The Fraenkel-Carnap Question

Posted May 28th, 2010 at 10:36 am.

Irena Penev
Mentor: Dr. George Weaver

This project attempts to answer an old, neglected, and open question of Fraenkel and Carnap: whether or not every finitely axiomatizable and semantically complete second-order theory is categorical. Assume that K is a finite set of non-logical constants and that L is the second-order language over K. A theory T in L is finitely axiomatizable provided that there is a finite set of sentences S such that T is the set of logical consequences of S. The theory T is semantically complete provided that given any sentence f in L, either f or its negation is in T. If T has a model, then T is the theory of (i.e. the set of sentences true on) any of its models. T is categorical provided that all of its models are isomorphic. If A is an interpretation for L, then A is finitely characterizable provided that there is a finite set of sentences true on A all of whose models are isomorphic. It is known that if A is finitely characterizable then the second-order theory of A is finitely axiomatizable. The Fraenkel-Carnap question is equivalent to whether or not the converse is true.

Carnap proposed a positive answer to the Fraenkel-Carnap question in the 20's, but his proof was flawed. It has, however, been shown that if K is empty, then the answer to the Fraenkel-Carnap question is positive. The answer is also positive when the question is restricted to finite interpretations. Results of Ajtai [1] imply that there are models of the Zermelo-Fraenkel set theory in which the Fraenkel-Carnap question restricted to countably infinite interpretations has a positive answer. It has been shown in [3] that the Fraenkel-Carnap question has a positive answer when restricted to the class of Dedekind algebras. Similar results have been obtained for other classes: infinite strict well orders, infinite abelian groups all of whose elements are of prime order p; uncountable divisible torsion-free abelian groups; uncountable algebraically closed fields of characteristic zero; uncountable algebraically closed fields of prime characteristic p; uncountable n-dimensional vector spaces over an algebraically closed field of prime characteristic p.

The method of procedure here is to investigate interpretations of a fixed infinite cardinality. The work relies on a characterization of finite axiomatizability implied by a back and forth characterization of equivalence in second-order languages and an improvement of an unpublished result of Kaplan (see [2], Lemma 2) established just this summer:

Theorem Assume that A and B infinite interpretations of L. If for all integers n 0 and n 1 , A and B agree on all sentences of quantifier rank (n 0 ,n 1 ,2), then they are equivalent with respect to L.

[1] M. Ajtai, "Isomorphism and higher order equivalence," Annals of Mathematical Logic, vol. 16, 1979, pp. 181-203.

[2] R. Montague, "Reduction in higher order logic," The theory of models; proceedings ( J.W. Addison, L. Henkin, and A. Tarski, editors), Amsterdam , North-Holland Pub. Co. , 1965, pp. 251-264.

[3] G. Weaver and B. George, "The Fraenkel-Carnap question for Dedekind algebras," Mathematical Logic Quarterly, vol. 49, 2003, pp. 92-96.

Filed under: 2004,Penev, Irena,Weaver, Dr. George by Ann Dixon

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