Finite Axiomatizability using additional predicates

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Abstract
By a theory we shall always mean one of first order, having finitely many non-logical constants. Then for theories with identity (as a logical constant, the theory being closed under deduction in first-order logic with identity), and also likewise for theories without identity, one may distinguish the following three notions of axiomatizability. First, a theory may be recursively axiomatizable, or, as we shall say, simply, axiomatizable. Second, a theory may be finitely axiomatizable using additional predicates (f. a.+), in the syntactical sense introduced by Kleene [9]. Finally, the italicized phrase may also be interpreted semantically. The resulting notion will be called s. f. a.+. It is closely related to the modeltheoretic notion PC introduced by Tarski [16], or rather, more strictly speaking, to PC∩ACδ.For arbitrary theories with or without identity, it is easily seen that s. f. a.+ implies f. a.+ and it is known that f. a.+ implies axiomatizability. Thus it is natural to ask under what conditions the converse implications hold, since then the notions concerned coincide and one can pass from one to the other.Kleene [9] has shown: (1) For arbitrary theories without identity, axiomatizability implies f. a.+. It also follows from his work that : (2) For theories with identity which have only infinite models, axiomatizability implies f. a.+.