On Logical Sentences in PA

LOGIC COLLOQLIIUM '82 G. Lolli, G. Long0 and A . Marcia (editors) 0 Elsevier Science Publishers B. V. (North-Holland), 1984

145

ON LOGICAL SENTENCES IN PA

Saharon Shelah Department of Mathematics The Hebrew University, Jerusalem, Israel Department of Mathematics Ohio S t a t e University, Columbus, Ohio, USA I n s t i t u t e ofAdvanced Studies The Hebrew University, Jerusalem, Israel Department of Mathematics University of California, Berkeley, Calif

.,

USA

Contents 5

1 . A representation of PH

5

2. On al-comprehension axiom 1 [We suggest a solution t o ? / n i - C A o

g 3 . A true 5

IT:

= Paris-Harrington/PAl

sentence in PA, n o t provable in PA.

4. On theories with incomparable consistency strength. [We show how t o produce such reasonable theories. We also draw the reader's attention t o reasonable examples where theconsistency strength areequal b u t t h e r e i s no interpretation].

ii 1 . A representation o f PH

We give in t h i s section a representation of Paris Harrington [PHI r e s u l t s , in a way which will be helpful 1.1. Definition: An

F

E

1)

M9"

2)

M

L,

later.

(L,n)-model i s a sequence

i s an L-model

I' Me

9"

5

n > such t h a t :

except t h a t functions a r e p a r t i a l (so M9" # 0 ) .

i s a submodel of M,+1

F

M =
M

e+ 1 ( f o r a . + l

5

n)

b u t f o r every function symbol

i s a t o t a l function (with range 5

I am very grateful t o Leon Henkin f o r saving the manuscript and t o Annalisa MarcJa f o r taking care of i t s typing. The author would l i k e t o t h a n k the NSF and the United States-Israel Binational Science Foundation f o r p a r t i a l l y supporting t h i s research.

S. SHELAH

146 1.1 .A. N o t a t i o n : Let

dp(cp)

-M =
a

5

5

-

n>,

1

v and 1 $ 1

be t h e q u a n t i f i e r depth o f

@(X)

1.2. D e f i n i t i o n : F o r a f o r m u l a

E

L

only,and o n l y atomic terms used,w.l.o.g.) and an (L,n)-model

(D

o f l e n g t h > dp(v)

5

M-ri~.il = .

n>.

i s t h e l e n g t h o f J,

.

( w i t h n e g a t i o n i n f r o n t o f atomic formulas

a

and a sequence (i.e.

n

t

dp(cp))

a(;)

M,,

from

= t(i),

we d e f i n e when

by i n d u c t i o n on t h e q u a n t i f i e r d e p t h o f cp :

fiI=cp[a] if

fi

: a.

M. =
atomic

: as u s u a l ( n o t e t h a t o n l y a t o m i c terms were used, and

t h e l e n g t h of

fi

i s > 0 = dep(cp),

hence we can compute

t h e terms and check t h e s a t i s f a c t i o n o f t h e r e l a t i o n in

1.3.

Claim: 1 )

a)

h a s a dp(+)-model

IJJ

I f a sentence

b) u, has an n-model

2)

J,

3) I f

fi

M,)

$

has an n-model t h e n

( i n f a c t an m-model whenever

5

m

5

n)

satisfying

has a model i f f i t has an n-model f o r e v e r y $

dp($)

has no n-model t h e n t h e r e i s a p r o o f o f

n i J,

. o f length

I+ln

2I'I

( t h e l e n g t h o f a p r o o f i s t h e number o f symbols i n i t ) .

4) If

J,

has an n-model t h e n t h e r e i s no c u t

free proof o f 1 $

o f length

5

n.

Remark. We have n o t t r i e d t o m i n i m i z e t h e f u n c t i o n and numbers n o r we s h a l l do i t elsewhere i n t h i s a r t i c l e .

Proof: 1) a) I f of

J,

i s an n-model o f

u, t h e n

fi[oydp(J,)li s

a

dp($)-model

(check t h e d e f i n i t i o n ) .

2 ) Suppose

i s an n-model o f J , . We d e f i n e by i n d u c t i o n on

a.

5

n,

A,

5 N,

On Logical Sentences in PA

147

such t h a t :

a ) l e t A, in

L

b) i f

At

appears in J , ) i s defined, 3 y (p(y,x) i s a subformula of

then f o r some b b

every non logical symbol

consist of a l l individual constants (w.1.o.g.

E

N2+1,

So clearly

IA,,

(we can forget

A

i s theunionof

and

E

ice’lsn’i= coCb,al;

“IA,I

a.

a

ji,

we demand t h a t there i s such

with a l l such b ’ s .

s Cnr.of individual constants in ~i I

+

”

,[Q,,tll I= ( 3 Y ) m ( Y i ) ,

c A e’

as the multiplication by

5

IJ,~

and

i s more than needed). As

models are non empty

Now l e t MI = Ne ? A

e’

and we can prove t h a t f o r every subformula

e(;)

of

JI, a c A R ’

n-a.

i C e s n l I= e [ a l

dp(e(i)) :

t

iff

i C e , n l I=

eG1

Me = M

, so

(just like Tarski Vaught c r i t e r i o n ) .

2) If

M

i s a model of

5

then l e t ( f o r a l l a. )

J,,

n> i s an n-model of n If f o r every n

J,. 5

n > i s an n-model of

which i s an w-model of J,

.

