The maximum principle in forcing and the axiom of choice

TL;DR

This paper demonstrates that the maximum principle in forcing is equivalent to the axiom of choice and examines its validity across different partial orders in the Basic Cohen model, revealing nuanced differences.

Contribution

It establishes the equivalence between the maximum principle in forcing and the axiom of choice, and analyzes its applicability in various models.

Findings

Maximum principle is equivalent to the axiom of choice.

The maximum principle holds for only one of three similar partial orders.

Differences in the maximum principle's validity depend on the specific partial order.

Abstract

In this paper we prove that the maximum principle in forcing is equivalent to the axiom of choice. The maximum principle is the property of forcing: p ||- exists x theta(x) iff for some name tau p ||- theta(tau). We also look at three similar partial orders in the Basic Cohen model for the failure of the axiom of choice. We show that despite their apparent similarity the maximum principle holds for only one of the three.