A program for the full axiom of choice

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.

A first-order completeness result about characteristic Boolean algebras in classical realizability

“…Miquel then adapted this classical realizability to higher-order arithmetic and explored its connections with forcing [10]. Work on interpreting reasoning that uses the full axiom of choice is ongoing [9].…”

“…One of its remarkable combinatorial properties is that in this model, there is a whole atomless Boolean algebra embedded in the poset of the cardinalities between the countable and the continuum; and the way this property was obtained was by first making the characteristic Boolean algebra is atomless, and then embedding it in this poset. As an other example, Krivine's construction of a particular classical realizability model that satisfies the axiom of choice [9] depends crucially on the ability to reliably force a realizability model's characteristic Boolean algebra to be isomorphic to any given finite Boolean algebra with at least 2 elements (in that case, the Boolean algebra with 4 elements).…”