Tag Archives: forcing

Indestructibility and the level-by-level agreement between strong compactness and supercompactness

Posted on by
Reply

[bibtex key=ApterHamkins2002:LevelByLevel]

Can a supercompact cardinal $\kappa$ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above $\kappa$, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or less level-by-level agreement, such as requiring it only on measure one sets, then yes, it can.

Indestructible weakly compact cardinals and the necessity of supercompactness for certain proof schemata

Posted on by
Reply

[bibtex key=ApterHamkins2001:IndestructibleWC]

We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.

Unfoldable cardinals and the GCH

Posted on by
Reply

[bibtex key=Hamkins2001:UnfoldableCardinals]

Introducing unfoldable cardinals last year, Andres Villaveces ingeniously extended the notion of weak compactness to a larger context, thereby producing a large cardinal notion, unfoldability, with some of the feel and flavor of weak compactness but with a greater consistency strength. Specifically, $\kappa$ is $\theta$-unfoldable when for any transitive structure $M$ of size $\kappa$ that contains $\kappa$ as an element, there is an elementary embedding $j:M\to N$ with critical point $\kappa$ for which $j(\kappa)$ is at least $\theta$. Define that $\kappa$ is fully unfoldable, then, when it is $\theta$-unfoldable for every $\theta$. In this paper I show that the embeddings associated with these unfoldable cardinals are amenable to some of the same lifting techniques that apply to weakly compact embeddings, augmented with methods from the strong cardinal context. Using these techniques, I show by set-forcing over any model of ZFC that any given unfoldable cardinal $\kappa$ can be made indestructible by the forcing to add any number of Cohen subsets to $\kappa$. This result contradicts expectations to the contrary that class forcing would be required.

The wholeness axioms and $V=\rm HOD$

Posted on by
Reply

[bibtex key=Hamkins2001:WholenessAxiom]

The Wholeness Axioms, proposed by Paul Corazza, axiomatize the existence of an elementary embedding $j:V\to V$. Formalized by augmenting the usual language of set theory with an additional unary function symbol j to represent the embedding, they avoid the Kunen inconsistency by restricting the base theory ZFC to the usual language of set theory. Thus, under the Wholeness Axioms one cannot appeal to the Replacement Axiom in the language with j as Kunen does in his famous inconsistency proof. Indeed, it is easy to see that the Wholeness Axioms have a consistency strength strictly below the existence of an $I_3$ cardinal. In this paper, I prove that if the Wholeness Axiom $WA_0$ is itself consistent, then it is consistent with $V=HOD$. A consequence of the proof is that the various Wholeness Axioms $WA_n$ are not all equivalent. Furthermore, the theory $ZFC+WA_0$ is finitely axiomatizable.

The lottery preparation

Posted on by
Reply

[bibtex key=Hamkins2000:LotteryPreparation]

The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal $\kappa$, for example, becomes fully indestructible by $\kappa$-directed closed forcing; a strong cardinal $\kappa$ becomes indestructible by less-than-or-equal-$\kappa$-strategically closed forcing; and a strongly compact cardinal $\kappa$ becomes indestructible by, among others, the forcing to add a Cohen subset to $\kappa$, the forcing to shoot a club $C$ in $\kappa$ which avoids the measurable cardinals and the forcing to add various long Prikry sequences. The lottery preparation works best when performed after fast function forcing, which adds a new completely general kind of Laver function for any large cardinal, thereby freeing the Laver function concept from the supercompact cardinal context.

Changing the heights of automorphism towers

Posted on by
2

[bibtex key=HamkinsThomas2000:ChangingHeights]

If $G$ is a centreless group, then $\tau(G)$ denotes the height of the automorphism tower of $G$. We prove that it is consistent that for every cardinal $\lambda$ and every ordinal $\alpha < \lambda$, there exists a centreless group $G$ such that (a) $\tau(G) = \alpha$; and (b) if $\beta$ is any ordinal such that $1 \leq \beta < \lambda$, then there exists a notion of forcing $P$, which preserves cofinalities and cardinalities, such that $\tau(G) = \beta$ in the corresponding generic extension $V^{P}$.

Small forcing creates neither strong nor Woodin cardinals

Posted on by
Reply

[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.

Gap forcing: generalizing the Lévy-Solovay theorem

Posted on by
Reply

[bibtex key=Hamkins99:GapForcingGen]

The landmark Levy-Solovay Theorem 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.

Universal indestructibility

Posted on by
Reply

[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

Posted on by
1

[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

Posted on by
Reply

[bibtex key=Hamkins98:SmallForcing]

Destruction or preservation as you like it

Posted on by
Reply

[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.

Canonical seeds and Prikry trees

Posted on by
Reply

[bibtex key=Hamkins97:Seeds]

Applying the seed concept to Prikry tree forcing $\mathbb{P}_\mu$, I investigate how well $\mathbb{P}_\mu$ preserves the maximality property of ordinary Prikry forcing and prove that $\mathbb{P}_\mu$ Prikry sequences are maximal exactly when $\mu$ admits no non-canonical seeds via a finite iteration.  In particular, I conclude that if $\mu$ is a strongly normal supercompactness measure, then $\mathbb{P}_\mu$ Prikry sequences are maximal, thereby proving, for a large class of measures, a conjecture of W. H. Woodin’s.

Fragile measurability

Posted on by
Reply

[bibtex key=Hamkins94:FragileMeasurability]

Lifting and extending measures; fragile measurability

Posted on by
Reply

[bibtex key=Hamkins94:Dissertation]

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