A program for the full axiom of choice

Jean-Louis Krivine

doi:10.46298/lmcs-17(3:21)2021

Abstract

The theory of classical realizability is a framework for the Curry-Howard correspondence which enables to associate a program with each proof in Zermelo-Fraenkel set theory. But, almost all the applications of mathematics in physics, probability, statistics, etc. use Analysis i.e. the axiom of dependent choice (DC) or even the (full) axiom of choice (AC). It is therefore important to find explicit programs for these axioms. Various solutions have been found for DC, for instance the lambda-term called "bar recursion" or the instruction "quote" of LISP. We present here the first program for AC.

Highlights

The theory of classical realizability is a framework for the Curry-Howard correspondence which enables to associate a program with each proof in Zermelo-Fraenkel set theory
Various solutions have been found for dependent choice (DC), for instance the lambda-term called ”bar recursion” or the instruction ”quote” of LISP
The Curry-Howard correspondence enables to associate a program with each proof in classical natural deduction

Summary

Introduction

The Curry-Howard correspondence enables to associate a program with each proof in classical natural deduction. Note that AC is realized for any r.a. associated with a set of forcing conditions (generic extension of M) In this case, there is only one proof-like term which is the greatest element 1. N a structure of Boolean model on the Boolean algebra ‫ג‬2, that we denote by M‫ג‬2 It is an elementary extension of M since the truth value of every closed formula of ZF with parameters in M is the same in M and M‫ג‬2. It is useless in the present paper, because ‫ג‬2 will be the four elements algebra, with two atoms a0, a1 which give the two trivial ultrafilters on ‫ג‬2 It is seen (Lemma 4.2) that one of them, say a0 gives a well founded model denoted by Ma0 which is the class a0N = MD.

Extensional generic extensions

The algebra A0