Proceedings of the Fourth International Congress for Logic, by Patrick Suppes

By Patrick Suppes

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By Patrick Suppes

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G is not unique. DEFINITION 37. Let R( ex, fJ, C) be a formula with two element variables and one set variable C and possibly additional variables of the language of SwS and let S(C) be a formula of one set variable and possibly additional parameters of this language. Let M be a class of structures defined by a formula F. We say that (S, R, A) is adequate for F if for every structure (S, R, 0, D t ) EM there exists an embedding g of (S*, R*, 0*) (of Definition 37) into N* with T being the image of s* and the restriction of < 1 to T being the image of R* and A the image of 0* such that the following holds (for some values of the parameters of Rand S): (al) S(T) holds.

When we speak of indiscernible sets below we shall always mean ones of the same kind as 1. However the index which always has type Pro may vary from one indiscernible set to another. Let I be an indiscernible set with index b. 2 (vii), for each n < w there exists rl E ro>w such that cfIn is defined and RanknV'n 1\ On(X; (tin) 1\ On(X; [jn) = RanknV'n. 2 (iii) and (vi), it follows that rl is unique. 1. Thus 'YJn :=> 'YJn-t. Let 'YJ = U {'YJn In < w}, we call aTJ a quasi-index of 1. A set J c M is called independent if each singleton subset is indiscernible with index in 8, and for any formula tp(x; y), a e J, and c e (J - {a}) u U {Rngc Ia E 8} such that F= tp [a; c] it is the case that Rank, V'n 1\ tp(x; c) = Rank, V'n for all n < w.

M}) uU {Rngala EEl. 1 (viii) P is expressible by a first-order formula. (k+l) "q;(x; c) = Rankk"Pk for all k < co, Let I', consist of all first-order formulas expressing propositions such as P. Note that the members of I', will contain parameters from B uU [Rngji la e E}. I)1 i < w}. From the definition of independent set and the ~ t-saturation of M it is sufficient to show that the set T = To uTt uT2 is consistent. 1 be a finite subset of T in which at most b t, b~, ... k occur from Bu{b t}. For n < w let m be chosen such that (G(m)ln, G(m-1)ln, ...

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