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Kenneth Kunen showed that thereis no non-trivial elementaryembedding of theuniverse V intoitself.We generalize thistheorem by proving that thereis no non-trivialgenericele- mentary embedding of V intoitself. The existence of certain elementary embeddings characterize various large cardinal axioms. And various authors have investigated elementary embeddings. One example isthefollowing theorem of Kunen: thereisno non-trivialelementary embedding of theuniverse V intoitself(3).One way to formulate the elementary embeddings of the theorem above is to restrictthem to definable ones. So we treatdefinable elementary embeddings throughout thispaper. We discusswhether or not,in a generic extension,thereis a non-trivialelementary embedding of the ground model V into itself.Following theproof of Woodin of Kunen's theorem (2,p. 320),we willshow thatthereisno such elementary embedding in a generic extension by a partiallyordered set. We reserve Greek lettersfor ordinal numbers, and k and k willbe used for cardinal numbers. By NSx we denote the non-stationary ideal on k. For a partiallyordered setP, we denote the class of F-names in V by Vp. If j:M ―>N is an elementary embedding where M and N are transitive models for ZFC, then the criticalpoint ofj (crit(y))is theleast ordinal a such that a <j(a), if such an a exists.An elementary embedding j :M ―+N is non- trivialifjisnot theidentitymap on M. We would liketoexpress our gratitudeto Yo Matsubara for many useful discussions.

Non-existence of generic elementary embeddings into the ground model