TL;DR
This paper introduces two new notions of effective reducibility for set-theoretical statements using Ordinal Turing Machines, exploring their applications and implications for the axiom of choice.
Contribution
It defines and compares two notions of effective reducibility based on OTMs, and applies them to algebraic constructions and the axiom of choice.
Findings
Certain algebraic constructions are not effective in the OTM-sense.
Different versions of the axiom of choice are effectively equivalent under these notions.
Abstract
We introduce two notions of effective reducibility for set-theoretical statements, based on computability with Ordinal Turing Machines (OTMs), one of which resembles Turing reducibility while the other is modelled after Weihrauch reducibility. We give sample applications by showing that certain (algebraic) constructions are not effective in the OTM-sense and considerung the effective equivalence of various versions of the axiom of choice.
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Taxonomy
TopicsComputability, Logic, AI Algorithms · semigroups and automata theory · Logic, Reasoning, and Knowledge