2 Scott's Isomorphism Theorem

2 Scott's Isomorphism Theorem

THEOREM 1 (SCOTT [1965]). Let U be a countable model for L. Then there is a sentence cp of L,,, such that for all countable models 23 for L, % k cp if and only if 23 2 a.

PROOF. For all a , , . . ., a,, E A and p < wl, define the formula cp,: . . . a m . . . x,,) inductively as follows: 0 'Pat . . . an = A { 8 ( x , . . . x,,): i?l k 8[a, . . . a,,] and 8 is either atomic or the negation of an atomic formula). If p is a limit ordinal, (xl

whenever y < p < w , . Since M is countable, for each a, , . . . a,, E A there exists that for all p 2 a,

a

(Vx1

*

*

xn)d, 7

...a. * V a8t ...a n *

tl

< w , such

8

(2

SCOTT'S ISOMORPHISM THEOREM

It follows that there exists all fi 1 z,

CI

< o,such that for all a , , . . .,a, E A and

'21 ( V X ~* . . X n b : 1 ...a , * ( ~B a ... l a,

*

Let cp be the sentence

A

CP; A

(VX,

* * *

..U,EA

xn)(qzi ... a,

a+ 1 +1.04

... a,).

,
Then 9 ' 1 k cp. Suppose 23 is countable and 93 k cp. We show '21 r 23 by a back and forth argument. It suffices to show that for all a , , . . ., a, E A and b, , . . ., b, E B such that 23 k cp,", . . .a,[bl . . b,],

.

(Van+, E A ) ( g b n + 1 E B ) m 1~:1...a,+j[bl

**.bn+11,

(1)

and (Vbn+,

E B)(Ian+,

€ A ) BC

q:1

...an+l[bl

. bn+,]-

(2)

To show (1): Since 23 k q ! B k cp::.f.,,[b, . . . b,], whence 5 3 k (3x,+,)qz1 ...a,+,[ bl . . . b,], and thus (1) holds. Proof of (2): Again . . . b,], using the fact that B k q,":.f.,,[b, 8 ( V x n + l ) Va,+lEA~zl...a,+l[bl - . b n l whence for Some an+, E A , 23 k yi1...a , + l C b l * . . b n + l I . d The above proof is due to CHANG[1968]. Scott's theorem has the following corollary.

COROLLARY. If 9X and 93 are countable models for L and '21 = B(LmIm), then '3 2 23. CHANG[1969] has shown that the above holds even if L has uncountably many symbols. On the other hand, the exercises show that it fails if we allow 23 to be uncountable. PROBLEMS

-

1. Let '3, 23 be models for L. Suppose that for each n c o there is a relation between A" and B" such that (i) If ( a , . . . a , ) ( b , . . . b,) then ( a , . . a,) satisfies exactly the same atomic formulas in '21 as ( b , . . b,) satisfies in 23.

-

.

.

21

-

,

(ii) If ( a , . . . a,) ( b , . . . b,) then (Van+ . . * an+,) (61 * . . b n + 1 ) (iii) If ( a , . . . a,) ( b , . . . b,) then (Vb,,, (a1

E A)(%,+

,

E B)

N

N

(01

9

SCOTT'S ISOMORPHISM THEOREM

an+,)

N

(b1-

* *

E B)(3a,+l E

A)

bn+l).

Prove that (21 = %(Leo,,).

2. Apply problem 1 to show that 9l = %(L,,,) when: (i) a, '23 are densely ordered structures without endpoints. (ii) 9l = and 23 = ( S ( Y ) , E) where X , Yare infinite. (iii) % = %(L) and a, 23 are w-saturated models. NOTE: is w-saturated means that for every sequence cpo(x, . . . x,), cpl(xl . . . x,), . . . of formulas of L,

a I=(VX, (iv)

* * * Xn-1)C

A

m
(%)(POA*

a, 23 are w-homogeneous and . . . x,), . . . of formulas of L,

-

* ~ c ~ + m () 3 X n )

A~ml-

m
for every sequence cpo(xl . . . x,),

'pl(xl

9lk(3xl*..x,)Acp, m
iff B k ( 3 x ,... x,)Acp,. mi,

NOTE:3 is w-homogeneous means that for all a,, . . . , a , + , , b , , . . ., b, E A , if (%, a, . . . a,) =- (a, b, . . .b,)(L) then (%,+ 1 E B)(%, a1 . . . a,,,) = (a, b , . . .b,+,)(L). 3. Let the language L have possibly uncountably many symbols. Suppose that %, % are countable models for L and % = %(L,,,). Then

9 z 23.

4. Let L have possibly uncountably many symbols, let

9l be a countable

model for L, and let P c A . Then the following are equivalent: (i) For any Q c A , if (a,P) z (a, Q) then P = Q. (ii) P is definable in L,,,, that is, there is a formula q(x) of L,,, that

(a, P)

vxrqx)

-

cp(X)l.

such