All large-cardinal axioms not known to be inconsistent with ZFC are justified

TL;DR

This paper provides a justification for a broad class of large-cardinal axioms, including new ones, showing they are consistent with ZFC and related theories, and resolves open questions about their strength.

Contribution

It introduces a new large-cardinal property equivalent to Vopenka scheme cardinals and demonstrates the consistency of various large-cardinal axioms with ZFC.

Findings

A new large-cardinal property is defined and shown to be equivalent to Vopenka scheme cardinals.

The theory B_0(V_0) implies the existence of a Vopenka scheme cardinal with V_κ ≺ V.

The paper resolves whether certain theories imply supercompact or extendible cardinals positively.

Abstract

In other work we have outlined how, building on ideas of Welch and Roberts, one can motivate believing in the existence of supercompact cardinals. After making this observation we strove to formulate a justification for large-cardinal axioms of greater strength, and arrived at a motivation for a new large-cardinal property, which we define here and prove to be equivalent to the property of being a Vop\v{e}nka scheme cardinal. Making use of this result, one can also show that a theory $B_0(V_0)$ described in a previous paper of Victoria Marshall implies the existence of a Vop\v{e}nka scheme cardinal $\kappa$ such that $V_{\kappa} \prec V$ (and therefore, in particular, a proper class of extendible cardinals as well). Marshall left as an open question whether her theory $B_0(V_0)$, whose consistency is implied by the existence of an almost huge cardinal, implied the existence of supercompact or extendible cardinals. Here both questions are resolved positively. In the final section we give an account of how one could plausibly motivate every large-cardinal axiom not known to be inconsistent with choice while stopping short of the point of inconsistency with choice.