

open

~

,

in

the

PA

set

formula

# ~

. For

of

language I b

M

iff

9(y)

that

the

A =

formula

{n

6 EO

there

3 x1 0

fixed

~ a

, tr

n

the

variables,

(3 e tr

E -truth-predicate n

gn:

(c,d,e)

~

~

~

g i v e n by (n)

[x,y]

~

(z+1) z

gn(a) be

tr

-

g b

the

with

3 e < b

n

relation

~

t h e n becomes

formula

saying:

gn(a)

(c,d,e)))

Hn-definable. ~

required

gn(min(X)) formula

a

n =

< Card(X)" ~

=

Let plus

Z

homogeneous

Ix,y]

E n

. The

(z+1) z

Z

"the

for

n

[x,y]

sets

X

(instead

of

z

(z+1) z)

can the

be

chosen

special

such

case

gn

that = id

in

the

. Set

(n) VxV

z 3y[x,y]

~

(z+1) z z

i

n

(n) is

then

a

Hn+2-sentence

, [x,y]

have

(z+1) z is z

a

H -formula. n

172

The paragraph

second

proof

3 generalizes

4.2 L e m m a : language

For

each

is an i n i t i a l

then

PA U Th

the

to

an

for

independence . First,

n 6 ~ :

of P A so t h a t

there

for

If

all

~

there

~(x,y,z)

a

given

is n o t h i n g

is a

M b PA:(if

substructure

(~)

of

new

in

En+1-formula

M k ~(a,b,c), I ~ PA,

I ~e M with

in

in the

c > ~ , then

I _ c M, n

a 6 I < b),

V x V z ~ y ~(x,y,z) .

~n+] Remark:

Such

indicator,

a

~

this

has

the m a i n

time for

property

of

~ -substructures n

a definition

which

of an

are m o d e l s

of

PA

.

(n) NOW we

show

that

[x,y]

~

(z+1)zz

indeed

defines

such

a

Z -indicator: n

(n) 4.3 L e m m a :

If

then

is an

there

M ~ PA,

a,b

I ~ PA,

£

M,

c 6 M \ ~

I --eC M,

I ~

, n > O, M ~

M

with

[a,b]

~

a 6 I

[a,b]

< b

~

c

(c+I) e,

.

n

(n) Proof:

[a,b]

sequence

c (c+I) c

~

Xo-indiscernibles

of

in 2.3, (ii).

Since

a homogeneous

set

[a,b] as

M

for

being

any

<eo,...,ei>

infinite

segment

in

of

6 ~ ),

partition g~(ei)

primitive

recursive

by

6 ~)

< b

, as

[[a,b]]

d

function,

condition

ei+1,

the

. By 2.3, (ii)

from

all

than

be

arises

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

for

I

a

described

, the

if w e w i s h .

Let

(eili

" So w e h a v e

sequence

is s m a l l e r

Zo-indiscernibility.

determined

this

on s o m e

< ei+1,

(c+I) c c

a ~ ei

2.3,(ii)

(sequence-number)

by

M

some

yields

dominates

i 6 ~

(eili

b y the p r o o ~

(~)* (c+1)Cc

gn

implies

all

ei

initial only

I ~

M n

remains

to b e p r o v e d .

We

show

I ~

M, m

~ n, b y

induction

on

m

.

~ 6 ~m+1

"

Zm By

I --eC M w e h a v e

already

I ~ Z M.

Let

m

< n,

I ~

O

~ =

H x ~ ( y I ..... Y t , X ) ,

I ~NmM .

So

let

Equivalently replacing 0

we only

sets

some

and b y i n d u c t i o n

to r e n d e r

For

for

entails

3x the

this

<e i + 2 t r n ( . . . ) , t same

b

holds

in

I .

we need

n in the p r o o f

from

of

an i n f i n i t e

< Card(X)

for

any

~

~ s

set,

the

so it is n o t

fixed

function

f:~.

obtain

n > O

an

Moreover,

if

is a t r u e M ~

PA,

Nn+2-sentence

M e

~

,

there

is

independent an

~n+1 I --eC M,

I

N PA,

so

that

I

~

M

and

I

~

~

~

n

n

Remark: case

i-6

For

the f i r s t

M ~ Th(~

Corollary:

For

of i n i t i a l

complete

extensions

stronger

all

Ehrenfeucht result

completions that

for

with

regard

of

n = O

of

4.5,4.3

is o n l y

needed

in the

special

)

infinity

Remark:

part

that PA 4.6

n 6 ~

any nonstandard

Zn-SUbStructures of

PA

and D.

satisfying

of

Th(~)

pairwise

has

an

different

.

Jensen

proved

a nonstandard satisfied

by

immediately

to the r e s u l t

model

model

in

[3] f o r

of P A h a s

n = 0 2

~o

its i n i t i a l

substructures.

generalizes

to m o d e l s

by Gaifman

quoted

in the

of

remark

the distinct Note PA

U Th

following

(~), 3.4.

174

References

[i]

J. Paris,

L. Harrington:

Ar i t h m e t i c ,

A Mathematical

in: H a n d b o o k

Incompleteness

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

Logic,

ed.

in Peano

Jon Barwise,

1133-1142. [2]

J. Paris:

Some

The Journal

[3]

Proc.

1971

of the

Lecture

H. Gaifman: Proc.

Lecture

J. F. Knight: The Journal

[7]

A Note

J.C.

of Number S6r.

Types

Astr.,

Bull.

vol.

43, No.

1976,

Summer

in Math.,

School vol.

vol.

40,

Model

XII,

of A r i t h m e t i c , London

255,

1975,

1970,

128-144.

Models

of A r i t h m e t i c ,

317-320.

Polonaise

No.

2,

Logic,

539-573.

for a Free V a r i a b l e

l'acad&mie vol.

in:

Logic,

in U n c o u n t a b l e vol.

arithmetics,

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

337,

in M a t h e m a t i c s ,

de

in e l e m e n t a r y

and Submodels

Logic,

725-731.

223-245.

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

Phys.,

4, 1978,

of Set Theories,

A Nonstandard

Theory,

Math.,

Models

Omitted

of S y m b o l i c

Sheperdson:

XCII,

on Models

Notes

for Peano A r i t h m e t i c ,

Some p r o b l e m s

Cambridge Notes,

of the C o n f e r e n c e

Sp r i n g e r [6]

Mathematical, Countable

Results

Logic,

D. Jensen:

H. Friedman:

Sp r i n g e r

[5]

of S y m b o l i c

A. E h r e n f e u c h t , Fundamenta

[4]

Independence

1964,

Fragment

des Sciences, 79-86.