Easily

3 ) Also immediate.

=

X ; ~ - ~ , ~

[ $ ] ( m i ) dp(’)

n-model of

J,,

X ,... ; , 3x nl , . . . , x n

and

e

J,,

Me

by compactness there i s

i s a model of

n

[There i s a quite short proof (of length <

3xi ,...,

easily

says t h a t

dn-1

e

( ~ i I l * l) ,

Now

J,

where m,, = 141, m

< { x im : m

e i s quantifier f r e e .

showing

5

m i ’ i 5 j};

e can be refuted

j

J,

.

1- 3 xX ... xm,-1 o i+i 5

by a

=

n>

i s an

truth table of

lJ,T 1.

size 2 1 ~ 1 1

4 ) Just l i k e the proof t h a t every model s a t i s f i e s any provable sentence.

* * * * *

148

S.

Let

PA

SHELAH

be Peano a r i t h m e t i c , and

o f t h e i n d u c t i o n scheme o f l e n g t h

PAk

are included ( b u t except t h e instances o f

k

5

be Peano a r i t h m e t i c when o n l y i n s t a n c e s

t h e i n d u c t i o n scheme t h e r e a r e o n l y f i n i t e l y many axioms, which a r e included,hence i s finite).

PAk

Let

PAPL

be l i k e

PA, b u t wetake o n l y t h e i n s t a n c e s o f t h e

i n d u c t i o n h y p o t h e s i s w i t h no parameters, and and

PAk.

I t i s known

(Friedman t h e s i s , I t h i n k )

s i s t e n t . It i s c l e a r t h a t the consistency o f n

2

k,

m o n o t o n i c i t y we can t a k e o n l y

=
.Q

5

n> where

If

rQ+l> r i ,

r Q> 1,

i.e. there i s

t h e axioms o f if

a r e equi-con-

n t k

,

PAFL has an n-model" ( b y

k-!=

r, < rl < r 2 <

... c

r

.

n

PA minus i n d u c t i o n h o l d s . So we want t h a t

fik ( 3 x ) c p ( x )

x < r o such t h a t

PAPL

i s equivalent t o " f o r every

t i o n and m u l t i p l i c a t i o n r e s t r i c t e d t o t h i s , and where

f o r any f o r m u l a cp(x),

PA,

PL

PApL? I t h i n k t h a t t h e obvious c h o i c e i s k r = {0,1,2, ..., r - 1 1 w i t h t h e u s u a l a d d i -

i s the set

M

that

PA

.

n = k)

What i s t h e most n a t u r a l n-model o f

R

PA

has an n-model "and t o " f o r e v e r y

PAk

i s the intersection o f

PA[L

cplxl

then A

R!=(3x)Cu(x)~(Vy)((~(y) + x s y ) l ,

(Vy)(cp(y)

+

x sy)

( s i s definable,

letting

r o= 2

o r can be added as a r e l a t i o n ) . T h i s suggests d e f i n i n g

..., r n )

Fw(rl,

= m i n { x < rl :
if F$ ( r 1 ,

. . .,r

r

: Q = O , n>i=cp(x) Q

F (rl,r3,..., rn+l) = Fcp(r2,r3,...,r cp

nti

otherwise Let value o f


is

+

n 0,

t

...

r be homogeneous f o r FC (where m > 2n). I f t h e c o n s t a n t m F i on i n c r e a s i n g sequences f r o m {r,, ...,rml i s 1 , t h e n n e c e s s a r i l y

ro<

<

..., r m )

m, so i f

m

: a. < m-nt2 > i s s t r i c t l y d e c r e a s i n g ( i n Q ) ; hence

i s l a r g e enough compare t o

r,,

t h e constant value o f

hence t h e i n s t a n c e s o f i n d u c t i o n h o l d i n t h e n-model

We can work w i t h a l l t h e f o r m u l a s o f l e n g t h s k,

F " ( r ,. . . ,rn t 2 ) = cp

So i f

t



0

if

F (rl ,...,rntl) 'i

1

say =

(Pi

: II = 0, n > . rQ (is i,,)and d e f i n e

Fcp.(rz 1

F&


,...,rn+z 1 f o r every

i

5

i,

otherwise

i s indiscernible f o r

F" and l a r g e enough
:isn>

i

149

On Logical Sentences in PA

PL k ' (Paris Harrington

i s an n-model o f PH

So

PA

partition

theorem)

i s enough t o p r o v e t h e c o n s i -

stency o f PA.

2. On t h e

Ti-comprehension axiom.

Simpson and Schmerl CSSl f o l l o w i n g M a c i n t y r e proved t h a t t h e c o n s i s t e n c y s t r e n g t h of PA a u g m e n t e d by t h e Ramsey q u a n t i f i e r ,

PA(QMM), and o f

1

nl-CAD

(see 2.2) a r e t h e same, i n f a c t t h e y a r e b i i n t e r p r e t a b l e . M a c i n t y r e ' s aim was t o s t r e n g t h e n P A so as t o e l i m i n a t e t h e well-known incompleteness f o r f i n i t e combina-

[MI and ISSI, i . e . p r e s e n t a

t o r i c s . Here we suggest an answer t o a q u e s t i o n f r o m finitary

combinatorial p r i n c i p l e capturing

IT:

a:-CA,,

based on end-homogeneity

i n s t e a d o f homogeneity (see t h e book CEHMRI). T h i s p r i n c i p l e was c o n s t r u c t e d from t h e p r o o f and n o t as u s u a l by m i n i t u a r i z i n g i n f i n i t a r y ones. I t was c l e a r t h a t i t s i n f i n i t a r y analogy i s " p a r a m e t r i z e d Galvin-Prikry"

so we have i t s c o n s i s t e n c y s t r e n g t h e q u i v a l e n t t o t h a t o f

ni-CA,,

.

