Destruction or Preservation As You Like It

TL;DR

This paper introduces the Gap Forcing Theorem, demonstrating that certain forcings cannot create new supercompact cardinals and can be finely controlled to preserve supercompactness, enabling tailored forcing constructions.

Contribution

It proves the Gap Forcing Theorem, develops exact preservation theorems, and establishes the `As You Like It' Theorem for controlling preservation of supercompact cardinals via forcing.

Findings

Forcing with closed posets cannot create new supercompact cardinals.

The class of preserving posets can be customized to any local definition.

Fragility and superdestructibility are shown to be orthogonal concepts.

Abstract

The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call exact preservation theorems, I use this new technology to perform a kind of partial Laver preparation, and thereby finely control the class of posets which preserve a supercompact cardinal. Eventually, I prove the `As You Like It' Theorem, which asserts that the class of ${<}\kappa$-directed closed posets which preserve a supercompact cardinal $\kappa$ can be made by forcing to conform with any pre-selected local definition which respects the equivalence of forcing. Along the way I separate completely the levels of the superdestructibility hierarchy, and, in an epilogue, prove that the notions of fragility and superdestructibility are orthogonal---all four combinations are possible.