On a Question of Brown and Simpson

Brown and Simpson (1993) considered two versions of the Baire Category Theorem, which they called BCT-I and BCT-II . They showed that BCT-I is provable in RCA0, the subsystem of second order arithmetic specifying some basic algebraic properties of the numbers, induction for Σ 1 sets and the recursive comprehension axiom. Additionally, they showed that BCT-II is not provable in RCA0. Brown and Simpson then introduced RCA + 0 , the extension of RCA0 which for each n includes the statement that for each subset X of the natural numbers N, there is a G ⊆ N such that G is Cohen generic for n-quantifier arithmetic relative to X. They showed that BCT-II is provable in RCA+0 and posed the question whether BCT-II and its consequence the Open Mapping Theorem are provable in a system weaker than RCA+0 . In this paper, we introduce an equivalent formulation of BCT-II , which we denote BCT-Π 1 . It states that the intersection of every sequence of dense open subsets of Baire space which has a uniformlyΠ 1 representation is dense. In the language of forcing, the set of reals which are generic for the sequence 〈On : n ∈ N〉 is dense. Using this recursion theoretic characterization, we show that there is an ω-model of BCT-Π 1 which is not a model of RCA + 0 (Corollary 4.3, thereby answering the question raised by Brown and Simpson.) Brown and Simpson compared BCT-II with WKL0, which consists of RCA0 and the assertion (of compactness) that every infinite binary tree has an infinite branch. They showed that neither of these two systems is a subtheory of the other, over the base system RCA0. We continue in this line by comparing BCT-Π 1 with the assertions of the various cases of Ramsey’s Theorem. See (Ramsey 1930).