We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a Π 1 2 sentence of a certain form is provable using E-HA ω along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and the Dialectica interpretation. These results allow us to use techniques of classical reverse mathematics to demonstrate the unprovability of several mathematical principles in subsystems of constructive analysis.

Abstract. We initiate the reverse m athem atics of generaltopology. We show thata certain m etrization theorem is equivalent to Π 1 2 com prehension. An MF space is defi ned to be a topologicalspace of the form MF(P) with the topology generated by {Np | p ∈ P}. Here P is a poset, MF(P) is the setof m axim alfilters on P, and Np = {F ∈ MF(P) | p ∈ F }.If the poset P is countable, the space MF(P) is said to be countably based. The cl ass of countably based MF spaces can be defi ned and discussed within the subsystem ACA 0 of second order arithm etic. One can prove within ACA 0 that every com plete separable m etric space is hom eom orphic to a countably based MF space which is regular. We show that the converse statem ent, "every countably based MF space which is regularis hom eom orphicto a com plete separable m etric space," is equivalent to Π 1 2 -CA 0 . The equivalence is proved in the weaker system Π 1 1 -CA 0 . This is the fi rstexam ple of a theorem of core m athem atics which is provable in second orderarithm etic and im plies Π 1 2 com prehension. In the foundations of m athem atics, there is an ongoing research program

This paper gives a formalization of general topology in second order arithmetic using countably based MF spaces. This formalization is used to study the reverse mathematics of general topology.For each poset P we let MF(P ) denote the set of maximal filters on P endowed with the topology generated by {N p | p ∈ P }, whereDefine a countably based MF space to be a space of the form MF(P ) for some countable poset P . The class of countably based MF spaces includes all complete separable metric spaces as well as many nonmetrizable spaces.The following reverse mathematics results are obtained. The proposition that every nonempty G δ subset of a countably based MF space is homeomorphic to a countably based MF space is equivalent to Π 1 1 -CA 0 over ACA 0 . The proposition that every uncountable closed subset of a countably based MF space contains a perfect set is equivalent over Π

Abstract. The implication relationship between subsystems in Reverse Mathematics has an underlying logic, which can be used to deduce certain new Reverse Mathematics results from existing ones in a routine way. We use techniques of modal logic to formalize the logic of Reverse Mathematics into a system that we name s-logic. We argue that s-logic captures precisely the "logical" content of the implication and nonimplication relations between subsystems in Reverse Mathematics. We present a sound, complete, decidable, and compact tableaustyle deductive system for s-logic, and explore in detail two fragments that are particularly relevant to Reverse Mathematics practice and automated theorem proving of Reverse Mathematics results.

Abstract. We study the logical content of several maximality principles related to the finite intersection principle (F IP) in set theory. Classically, these are all equivalent to the axiom of choice, but in the context of reverse mathematics their strengths vary: some are equivalent to ACA 0 over RCA 0 , while others are strictly weaker, and incomparable with WKL 0 . We show that there is a computable instance of F IP all of whose solutions have hyperimmune degree, and that every computable instance has a solution in every nonzero c.e. degree. In terms of other weak principles previously studied in the literature, the fomer result translates to F IP implying the omitting partial types principle (OPT). We also show that, modulo Σ 0 2 induction, F IP lies strictly below the atomic model theorem (AMT).