T h i s n a t u r a l l y h i n t s t h a t t h e y a r e a c t u a l l y e q u i v a l e n t . Simpson takes i t on h i m s e l f t o prove t h i s and succeeds. T h i s work was done i n summer 1979, a f t e r h e a r i n g on t h e r e s u l t s o f Friedman I F 1 1 and j u s t a f t e r Simpson has e x p l a i n e d us [MI and [SSI (and b e f o r e Friedman and Simpson s t a r t e d i n v e s t i g a t i n g t h e q u a s i - o r d e r o f t r e e s ) , and we would l i k e t o thank Simpson f o r t h e c o n v e r s a t i o n . 2.1. N o t a t i o n : F o r a s e t

A

o f n a t u r a l numbers, we c a l l

i f i t i s a f u n c t i o n f r o m i n c r e a s i n g sequences f r o m

A

o f power k ) For B

2.2.

A,,

c

f, A

A

into

..., Am E n, ,...,A,-,,D)

=

H

Remark: n:-CA,

f

i s c o n s t a n t on

k

( o r subsets o f

B.

H a hereditary function f o r

H(p,A,

H(p,A,

2.2A.

o f length

A

as above

i s f-homogeneous i f

As

a k-colouring o f

{O,l}.

N o t a t i o n : We c a l l

variable).

A

f

0

,...,Am) f o r every

belongs t o Am

n

for

p < n,

m,

and i f i t i s z e r o t h e n

{O,ll, (i.e.

if

H

i s monotonic i n t h e l a s t

i s always l i k e t h i s we would n o t r e q u i r e h e r e d i t a r i t y e x p l i c i t e l y . i s a t h e o r y "speaking" on n a t u r a l numbers and r e a l s ( i . e

150

S.

SHELAH

sets o f n a t u r a l numbers) s a t i s f y i n g : i f

$(Y,x)

i s a formula, p o s s i b l y w i t h r e a l

I x : (3Y) $ ( Y , x ) l

parameters b u t a l l q u a n t i f i c a t i o n a r e o n n a t u r a l numbers, t h e n

( x v a r i e s on n a t u r a l numbers,

i s a real

Y

on r e a l s ) .

D e f i n i t i o n : We d e f i n e a combinatorial p r i n c i p l e which corresponds t o t h e e x i -

2.3.

stence o f "end homogeneous sequence" i n Erdos-Rado terminology: : For any f u n c t i o n

CPeh(n,k,e)

whose values are k - c o l o u r i n g o f

-homo for

when m

Am/p

c

1

r

... < ik

p < Min Ar-,,

5 &,

Am

is

i, <

5

-

and f u n c t i o n

n,

n such t h a t ( a ) (A,(

subsets o f

F, k-place, defined on subsets on

2

m s a

<

H(p,A,

,

or

Am

each

( c )
,...,Am_,,Am/p)

= 0

ii : ( p , i ) n A,

H, t h e r e are A,

and ( b )

Min A,

i

5

CPCh (n,k)

n = {O,..,,n-ll

subsets o f

there a r e A, 2 A, Am i s

i s H-end-homo,

,...,Am-l,Ar)

= H(p,A,

2.5.

Claim:

...IAt

i.e.

'k

)-

where

k, a.

f o r every

F d e f i n e d on sequences o f

whose range a r e k - c o l o u r i n g o f

A, 5 n such t h a t (a)

...,Am- 1 )-homogeneous

F(A,,

A,?

2

F(A i,,...,A.

p}.

means: f o r every f u n c t i o n

... 2 Ak,

2

and

e ) holds.

CPeh (n, k,

Definition:

2.4.

2 A,

is

Remark: So t h e combinatorial statement we are i n t e r e s t e d i n i s : f o r some n

n,

lAkl

and ( c ) 1

n 2

and f u n c t i o n

Min A k

H

and ( b ) each

i s H-end-homo.

The statements

V k

Ya 3 n

CPeh

(n,k,a)

Yk

3 n

CPLh

(n,k)

a r e equivalent. Proof: Immediate. 2.6.

Claim:

Suppose

PA

+ Vk3n

CPIeh (n,k)

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

is consis-

T:-CA,

tent. of

Proof: So t h e r e i s a non standard model non-standard

k

~ eand, choose

(? kcp&

n

E

PA

+

Yk 3 n CPBh

(n,k).

Choose a

such t h a t

(ny3k)

Now we s h a l l d e f i n e

F,

so l e t

shall define a (3k)-colouring o f n,

be a sequence o f subsets o f F(A).

n,

and we

151

On Logical Sentences in PA

r s n,

F o r t h i s f o r e v e r y n a t u r a l numbers

r = {0,1,2,3,..

universe

t o r ) ,individual

c o n s t a n t s 0,l

p l a c e r e l a t i o n ) . The language

and r e l a t i o n s
m,

a sentence o f l e n g t h

L depends on ,(A)

involving only

m, t m 2 . Now we d e f i n e

be t h e model w i t h

<

&(a))

(i.e. A, i s a one

only. many ways, choose one so t h a t

i < m2, i s n o t t o o l a r g e compare t o

,...,i 3 k - 1 ) :

F(A) (io,il

G ( a ) (io ,...,i2k-1 ) =

Ai,

(a.

a.

