Paradox on the Countable Axiom of Choice

TL;DR

This paper presents a paradox demonstrating that using the Countable Axiom of Choice and the Axiom of Dependent Choice within Bishop's constructive mathematics leads to inconsistency, challenging the compatibility of these principles.

Contribution

It introduces a paradoxical example showing the inconsistency of combining the Countable Axiom of Choice with constructive mathematics principles.

Findings

The paradox invalidates the consistency of certain choice principles in constructive math.

The Countable Axiom of Choice implies the Axiom of Dependent Choice.

Constructive mathematics cannot safely incorporate these choice axioms without inconsistency.

Abstract

Bishop's constructive mathematics school rejects the Law of Excluded Middle, but instead vastly makes use of weaker versions of the Choice. In this paper we pioneer an example, which shows that this road is not consistent, as our example provides a paradox. Therefore, rejecting the Law of Excluded Middle, and as an alternative using the Countable Axiom of Choice and the Axiom of Dependent Choice, still does not create a consistent structure. Actually, constructively; the Countable Axiom of Choice is an implication of the Axiom of Dependent Choice.