If We Add Axiom of Choice to Constructive Analysis, We Get Classical Arithmetic: An Exercise in Reverse Constructive Mathematics

Authors

Olga Kosheleva, The University of Texas at El PasoFollow
Vladik Kreinovich, The University of Texas at El PasoFollow

Publication Date

12-1-2023

Comments

Technical Report: UTEP-CS-23-70

Abstract

A recent paper in Bulletin of Symbolic Logic reminded that the Axiom of Choice is, in general, false in constructive analysis. This result is an immediate consequence of a theorem -- first proved by Tseytin -- that every computable function is continuous. In this paper, we strengthen the result about the Axiom of Choice by proving that this axiom is as non-constructive as possible: namely, that if we add this axiom to constructive analysis, then we get full classical arithmetic.