We can code t h e sentences o f t h i s language i n

if

ar =IfTrlJ\l

let

.,r-ll, f u n c t i o n s : a d d i t i o n and m u l t i p l i c a t i o n ( r e s t r i c t e d

i t i s zero

G(h) (ik ,...,i3k-1 )

and one o t h e r w i s e , where

G(1) L-sentece

i s the f i r s t (iO,...,izk-,) and

J,


Cil, ...,ori

[A1,Oci 0

CAI> k k- 1

1

We now w i l l d e f i n e a f u n c t i o n i.e.

we w i l l d e f i n e i t i s zero i f

whenever

p s i, which i s

H(p, A,

i*,io ,...,im E A +,.


H.

,...,A m+ 1)

...
..., Ai [ h ] > m

B

which i s an

*

:

e(-,

p codesa f i r s t o r d e r f o r m u l a

< p < B < i,
i, o r

m = dp(e) t 8

+

-, A,

,...,A,-,),

t "8 d e f i n e a t r e e and f o r some

x , y < i

D the i n t e r v a l

and

8

A+,,

X,Y

(x,y)

t * f o r every

x < Y

5

has a member below

t*

( i n the tree)" o t h e r w i s e t h e v a l u e i s one

A

(so i f As

m+ 1

i s empty t h e v a l u e i s zero, n o t e

C P I e h (n,3k)

such t h a t ( f o r each

holds, m)

Am

H

i s hereditary).

A, 2 A, 2 ... 2 Ak, subsets o f n , ...,Am- 1)-homogeneous, and

there are is

F(A,,

i s H-end-homo. Let f o r n

a.

a.

< k

= Min A,

Now we d e f i n e a model

. c%

which w i l l be a model o f

a;-CAo:

t h e n a t u r a l numbers

152 of

S. SHELAH

63 a r e

(remember

k

i s not standard).

A d d i t i o n , m u l t i p l i c a t i o n 0, 1, < The f a m i l y o f r e a l s

QR = {B 5

a r e i n h e r i t e d from

c.

( = s e t s o f n a t u r a l numbers) i s t h e f a m i l y

aN:

f o r some s t a n d a r d

il,

i s f i r s t order definable

B

( w i t h parameters) i n t h e model

(a3,,+,

X,

Al,...,Ail)}

ni-CA,.

The o n l y n o n - t r i v i a l p a r t i s So we have t o p r o v e t h a t

tP

(FO) where

aN:

E

63

i=

( 3 Y ) j,(Y,p,i)}

i s a f i n i t e sequence of s e t s f r o m

. I t i s w e l l known t h a t w.1.o.g.

e(-,-;

a tree

p,

i)

@I

G3 t j,CY,P,q,A1

9 . .

As

(F1) i.e. (F2)

u s i n g a parameter Am(*J+l

is

F(A,,

( V X9Y) [ x

we can r e p l a c e f o r some

Y v a r i e s on

61,.

by

m(*),

i s f i r s t order definable i n

Y

...,Am ( * ) A

~

)-homogeneous E A m( * ) + I

(aN,A, ,...,Am(*)+1 )

a E A m( * ) + I

there i s

Am(*)+17 Y an element below By t h e H-end-homo,

>

f o r some

in

(@"A1,

...,

cBN .

q(*)

Am(*)+1

The model

...,A

"
Y

A X

t h e branch has a member i n each i n t e r v a l o f

(remembering K6nig

order,

Y i s an i n f i n i t e branch i n

..Am]

Then f o r some t r u e l y f i n i t e

)

,

€aN.Suppose

t r u e l y f i n i t e m, q

m(*)

J,

"says" t h a t

(whose o r d e r i s i n c l u d e d i n t h e n a t u r a l o r d e r o f & N ) .

By t h e d e f i n i t i o n o f

A

j,

€aR a,, f i r s t

<

y

A

(X < 2 <

y)]

s a t i s f i e s " f o r every

e(-,-,p,q,A,

we can r e p l a c e

lemma) t h e s e t

Y)

Hence

m(*)+i'

i n the interval

Am(*)+i t* ( i n t h e t r e e

( 3 2 E

-t

t* such t h a t f o r e v e r y

,

(F2)

i t i s easy t o see t h a t

m(*)+l

x < y < a (x,y)

there i s

,...,Am)". by

m+l,

hence c l e a r l y

153

On Logical Sentences in PA

ip

€ f i N:

{p

E

a k ( 3 Y ) *(Y,

aN:

f o r every a

P, ill = E

y

...,Am )

BR a s

which belongs t o 2.7.

t* ( i n t h e t r e e

(x,y)

B(-,-,p,q,

l.

required.

m: For every n a t u r a l number

3 n CPQh ( n ,

in t h e i n t e r v a l

Am+1

E

t h e r e i s an element below Al,

t* such t h a t f o r every

there i s

A,+,

x < y < a , X E Am + l ,

k we can prove i n PA(Q

MM

) t h e statement

k).

