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.