A 5-quantifier (\in,=)-expression ZF-equivalent to the Axiom of Choice

TL;DR

This paper introduces a concise (,)-sentence with only five quantifiers that is equivalent to the Axiom of Choice, reducing the known minimal quantifier count needed for such equivalence.

Contribution

It presents the first (,)-sentence with five quantifiers equivalent to AC, narrowing the gap from previous minimums and advancing understanding of logical complexity of AC.

Findings

A 5-quantifier (,)-sentence implies AC.

AC implies the 5-quantifier sentence.

The minimal quantifier count for such equivalence is now narrowed to 4.

Abstract

In this paper I present an (\in, =)-sentence, AC**, with only 5 quantifiers, that logically implies the axiom of choice, AC. Furthermore, using a weak fragment of ZF set theory, I prove that AC implies AC**. Up to now 6 quantifiers were the minimum and 3 quantifiers don't suffice since all 3-quantifier (\in, =)-sentences are decided in a weak fragment of ZF set theory. Thus the gap is reduced to the undecided case of a 4 quantifier sentence ZF-equivalent to AC.