Proof: F i r s t we prove t h a t t h e conclusion i s t r u e , i . e . t r u e i n t h e universe

6?)if

i t i s a model of say second o r d e r Peano a r i t h m e t i c .

Then f o r every n , t h e r e i s a p a i r

Suppose t h a t the conclusion f a i l s f o r k.

(Fn, H ) which forms acounterexample t o CP'

of functions

eh

n

We now d e f i n e by induction on i For

5

k i n f i n i t e sets

so t h a t R i + ,

Ai,

5 Ai.

i = 0 t h e r e i s no problem. Let A, = B, = t h e set of a l l natural numbers.

So suppose we have defined

A

for j s i

j

and we s h a l l define

By t h e i n f i n i t e Ramsey theorem we can get an i n f i n i t e ( a ) f o r every y Ai n y )

Now

(n,k).

z

<

in

A:,

f

j

for j = i+l.

Af 5 Ai such t h a t : F ( A , n y , A, n y , ...,

i s t h e function

Y¶Z

A

Z

restricted t o y.

f

does not depend on

z, i . e . i f

y

z,

< z,,

E

A

then

YSZ

SO l e t

f

= f

Y

for y

Y,Z

( b ) For

Y,

<

<

A' i '

z

Y, < y 2 i n

f

= YlZ,

f

YJ,

.

1

fy,ry, = fy,ry0 [Simply apply Ramsey theorem t o t h e natural t h r e e place c o l o u r i n g , f i r s t f o r ( a ) then f o r ( b ) l So

f

=

U If

Y

rz

are i n A i l

: z < y

i s a k-colouring ( o f t h e natural num-

b e r s ) . By t h e i n f i n i t e Ramsey theorem t h e r e i s i n

&3,

an i n f i n i t e s e t A:

A:,

which i s f-homogenous. Now we s h a l l deal with

H.

Again t h e r e is an i n f i n i t e

A: 5 A f

(A;

of course) such t h a t :

(c) i f H Z ( p , A, n y ,

y

<

z

. .., Ai

are i n

A:,

ny,

9)

< z

a r e in

95 Ai

n

y,

p

does not depend on

<

y

t h e n t h e value of

z.

Moreover (d) i f

x

<

y

A;,

9 5 Ai

nx,

p

<

x then t h e value of

BR,

S. SHELAH

154

..., Ai

HZ(p, A, n y.

For every p

9)

n y,

63,

E

and i n f i n i t e s e t

zero i f f f o r a r b i t r a r i l y large q HZ(p, A,

..., Ai

y,

n

<

. 9):

z in A i / q

i t is

the value of

i s zero. Clearly a f i n i t e change i n

HZ, and "for a r b i t r a r i l y large q

"for every large enough q "

z )

we define T ( p ,

5 A:

f o r every y

E ( % ~

n y, D n y - q )

n o t change the value of

(and on

does not depend on y

3

does

can be replaced by

"

(see 2 . 2 ) .

Now we want t o apply the Galvin Prikry theorem t o T, more exactly t o a parametrized version of i t

( p as the parameter). Simply i t e r a t e the usual Galvin-

Prikry theorem on the natural numbers, and take the diagonal intersection. What we get we call

.

A;

Again remembering 2.2, and the conclusion of the Galvin-Prikry theorem, we can

so t h a t f o r every p,

find Aitl 5 A:, Ai/q

the value of

i f f o r some q > p ,

..., A 1.

HZ(p,A, n y ,

n Y, Ai+,nY-q)

f o r every y

i s zero,

<

z in

then 9 = P

will serve. If

..., Ak

A,, n

A,,

9

Aktl

are defined, choose y

y > contradicts thechoice of

Fz, HZ

<

z in Aktl

and < A , n y,

.

However we want t o prove t h i s in PA(QMM) (and not in ZFC o r i n second order number theory). The proof i s the same, replacing "a s e t of natural numbers" by "a definable s e t of natural numbers". Why i s < < F , H > : n < W > definable? We canchoose for each n , a minimal n n pair < F n , H n > (by a simple enough coding). We can apply the i n f i n i t e Ramsey theoMM

rem as Macintyre [MI proves theparallel statement holds (from PA(Q ), of course the proof depends on the formula defining the colouring and the i n f i n i t e s e t ) . Wore exactly he proves that i f

,...,xn,

~(x,

-

p, z)

A

i s i n f i n i t e and definable,

i s such that

then there i s a definable i n f i n i t e

V x1

B c A and co

<

,...,xn

( 3 2 < c ) ~ ( x ,..., , xn,p,z)

c such that

We are l e f t with the "parametrized Galvin-Prikry". Let a < * b mean:
<

a , b code f i n i t e increasing sequences, ,

e ( a ) > respectively, and ca(m) : m < a ( a ) > i s a proper i n i t i a l seg-

ment of < b ( m ) : m

<

e(b)>.

155

On Logical Sentences in PA We d e f i n e

A:

A:

(after

has been d e f i n e d )

by d e f i n i n g by i n d u c t i o n on

I, kk, 6, such t h a t k,

(a)

<

... <

k

2- 1

L o , C1,

and

..., sI-l

{0,11

E

> < * a < b and f o r e v e r y p < k f o r some q, (QMMa,b) C < k o , ..., k f i - l y, z i f q < y < z, y E A?, z E A? t h e n HZ(p, A, n y, A . n y.

