Reverse mathematics and uniformity in proofs without excluded middle

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 $\Pi^1_2$ sentence of a certain form is provable using E-HA${}^\omega$ 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 \textit{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.Comment: Accepted, Notre Dame Journal of Formal Logi

Banach's theorem in higher order reverse mathematics

In this paper, methods of second order and higher order reverse mathematics are applied to versions of a theorem of Banach that extends the Schroeder-Bernstein theorem. Some additional results address statements in higher order arithmetic formalizing the uncountability of the power set of the natural numbers. In general, the formalizations of higher order principles here have a Skolemized form asserting the existence of functionals that solve problems uniformly. This facilitates proofs of reversals in axiom systems with restricted choice.Comment: Version: count-20230393arxiv.tex adding publication info in header and listing modification

Comparing the strength of diagonally non-recursive functions in the absence of Σ02 induction

We prove that the statement there is a k such that for every f there is a k-bounded diagonally non-recursive function relative to f does not imply weak K\ onig\u27s lemma over RCA0+BΣ02. This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that every k-bounded diagonally non-recursive function computes a 2-bounded diagonally non-recursive function may fail in the absence of IΣ02

Reverse Mathematics, Computability, and Partitions of Trees

We examine the reverse mathematics and computability theory of a form of Ramsey’s theorem in which the linear n-tuples of a binary tree are colored