Gap Forcing

After forcing which admits a very low gap—and this includes many of the forcing iterations, such as the Laver preparation, which are commonly found in the large cardinal context—every embedding j : V [G] → M [j(G)] in the extension which satisfies a mild closure condition is the lift of an embedding j : V → M in the ground model. In particular, every ultrapower embedding in the extension lifts an embedding from the ground model and every measure in the extension which concentrates on a set in the ground model extends a measure in the ground model. It follows that gap forcing cannot create new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on. When forcing with a large cardinal κ whose largeness is witnessed by the existence of a certain kind of elementary embedding j : V → M , one often proceeds by lifting the embedding to the forcing extension j : V [G] → M [j(G)] and arguing that the lifted embedding witnesses that κ retains the desired large cardinal property in V [G]. Surprisingly, for a large class of forcing notions which includes many of the reverse Easton iterations one commonly finds in the large cardinal literature, such as Silver forcing [Sil71] and the Laver preparation [Lav78], the converse also holds. Namely, after such forcing every embedding j : V [G] → M [j(G)] satisfying a mild closure condition is the lift of an embedding in the ground model. That is, M ⊆ V and j ↾V : V → M is definable in V . Since these restricted embeddings witness the large cardinal property on κ in V , it follows that the forcing cannot have created new measurable cardinals, strong cardinals, supercompact cardinals, and so on. The class of forcing notions for which this theorem holds pervades the large cardinal literature. All that is required is that the forcing admit a gap at some δ below the cardinal κ in question in the sense that the forcing factors as P ∗ Q̇ where My research has been supported in part by grants from the PSC-CUNY Research Foundation and from the Japan Society for the Promotion of Science. I would like to thank my gracious hosts here at Kobe University in Japan for their generous hospitality.