A Mid Version of Hamkins' Maximality Principle

TL;DR

This paper introduces a streamlined approach to maximality principles in set theory and proposes an intermediate principle linked to the existence of a weakly compact cardinal with specific elementary substructure properties.

Contribution

It provides simplified proofs of existing maximality principles and formulates a new intermediate principle with equiconsistency results involving weakly compact cardinals.

Findings

Streamlined proofs of maximality principles by Hamkins and Woodin.

Introduction of an intermediate maximality principle.

Equiconsistency with a weakly compact cardinal and $V_{\kappa} ext{ is elementary in }V$.

Abstract

We present new, streamlined proofs of certain maximality principles studied by Hamkins and Woodin. Moreover, we formulate an intermediate maximality principle, which is shown here to be equiconsistent with the existence of a weakly compact cardinal $\kappa$ such that $V_{\kappa}\prec V$.