Gap forcing: Generalizing the Lévy-Solovay theorem

The Levy-Solovay Theorem [LevSol67] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on. Large cardinal set theorists today generally look upon small forcing—that is, forcing with a poset P of cardinality less than whatever large cardinal κ is under consideration—as benign. This outlook is largely due to the Levy-Solovay theorem [LevSol67], which asserts that small forcing does not affect the measurability of any cardinal. (Specifically, the theorem says that if a forcing notion P has size less than κ, then the ground model V and the forcing extension V P agree on the measurability of κ in a strong way: the ground model measures on κ all generate as filters measures in the forcing extension, the corresponding ultrapower embeddings lift uniquely from the ground model to the forcing extension and all the measures and ultrapower embeddings in the forcing extension arise in this way.) Since the Levy-Solovay argument generalizes to the other large cardinals whose existence is witnessed by certain kinds of measures or ultrapowers, such as strongly compact cardinals, supercompact cardinals, almost huge cardinals and so on, one is led to the broad conclusion that small forcing is harmless; one can understand the measures in a small forcing extension by their relation to the measures existing already in the ground model. Here in this Communication I would like to announce a generalization of the Levy-Solovay Theorem to a broad new class of forcing notions. 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.