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Effective Mathematics of the Uncountable by Joel David Hamkins, Denis Hirschfeldt, Noam Greenberg

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REVERSE MATHEMATICS, COUNTABLE AND UNCOUNTABLE

RICHARD A. SHORE

Abstract Reverse mathematics analyzes the complexity of mathematical statements in terms of the strength of axiomatic systems needed to prove them. Its setting is countable mathematics and subsystems of second order arithmetic. We present a similar analysis based on (recursion-theoretic) computational complexity instead. In the countable case, this view is implicit in many of results in the area. By making it explicit and precise, we provide an alternate approach to this type of analysis for countable mathematics. It may be more intelligible to some mathematicians in that it replaces logic and proof systems with relative computability. In the uncountable case, second order arithmetic ...

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