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In this paper, methods of second order and higher order reverse mathematicsare applied to versions of a theorem of Banach that extends theSchroeder-Bernstein theorem. Some additional results address statements inhigher order arithmetic formalizing the uncountability of the power set of thenatural numbers. In general, the formalizations of higher order principles herehave a Skolemized form asserting the existence of functionals that solveproblems uniformly. This facilitates proofs of reversals in axiom systems withrestricted choice.

Banach's theorem in higher order reverse mathematics