The Baire Category Theorem in Weak Subsystems of Second-Order Arithmetic

Working within weak subsystems of second-order arithmetic Z2 we consider two versions of the Baire Category theorem which are not equivalent over the base system RCAo. We show that one version (B.C.T.I) is provable in RCAo while the second version (B.C.T.II) requires a stronger system. We introduce two new subsystems of Z2, which we call RCA' and WKL', and show that RCA' suffices to prove B.C.T.II. Some model theory of WKL' and its importance in view of Hilbert's program is discussed, as well as applications of our results to functional analysis. ?0. Introduction. This paper consists of some of the material contained in [2], which is concerned with the development of the basic definitions and theorems of functional analysis within second-order arithmetic, Z2. Such studies take place within a broader program initiated by Friedman and carried forward by Friedman, Simpson, and others. The goal of this program is to examine the Main Question: Which set existence axioms are needed to prove the theorems of "ordinary mathematics?" An exposition of the meaning of "ordinary mathematics" can be found in [22, 21, 2] -for the purposes of this paper it suffices to note that the theory of complete separable metric spaces is an example of ordinary mathematics. The language of second-order arithmetic is a two sorted language with number variables i, j] k, m, n, ... and set variables X, Y, Z..... Numerical terms are built up as usual from number variables, constant symbols 0 and 1, and the binary operations of addition (+) and multiplication (.). Atomic formulas are t1 = t2, t1 < t2, and t1 e X where t1 and t2 are numerical terms. Formulas are built up as usual from atomic formulas by means of propositional connective A, V, -, , -+, number quantifiers Vn and 3n, and set quantifiers VX and ]X. The formal system Z2 includes the ordered semiring axioms for N, +, *, 0, 1, < as well as the induction axiom (OeX A Vn(neX-*n + 1 eX))-*Vn(neX) and the comprehension scheme 3XVn(n E X *-(p(n)), where (p(n) is any formula in which X does not occur freely. Received September 16, 1991. Research of the second author was partially supported by NSF grant DMS-8701481. ?