Small forcing creates neither strong nor Woodin cardinals

[bibtex key=HamkinsWoodin2000:SmallForcing]

After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin cardinals.

Universal indestructibility

[bibtex key=ApterHamkins99:UniversalIndestructibility]

From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal kappa is fully indestructible by kappa-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal.

Superdestructibility: a dual to Laver's indestructibility

[bibtex key=HamkinsShelah98:Dual]

After small forcing, any $<\kappa$-closed forcing will destroy the supercompactness, even the strong compactness, of $\kappa$.

Small forcing makes any cardinal superdestructible

[bibtex key=Hamkins98:SmallForcing]

Destruction or preservation as you like it

[bibtex key=Hamkins98:AsYouLikeIt]

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.

Fragile measurability

[bibtex key=Hamkins94:FragileMeasurability]

Lifting and extending measures; fragile measurability

[bibtex key=Hamkins94:Dissertation]

A scan of the dissertation is available:  Lifting and extending measures; fragile measurability (15 Mb)


A class of strong diamond principles

[bibtex key=Hamkins:LaverDiamond]

In the context of large cardinals, the classical diamond principle $\Diamond_\kappa$ is easily strengthened in natural ways. When $\kappa$ is a measurable cardinal, for example, one might ask that a $\Diamond_\kappa$ sequence anticipate every subset of $\kappa$ not merely on a stationary set, but on a set of normal measure one. This is equivalent to the existence of a function $\ell:\kappa\to V_\kappa$ such that for any $A\in H(\kappa^+)$ there is an embedding $j:V\to M$ having critical point $\kappa$ with $j(\ell)(\kappa)=A$. This and similar principles formulated for many other large cardinal notions, including weakly compact, indescribable, unfoldable, Ramsey, strongly unfoldable and strongly compact cardinals, are best conceived as an expression of the Laver function concept from supercompact cardinals for these weaker large cardinal notions. The resulting Laver diamond principles can hold or fail in a variety of interesting ways.