Certain very large cardinals are not created in small forcing extensions

Abstract

AbstractThe large cardinal axioms of the title assert, respectively, the existence of a nontrivial elementary embedding j:Vλ→Vλ, the existence of such a j which is moreover Σn1, and the existence of such a j which extends to an elementary j:Vλ+1→Vλ+1. It is known that these axioms are preserved in passing from a ground model to a small forcing extension. In this paper the reverse directions of these preservations are proved. Also the following is shown (and used in the above proofs in place of using a standard fact): if V is a model of ZFC and V[G] is a P-generic forcing extension of V, then in V[G], V is definable using the parameter Vδ+1, where δ=P=+