Indestructibility of Wholeness
Volume 252 / 2021
Abstract
The Wholeness Axiom (WA) is an axiom schema that asserts the existence of a nontrivial elementary embedding from $V$ to itself. The schema is formulated in the language $\{\in ,{\bf j} \}$, where ${\bf j} $ is a unary function symbol intended to stand for the embedding. WA consists of an Elementarity schema that asserts ${\bf j} $ is an elementary embedding, a Critical Point axiom that asserts existence of a least ordinal moved, and a schema ${\rm Separation}_\mathbf{j} $ that asserts Separation holds for all instances of ${\bf j} $-formulas. The theory $\rm ZFC +WA $ has been proposed in the author’s earlier papers as a natural axiomatic extension of $\rm ZFC $ to account for most of the known large cardinals. In this paper we offer evidence for the naturalness of this theory by showing that it is, like ZFC itself, indestructible by set forcing. We show first that if $\kappa $ is the critical point of the embedding, then $\rm ZFC +WA $ is preserved by any notion of forcing that belongs to $V_\kappa $. This step is nontrivial because to prove ${\rm Separation}_\mathbf{j} $ holds in the forcing extension after lifting the embedding, it is necessary to incorporate ${\bf j} $ into the definition of the forcing relation. Then for arbitrary notions of forcing, we introduce a different technique of lifting that lifts one of the original embedding’s applicative iterates.