(b) f o r every

...,

1

Ia(m) : m < z ( a ) > / q ) =

1

cPl.

c ~ =- 0~ . Now we can c a r r y t h e d e f i n i t i o n (as i n [ M I n o t i n g t h e f o r m u l a s we use have (c) i f compatible w i t h

(a) t (b),

bounded complexity).So we f i n i s h t h e p r o o f o f 2 . 7 .

5 3. A t r u e

ny-sentence o f P A n o t p r o v a b l e i n PA

I n summer '80 Friedman and H a r r i n g t o n o f f e r e d h o t l y t h e i r view t h a t i t i s one o f t h e main problems o f contemporary l o g i c t o f i n d mathematical sentences as ment i o n e d i n t h e t y t l e , as w e l l as t o f i n d n a t u r a l t h e o r i e s w i t h incomparable c o n s i stency s t r e n g t h . The " t e c h n i c a l d i f f i c u l t y " i s i m m a t e r i a l ; i n f a c t t h e easiness o f t h e p r o o f may i n d i c a t e t h e profoundness and n a t u r a l i t y o f t h e sentence. Now an answer t o such q u e s t i o n i s n a t u r a l l y more open t o debate t h a n t h e usual mathematic a l problem.

A s t h e a u t o r d i d n o t want t o go i n t o such d i s c u s s i o n , and H a r r i n g t o n wanted a s o l u t i o n , an agreement was reached: i f t h e a u t h o r c o u l d f i n d a s o l u t i o n which H a r r i n g t o n would t h i n k i s O.K.,

he would w r i t e i t up, d i s c u s s i t and p u b l i s h i t .

The c o n t e n t o f t h i s s e c t i o n was done i n s p r i n g '81, H a r r i n g t o n O.K.ed i t , as w e l l as I 4 ( w h i c h was done i n summer '80) b u t was t o o l a z y t o f u l f i l l h i s promise. 3.1. C o n t e x t :

N be a n a t u r a l number

1) Let

r

= < r : I < I(;)> 9"

a finitesequence

o f n a t u r a l numbers.-

2) L e t

K = K i = {(A,

<,

R)

: A

6 R

a subset o f

N,

<

a l i n e a r order o f

A

a sequence o f r e l a t i o n s o v e r an

A

,

,

r -place

Ik e ( R ) = I(;)}.

3 ) Members o f

K

a r e denoted by

A, B

,

letting

power o f t h e s e t o f t h e s e t o f elements o f 4) I n (4a)

K

A =

/ ~ ,l

so

i s the

llAll

A.

we d e f i n e

A
( B an end e x t e n s i o n o f

A) i f A

i s a submodel o f

B

and

S. SHELAH

156 x

IBI

E

,

- 1.4

(4b) A < B

y

satisfying

implies

y < x.

of

A'

into G

IIb/l 5 IlAll

A',

A <

en B over

i n t o @(N)

Let

+

1,

A < B and f o r any en t h e r e i s an embedding

.

A

f r o m @(N)

3.2. E ( N , r , k , n ) - p r i n c i p l e : F

PI

( B an u n i v e r s a l end e x t e n s i o n o f A) i f

A'

5) A function

1 ) Domain:

E

F

is

be a k - p l a c e f u n c t i o n s a t i s f y i n g

i s d e f i n e d on i n c r e a s i n g sequences o f l e n g t h

2) Choice F u n c t i o n :

..., A k )

F(A,,

...

3 ) Isomorphism I n v a r i a n c y : i f A 1 < i s an isomorphism

g

from A k

< Ak

onto

k

K

from

.

IA,[

E

if / G ( A ) I z f ( 1 A I ) .

f-small

B

k

E

K,

B1 <

mapping

... < B k

At

onto

E

Bil

K , and F

then

there and

g

commute, i .e. F(B~.

..., B k ) =

4 ) Weak H e r e d i t a r i t y : F o r e v e r y such t h a t : i f

f

B

-1

=

0)

then

Then - there

A,

+)

A2 <

<

i s a submodel o f

B~

F(A,

...,

g(F(Al,

,...,A k )

= F(Bl

...

. k t h e r e i s an x -small f u n c t i o n

< Ak,

f(Bil-l)

Ai,l

n Ail

5 Bil

(stipulating

,..., B k ) .

i s an i n c r e a s i n g sequence < A :~ il < n > on which

F

depends on

the f i r s t structure only. 3.3.

Fact

F to

F'

if

N > 22 k+n+'(r(i)+l),

F a s i n 3.2 and

N' >

a k - p l a c e f u n c t i o n s a t i s f y i n g 1 ) - 4 ) o f 3.2 f o r

N,

t h e n we can e x t e n d

N ' , i n one and o n l y one

way. __ P r o o f . By 3.2 ( 4 ) ( 3 ) (as i n t h e p r o o f o f 1.3 ( 1 ) ( b ) ) . 3.4.

Claim: I n

PA+PH

$* = ( V

r,

we can p r o v e (the

k, il)

r,

(N,

k , n ) - p r i n c i p l e h o l d s f o r e.g.

,k+n+ f C r ( i ) + l l

).

N = 2 __ P r o o f : D e f i n e by i n d u c t i o n on

m

e,

i s s m a l l e r enough t h a n

putation). Applying

PH

and i n d i s c e r n i b l e f o r F ' ( A ~, 1

e,

a model

Am<*A

e

we g e t a s e t (F'

as i n 3.2

..., A .

