Certain very large cardinals are not created in small forcing extensions

The 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 Σn, 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. A property which has been verified for most large cardinal axioms is that the satisfaction of the axiom for a cardinal κ cannot be created or destroyed in a small forcing extension — if V is a model of ZFC in which κ is a cardinal and P is a partial ordering with = P < κ, LC(κ) is a large cardinal axiom about κ, and G ⊆ P is V -generic, then (∗) V |= LC(κ) iff V [G] |= LC(κ) The proof of (∗) when LC(κ) is “κ is a measurable cardinal” is Levy–Solovay [LS]. 1Partially supported by NSF Grant DMS 9972257. 1