Extender-based Radin forcing

We define extender sequences, generalizing measure sequences of Radin forcing.

Using the extender sequences, we show how to combine the Gitik-Magidor forcing for adding many Prikry sequences with Radin forcing.

We show that this forcing satisfies a Prikry-like condition, destroys no cardinals, and has a kind of properness.

Depending on the large cardinals we start with, this forcing can blow the power of a cardinal together with changing its cofinality to a prescribed value. It can even blow the power of a cardinal while keeping it regular or measurable.

     

Indestructibility and stationary reflection

If κ < λ are such that κ is a strong cardinal whose strongness is indestructible under κ ‐strategically closed forcing and λ is weakly compact, then we show that A = {δ < κ | δ is a non‐weakly compact Mahlo cardinal which reflects stationary sets} must be unbounded in κ. This phenomenon, however, need not occur in a universe with relatively few large cardinals. In particular, we show how to construct a model where no cardinal is supercompact up to a Mahlo cardinal in which the least supercompact cardinal κ is also the least strongly compact cardinal, κ 's strongness is indestructible under κ ‐strategically closed forcing, κ 's supercompactness is indestructible under κ ‐directed closed forcing not adding any new subsets of κ, and δ is Mahlo and reflects stationary sets iff δ is weakly compact. In this model, no strong cardinal δ < κ is indestructible under δ ‐strategically closed forcing. It therefore follows that it is relatively consistent for the least strong cardinal κ whose strongness is indestructible under κ ‐strategically closed forcing to be the same as the least supercompact cardinal, which also has its supercompactness indestructible under κ ‐directed closed forcing not adding any new subsets of κ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Universal partial indestructibility and strong compactness

For any ordinal δ, let λ_δ be the least inaccessible cardinal above δ. We force and construct a model in which the least supercompact cardinal κ is indestructible under κ‐directed closed forcing and in which every measurable cardinal δ < κ is < λ_δ strongly compact and has its < λ_δ strong compactness indestructible under δ‐directed closed forcing of rank less than λ_δ. In this model, κ is also the least strongly compact cardinal. We also establish versions of this result in which κ is the least strongly compact cardinal but is not supercompact. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Strong Cardinals can be Fully Laver Indestructible

We prove three theorems which show that it is relatively consistent for any strong cardinal κ to be fully Laver indestructible under κ‐directed closed forcing.     

More on the Least Strongly Compact Cardinal

We show that it is consistent, relative to a supercompact limit of supercompact cardinals, for the least strongly compact cardinal k to be both the least measurable cardinal and to be > 2^k supercompact.     

Weakly measurable cardinals

In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal κ is weakly measurable if for any collection $\mathcal {A}$ containing at most κ⁺ many subsets of κ, there exists a nonprincipal κ‐complete filter on κ measuring all sets in $\mathcal {A}$. Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if κ is measurable, then we can make its weak measurability indestructible by the forcing Add(κ, η) for any η while forcing the GCH to hold below κ. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Power function on stationary classes

We show that under certain large cardinal requirements there is a generic extension in which the power function behaves differently on different stationary classes. We achieve this by doing an Easton support iteration of the Radin on extenders forcing.     

Locally-generic Boolean algebras and cardinal sequences

Given a sequence of cardinals of length less than , with each cardinal in the sequence being either or , we construct a -poset (see Defnition 1 below) which, with a natural topology, becomes a locally-compact, Hausdorff, scattered space with cardinal sequence . The algebra of the clopen subsets of its one-point compactification yields, in turn, a superatomic Boolean algebra with as its cardinal sequence. The posets are locallygeneric, that is, they are constructed generically over countable sets. This gives them additional chain properties, specially under Under Martin's Axiom, the construction allows any cardinals in the sequence, provided it has length Finally, we modify a forcing argument of Baumgartner-Shelah [B-S], to build -posets for any given cardinal sequence of length with each cardinal in the sequence being either or . Received September 2, 1998; accepted in final form September 13, 2001.     

