Incompatibility of generic hugeness principles

TL;DR

This paper demonstrates that the weakest forms of Foreman's minimal generic hugeness axioms are mutually incompatible on neighboring cardinals and cannot be achieved through standard forcing methods.

Contribution

It establishes the fundamental incompatibility of certain minimal generic hugeness axioms and shows their limitations within conventional forcing frameworks.

Findings

Weakest versions of the axioms cannot hold simultaneously on adjacent cardinals.

Standard forcing techniques cannot produce models satisfying these axioms.

The results highlight inherent limitations in the axiomatization of generic hugeness.

Abstract

We show that the weakest versions of Foreman's minimal generic hugeness axioms cannot hold simultaneously on adjacent cardinals. Moreover, conventional forcing techniques cannot produce a model of one of these axioms.