A Generalization of the Gap Forcing Theorem

The Main Theorem of this article asserts in part that if an extension V ⊆ V satisfies the δ approximation and covering properties, then every embedding j : V → N definable in V with critical point above δ is the lift of an embedding j ↾ V : V → N definable in the ground model V . It follows that in such extensions there can be no new weakly compact cardinals, totally indescribable cardinals, strongly unfoldable cardinals, measurable cardinals, tall cardinals, strong cardinals, Woodin cardinals, supercompact cardinals, almost huge cardinals and so on. This result generalizes the Gap Forcing Theorem of [Ham01] to a broader class of extensions and to a broader class of embeddings within those extensions.