The Baire category theorem in weak subsystems of second-order arithmetic

Working within weak subsystems of second-order arithmetic Z2we consider two versions of the Baire Category theorem which are not equivalent over the base system RCA0. We show that one version (B.C.T.I) is provable in RCA0while the second version (B.C.T.II) requires a stronger system. We introduce two new subsystems of Z2, which we call RCA+ 0and WKL+ 0, and show that RCA+ 0suffices to prove B.C.T.II. Some model theory of WKL+ 0and its importance in view of Hilbert's program is discussed, as well as applications of our results to functional analysis.