Indestructible strong compactness and level by level inequivalence

If are such that δ is indestructibly supercompact and γ is measurable, then it must be the case that level by level inequivalence between strong compactness and supercompactness fails. We prove a theorem which points to this result being best possible. Specifically, we show that relative to the existence of cardinals such that κ₁ is λ‐supercompact and λ is inaccessible, there is a model for level by level inequivalence between strong compactness and supercompactness containing a supercompact cardinal in which κ’s strong compactness, but not supercompactness, is indestructible under κ‐directed closed forcing. In this model, κ is the least strongly compact cardinal, and no cardinal is supercompact up to an inaccessible cardinal.     

The Necessary Maximality Principle for c. c. c. forcing is equiconsistent with a weakly compact cardinal

The Necessary Maximality Principle for c. c. c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the consistency strength of the proper forcing axiom

In recent work, the second author extended combinatorial principles due to Jech and Magidor that characterize certain large cardinal properties so that they can also hold true for small cardinals. For inaccessible cardinals, these modifications have no effect, and the resulting principles still give the same characterization of large cardinals. We prove that the proper forcing axiom PFA implies these principles hold for ω₂. Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary.     

Supercompactness and level by level equivalence are compatible with indestructibility for strong compactness

It is known that if are such that κ is indestructibly supercompact and λ is 2^λ supercompact, then level by level equivalence between strong compactness and supercompactness fails. We prove a theorem which points towards this result being best possible. Specifically, we show that relative to the existence of a supercompact cardinal, there is a model for level by level equivalence between strong compactness and supercompactness containing a supercompact cardinal κ in which κ’s strong compactness is indestructible under κ-directed closed forcing. The author’s research was partially supported by PSC-CUNY Grant 66489-00-35 and a CUNY Collaborative Incentive Grant.     

Universal forcing notions and ideals

Our main result states that a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the σ-ideals determined by those universality parameters. Both authors acknowledge support from the United States-Israel Binational Science Foundation (Grant no. 2002323). Also, we would like to thank the referee for valuable comments and suggestions concerning the exposition of the paper. This is publication 845 of the second author     

Removing Laver functions from supercompactness arguments

We show how the use of a Laver function in the proof of the consistency, relative to the existence of a supercompact cardinal, of both the Proper Forcing Axiom and the Semiproper Forcing Axiom can be eliminated via the use of lottery sums of the appropriate partial orderings. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A consistency result on cardinal sequences of scattered Boolean spaces

We prove that if GCH holds and τ = 〈κ_α : α < η 〉 is a sequence of infinite cardinals such that κ_α ≥ |η | for each α < η, then there is a cardinal‐preserving partial order that forces the existence of a scattered Boolean space whose cardinal sequence is τ. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Weak saturation of ideals on Pκ(λ)

We show that if κ is an infinite successor cardinal, and λ > κ a cardinal of cofinality less than κ satisfying certain conditions, then no (proper, fine, κ‐complete) ideal on P_κ(λ) is weakly λ⁺‐saturated. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Can a large cardinal be forced from a condition implying its negation?

In this note, we provide an affirmative answer to the title question by giving two examples of cardinals satisfying conditions implying they are non-Rowbottom which can be turned into Rowbottom cardinals via forcing. In our second example, our cardinal is also non-Jonsson.

     

The Bounded Axiom A Forcing Axiom

We introduce the Bounded Axiom A Forcing Axiom (BAAFA). It turns out that it is equiconsistent with the existence of a regular ∑₂‐correct cardinal and hence also equiconsistent with BPFA. Furthermore we show that, if consistent, it does not imply the Bounded Proper Forcing Axiom (BPFA) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A FORMALISM FOR SOME CLASS OF FORCING NOTIONS

We introduce a class of forcing notions, called forcing notions of type S, which contains among other Sacks forcing, Prikry-Silver forcing and their iterations and products with countable supports. We construct and investigate some formalism suitable for this forcing notions, which allows all standard tricks for iterations or products with countable supports of Sacks forcing. On the other hand it does not involve internal combinatorial structure of conditions of iterations or products. We prove that the class of forcing notions of type S is closed under products and certain iterations with countable supports.     

Forcing the Least Measurable to Violate GCH

Starting with a model for “GCH + k is k⁺ supercompact”, we force and construct a model for “k is the least measurable cardinal + 2^k = K⁺”. This model has the property that forcing over it with Add(k,k⁺⁺) preserves the fact k is the least measurable cardinal.