'k

) = F'(A. Ji

.

e

c

K

w i t h universe

il,

so t h a t i f

( j u s t t a k e c a r e o f 3.1 ( 4 ) ( b ) , easy com-

C

o f n a t u r a l numbers

for

,

A

N'

..., Ajk)

(V x < y ~ C ) C 2 2 ~ < y l ,

l a r g e enough)

,

and

ICI

>

4 Min C

157

On Logical Sentences in PA (we use an e q u i v a l e n t v a r i a n t o f P.H.).

: i


F

C>,

E

As i n 5 1

depends on t h e f i r s t v a r i a b l e o n l y . Now as i n t h e p r o o f o f 1.3

( 1 ) ( b ) we c o l l a p s e t h e s o l u t i o n below 3.5.

Claim: I n

PA+$*

N

.

we can p r o v e t h e c o n s i s t e n c y o f PA (hence

Proof. We can b u i l d a non-standard model o f dard,

N

l a r g e enough, and

..., Ak)

F(A,,

mal code f o r which

1

( v y)

e a s i l y f o r subsequences o f

r

PA,+$*,

<4> d e f i n e

F :

i s d e f i n e d as f o l l o w s :

let


{3Xcp(X.y)

=

..., A k > / = +

M, choose

cp(x,y)

3X[cp(X,y)

A

(v

z

< X)

and t h e n

( i f t h e r e i s one).

A,,

cp(x,y)

non s t a n -

be a f o r m u l a w i t h m i n i -

ic p ( Z , j ) l

(by the lexicographic order o f

E

k, n

"induction fails,i.e.

and t h e n t a k e minimal x

PAP$*).

1''


y.

y o , yl,

The r e s t i s as i n I 1

...>

)

the only

a d d i t i o n a l p o i n t i s why a r e a d d i t i o n and m u l t i p l i c a t i o n d e f i n a b l e ? T h i s i s by 3.1 ( 4 ) ( b ) .

5

4. On c o n s i s t e n c y s t r e n g t h

Let extending CON(T)

T PA

denote h e r e a ( r e c u r s i v e ) t h e o r y ( w i t h f i n i t e - s o r t , f i n i t e - l a n g u a g e ) b u t i t may a l s o "speak" on r e a l s and even a r b i t r a r y s e t s .

be t h e sentence ( i n

D e f i n i t i o n : We say

PA

T, scs T,

language) s a y i n g

T

i s consistent.

( t h e consistency strength o f

equal) than t h e consistency strength o f

T2)

if

PA

Let

t- CON(T,)

T,

i s smaller (or -f

CON(T,):

I t was observed t h a t e s s e n t i a l l y " l a r g e c a r d i n a l axioms a r e l i n e a r l y ordered"

(though i n some cases t h i s "has n o t y e t been proved").More e x a c t l y i t seems t h a t a l l s e t t h e o r i e s which has been c o n s i d e r e d so f a r , a r e l i n e a r l y o r d e r e d by Solovay

( I t h i n k ) has

found

T's

which a r e

5

-incomparable,

s

cs b u t t h e y were

.

cs " p a r a d o x i a l " (i .e. have s e l f - r e f e r e n t i a l sentences). We s h a l l t r y t o g e t more r e a sonable ones.

* * * * * * * Let

PA+

be

PA t C O N ( P A ) .

We work i n s i d e

P A . A model w i l l mean one which

i s definable. Let

T,,

T,

be c o n s i s t e n t t h e o r i e s ( i n o u r " u n i v e r s e " which s a t i s f i e s PA).

s. SHELAH

158

T,

" s a y i n g " t h e r e i s a model o f

(think o f

PA+,

PA

+

CFMSI, o r o f course As T,

+

ZFC,

T, " s a y i n g " t h e r e i s a model o f

ATR, see Friedman, McAloon and Simpson

T,tCON(TI)

hence t h e r e i s MI

( 1 ) q Q ( n ) says

n

i s c o n s i s t e n t , hence ( b y Godel incompleteness has a model M,

M,

E

f ''$2(n)''

I=

such t h a t

(n

. By

t h e r e q u i r e m e n t on T,

i s a non-standard i n t e g e r )

where

i s a n a t u r a l number and t h e r e i s a p r o o f o f s i z e

n

PA

ZFC+large c a r d i n a l s ) .

T, +CON(T,) + iCON(T, +CONTI)

iCON(T,),

n

of

t 1 CON ( Te)

PA

f ( 2 ) +,(n)

says

$

Q

(n)

but

f ( 3 ) T~ = PA + ( 3 n ) r$,(n) i s consistent with

i s t h e f i r s t such number.

n

As we have assumed t h a t

with

TI,

o r use

or

i s consistent,

theorem) M,

I rrl-CAo;

T, says t h a t

has a model, c l e a r l y

+-,~,,(2~")1 f

+ (Vm) l$,(m)

PA + 3 n$,(n)

( a s even

PA+

PA

i s consistent

.

PA+)

By a theorem o f Friedman (based on a n a l y z i n g Godel incompletness theorem) f

( 4 ) Tb = PA + ( 3 n ) C$,(n)

'

( 5 ) PA+

+

Ta, Tb

are

s

cs

-incomparable.

