On the effectiveness of the Schröder-Bernstein theorem

Abstract

Abstract. The effectiveness of the classical equivalence theorem of Schröder and Bernstein is investigated using the tools of recursion theory. We prove one result which generalizes all the effective versions of the Schröder-Bernstein theorem which occur in the literature. In contrast, we show that Banach's strengthening of the Schröder-Bernstein theorem fails to be effective.Introduction.

Banach's theorem in higher order reverse mathematics

“…A brief history of Banach's Theorem and the Schröder-Bernstein theorem is given by Remmel [12,Introduction]. An analysis of Banach's Theorem for subsets of N, using subsystems of second order arithmetic, appears in Hirst's thesis [6, §3.2] and a related article [7].…”