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# Finite Axiomatizability using additional predicates

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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 PCACδ.

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.+.

References
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[1]Asser, G., Das Repräsentantenproblem im Prädikatenkalkül der ersten Stufe mit Identität, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 252263.
[2]Church, A., Introduction to mathematical logic, vol. 1, Princeton (Princeton Univ. Press), 1956.
[3]Craig, W., On axiomatizability within a system, this Journal, vol. 18 (1953), pp. 3032.
[4]Craig, W., Further remarks on finite axiomatizability using additional predicates, Bulletin of the American Mathematical Society, vol. 62 (1956), p. 411.
[5]Ehrenfeucht, A., Two theories with axioms built by means of pleonasms, this Journal, vol. 22 (1957), pp. 3638.
[6]Gödel, K., Die Vollständigkeit des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349360.
[7]Henkin, L., On a theorem of Vaught, Indagationes mathematicae, vol. 17 (1955), pp. 326328.
[8]Kleene, S. C., Introduction to metamathematics, Amsterdam (North Holland) Groningen (Noordhoff), New York and Toronto (Van Nostrand), 1952.
[9]Kleene, S. C., Finite axiomatizability of theories in the predicate calculus using additional predicate symbols, Memoirs of the American Mathematical Society, no. 10 (Two papers on the predicate calculus), Providence, 1952, pp. 2768.
[10]Łoś, J., On the categoricity in power of elementary deductive systems and some related problems, Colloquium mathematicum, vol. 3 (1955), pp. 5862.
[11]Mostowski, A., Concerning a problem of H. Scholz, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 2 (1956), pp. 210214.
[12]Scholz, H., Ein ungelöstes Problem in der symbolischen Logik, this Journal, vol. 17 (1952), p. 160, Problem 1.
[13]Tarski, A., Der Wahrheitsbegriff in den formalisierten Sprachen, Studio philosophica, vol. 1 (1936), pp. 261405; English translation in Logic, semantics, metamathematics, Oxford (Oxford University Press) 1956, pp. 152–278.
[14]Tarski, A., A problem concerning the notion of definability, this Journal, vol. 13 (1948), pp. 107111.
[15]Tarski, A., Some notions and methods on the borderline of algebra and metamathematics, Proceedings of the International Congress of mathematicians, (Cambridge, Mass. USA, Aug. 30-Sept. 6, 1950), 1952, vol. 1, pp. 705720.
[16]Tarski, A., Contributions to the theory of models, Parts I and II, Indagationes mathematicae, vol. 16 (1954), pp. 572588.
[17]Tarski, A., A general theorem concerning the reduction of primitive notions, this Journal, vol. 19 (1954), p. 158.
[18]Tarski, A., Mostowski, A., and Robinson, R. M., Undecidable theories, Amsterdam (North Holland Pub. Co.), 1953.
[19]Vaught, R. L., Applications of the Löwenheim-Skolem-Tarski theorem to problems of completeness and decidability. Indagationes mathematicae, vol. 16 (1954), pp. 467472.
[20]Vaught, R. L., Finite axiomatizability using additional predicates, I and II, Bulletin of the American Mathematical Society, vol. 62 (1956), pp. 412413.
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