L e t us p r o v e e.q.

*

Ta 1- CON(Ta)

rBecause f o r any model nition

11

PA

We s h a l l p r o v e t h a t csTb

217

+.

i s consistent with

Ta

$,(2

f

o f a mode:

phism from

No

Clearly

N,

of

N o of PA

(i.e.

PA++Ta, No

b e i n g a model o f

1: "N,

i n t o a p r o p e r i n i t i a l segment o f N,

satisfies

of bounded f o r m u l a s and

Ta -PA,

has a d e f i -

i s a model o f P A " ) and an isomorN,.

as end e x t e n s i o n s p r e s e r v e t h e s a t i s f a c t i o n

f

$,(x),

PA+

$,(x)

a r e such f o r m u l a s . 1

Note a l s o t h a t (6)

PA

+

.

Ta I- 7CON(Tb)

[Otherwise t h e r e i s a model a d e f i n i t i o n o f a model segment o f No I= T

a' formulas,

N, No

(i.e. f "$,(n)

and

f

N, o f No

No

of

PA+Ta+CON(Tb)

Tb a n d a n i s o m o r p h i s m g of

hence i n

No

No

there i s

onto a proper i n i t i a l

s a t i s f i e s t h e sentences s a y i n g t h o s e t h i n g s ) . AS 211 f and -I $,(2 ) " f o r some n, b u t as $, $1 a r e bounded f commute w i t h e x p o n e n t i a t i o n , c l e a r l y N, I. "$,(g(n)) and

159

On Logical Sentences in PA 1 $,(22g(n))".

So

f 2m N, I= Tb hence f o r some m, N , I= "$,(m) and $ , ( 2 ) " . f f and $,(g(n))",hence by q 2 ' s definition g ( n ) , m have t o be

But

f

N , k "$,(m)

equal. B u t

N, k "$,(2

2m

)

and 7 $ 1 ( 2 ' g ( m ) ) " hence

g(n),

rn

should be unequal,

contraddictionl. By ( 5 ) and ( 6 ) clearly

PA+

+

C O N ( T a ) I$

(as PA+ + T~

CON(Tb)

is

consistent (by ( 3 ) ) ; t h i s implies

PA I f CON(Ta)

(7)

+

So Ta $ csTb.

CON(Tb)

. Tb $ csTa

i s t o t a l l y analogous. n f NOW (Woodin suggests) wecan replace the sentences ( 3n ) r $ , ( n ) + - 1 $ ~ ( 2 ' ) I n f ( 3 n ) r $ , ( n ) + $ J 2 ' ) I by inequalities of the indicator functions corresponding t o

and

The proof of

f , , f,

T I and T, ( i . e . the function T , , T,).

t o the consistency of

exhibiting the

E.g. ( V n ) [ i f

f,(n)

ITsentences :

corresponding

i s defined then

so i s

f , ( f , ( f , ( n ) ) 1 and i t s negation. So i f we accept those functions as "mathematical"

( n o t j u s t reasonable methamathematical ) we get mathematical theories of incomparable consistency strength. (Originally we have used three t h e o r i e s ) . We a r r i v e t o the dangerous question o f which PA

function: on

see Paris and Harrington [ P H I ,

T's

have matheratical indicator

on many theories ( l i k e ZFC+large

cardinal) see Friedman rFl1 on ATR, see Friedman McAloon and Simpson rFMSl on 1

IT,-CA,

see 5 2. Alternatively f o r an indicator function

f ( f * ( n ) + l ) , and use

$,

$3

where

$,

f

f*

define

= "the f i r s t

by

f*(O) = 0

n f o r which f ( n ) i s

:Q

f*(n+l)=

mod 4".

+ * * * * * * * Notice the following two phenomena

( A ) For any two natural s e t theories, not only they are

5

cs

-comparable, b u t one

i s interpretable in the other ( o r expected t o be s o ) . ( 6 ) Similarly, f o r any theories, e.q. undecidab l i t y r e s u l t s are gotten by i n t e r -

pretation. Friedman rFr21 proved a theorem saying ( A ) i s really true. Concerning ( B ) however, in CSh1, (under C H )

the monadic theory of the re 1 order i s proven undecidable

without the usual interpretation. I n Gurevich and Shelah

CGSl1 t h i s

i s explained i t i s a Boolean-valued i n t e r p r e t a t i o n , and by CGS21 the usual i n t e r pretation i s impossible. Now we can t r a n s l a t e i t t o ( A ) : l i s t the reasonable axioms f o r the monadic theory of the real order (considered as a two-sort model).

S. SHELAH

160

REFERENCES

CEHMRl P. Erdos, A. H a j n a l , A. Mate and R. Rado, C o m b i n a t o r i a l s e t t h e o r y , N o r t h H o l l a n d P u b l . Co.

CF11

H. Friedman, On t h e necessary use o f A b s t r a c t s e t t h e o r y , Advances i n Mathem a t i c s 41 (1981), 209-280.

TF21

H. Friedman, T r a n s l a t a b i l i t y and r e l a t i v e c o n s i s t e n c y .

C FMS 1 H. Friedman, K. McAloon and S.G. Simpson, A f i n i t e c o m b i n a t o r i a l p r i n c i p l e which i s e q u i v a l e n t t o t h e I - c o n s i s t e n c y o f p r e d i c a t i v e a n a l y s i s . CGSll

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