Maximality Principles and Ressurection Axioms under a Laver-generic large cardinal

TL;DR

This paper explores the implications and independence results of Maximality Principles and Resurrection Axioms in the context of Laver-generic large cardinals, revealing their complex interactions and limitations within set theory.

Contribution

It introduces a classification of Laver-genericity axioms and demonstrates their independence from Maximality Principles, while also establishing conditions under which local principles follow from large cardinal assumptions.

Findings

Maximality Principle without parameters is independent of Laver-genericity axioms.

Local Maximality Principles follow from strong Laver-generic large cardinal assumptions.

Resurrection Axioms are consistent with certain Laver-generic large cardinal hypotheses.

Abstract

Set-theoretic axioms formulated in terms of existence of a Laver-generic large cardinal were introduced in [16] and studied further in [17], [18], [20]. These axioms, let us call them Laver-genericity axioms, claim the existence of a $\mathcal{P}$-Laver generic large cardinal for various classes $\mathcal{P}$-of proper or semi-proper posets, and they still vary depending on the notions of large cardinal involved, and a modification (tightness) of the definition of Laver-genericity. Laver-genericity axioms we consider here are divided into three groups depending on whether they imply that the Laver generic large cardinal $\kappa$ is either $\aleph_2=(2^{\aleph_0})^+$, or it is $\aleph_2 = 2^{\aleph_0}$, or else it is very large and $= 2^{\aleph_0}$ (see the Trichotomy Theorem (Theorem 3.5)). Many set-theoretic axioms and principles considered in the recent development of set theory follow from a Laver-genericity axiom in one of these three groups, and by this, they are placed uniformly in a global context (see Figure 3). In spite of this very strong unifying feature of the Laver genericity axioms, we show that Maximality Principle (MP) without parameters is independent over ZFC with any of the Laver-genericity axioms we consider in our present context (Theorem 4.8, Theorem 5.11). Similar independence is also shown for parameterized versions of Maximality Principles (Theorem 6.1, Theorem 6.5). In contrast to these independence results, we can show that local versions of Maximality Principle as well as versions of Resurrection Axioms including the Unbounded Resurrection Axioms of Tsaprounis follow from the existence of a tightly Laver-generic large cardinal for a strong enough notion of large cardinal (Theorem 6.6, Theorem 7.1, Theorem 7.2).