Gap forcing

In this paper, I generalize the landmark Levy-Solovay Theorem [LevSol67], w hich limits the kind of large cardinal embeddings that can exist in a small forcing extension, to a broad new class of forcing notions, a class that i ncludes many of the forcing iterations most commonly found in the large car dinal literature. After such forcing, the fact is that every embedding sati sfying a mild closure requirement lifts an embedding from the ground model. Such forcing, consequently, can create no new weakly compact cardinals, me asurable cardinals, strong cardinals, Woodin cardinals, strongly compact ca rdinals, supercompact cardinals, almost huge cardinals, or huge cardinals, and so on.