Banach’s theorem in higher-order reverse mathematics

Abstract

In this paper, methods of second-order and higher-order reverse mathematics are applied to versions of a theorem of Banach that extends the Schröder–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.