Effectiveness and Vaught’s gap $\omega $ two-cardinal theorem

Abstract

We consider functions f with the property that whenever a is a sentence in L,,,,, then f(a) f(a) model, then a admits all types. A question of Barwise is answered by showing that no such f is recursive, and that the least such f is not co-r.e. Barwise proves in [1] an effective version of Vaught's gap X two-cardinal theorem [7] for a class of sentences which have a certain syntactic form. He then asks (Problem 2.12 of [1]) the following question concerning the possibility of extending this result to the set of all sentences: What can be said about the effectiveness of a function f (and, in particular, the least J) such that for any sentence a, if a has a gap > f (a) model, then a admits all types? Note that we have reformulated Barwise's question in a way that best fits our answer. First, some notation and definitions. Let e be a sufficiently rich, recursive, first-order language which has among its symbols a distinguished unary predicate symbol U. We will deal throughout with this language P. We let E denote the set of all 6-sentences. A type is a pair (K, X) of infinite cardinals such that K >? X, and we say that the X-structure W = (A, U,... ) has type (K, X) if card (A) = K and card (U) = X. A sentence a admits the type (K, X) if it has a model of type (K, X). For each n n structure iff K > Dn(X) We can now state Vaught's gap X theorem. VAUGHT'S THEOREM. If a has a gap > n model for each n > X, then a admits all types. Barwise defines, for each n < t, a class of formulas which he calls 3V( )mod U formulas. By induction on n, we define when 4 is a V(n)-mod U formula: 0 is a V(?)-mod U formula iff it is universal, and 4 is a V(n+')-mod U formula iff it is in the form Vx (3y, E U) (3Y2 E U)... (3ymEe U)01, where 41 is a V(n)-mod U formula. Then, a formula is a Vg(n)-mod U formula if it is obtained from a V(n)-mod U formula by means of existential quantification. Received by the editors December 23, 1975. AMS (MOS) subject classifications (1970). Primary 02H05, 02G05.