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

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.