By R. 0. Gandy, C. M. E. Yates
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Additional resources for Logic Colloquium '69
U, E f l Then B = U B,, and there are at most h such B,’s. Hence our result will follow if we provd: (a) P is a union of B,’s; (b) I B, I = 1 or I B, 12K . < a). , a), where Q is independent of the ai. This proves (a). Fcr (b), we define and consider two cases. Case 1. Q 91 is false for increasing 2n-tuples in H . Since p ( H ) = 1, we have 1 HI = K ;so H may be split into K increasing n-tuples. Applying f to these gives K distinct members of B,. Case 2. Q is true for increasing 2n-tuples in H .
The question which we discuss here is: How similar can we make T h ( W ) and Th(%)? For 2 such that % W ? The following lemma example, can we find such an 9 shows various ways in which this is too much to expect. 1. Let %= (M, E ) and % = (N, F ) be structures. (a) (b) (c) (d) If % ZF and % is pointwise definable then % k V=OD. If % and % are pointwise definable and 311 % then there is a unique isomorphism of % onto % . If % and % are pointwise definable models of ZF and % % then sp(%) = sp(%).
A) follows from the reflection principle and the definition of being ordinal definable. Any isomorphism f of W onto 92 must satisfy: able and% for all formulas q ( x ) such that % k 3 ! x q ( x ) . But since W and 92 are pointwise definable this condition uniquely defines a function from M onto N which is clearly an isomorphism. This isomorphism takes the well founded part of % onto the well founded part of 92 and hence sp(%) is isomorphic to sp(%). But if x and y are transitive sets and (x, is isomorphic to (y, E,) then x = y .