We demonstrate that the technology of Radin forcing can be used to transfer compactness properties at a weakly inaccessible but not strong limit cardinal to a strongly inaccessible cardinal.
As an application, relative to the existence of large cardinals, we construct a model of set theory in which there is a cardinal $κ$ that is $n$-$d$-stationary for all $n\in ω$ but not weakly compact. This is in sharp contrast to the situation in the constructible universe $L$, where $κ$ being $(n+1)$-$d$-stationary is equivalent to $κ$ being $\mathbfΠ^1_n$-indescribable. We also show that it is consistent that there is a cardinal $κ\leq 2^ω$ such that $P_κ(λ)$ is $n$-stationary for all $λ\geq κ$ and $n\in ω$, answering a question of Sakai.
We give the definition of Woodin for strong compactness cardinals, the Woodinised version of strong compactness, and we prove an analogue of Magidor's identity crisis theorem for the first strongly compact cardinal.
We introduce the notion of weakly extendible cardinals and show that these cardinals are characterized in terms of weak compactness of second order logic. The consistency strength and largeness of weakly extendible cardinals are located strictly between that of strongly unfoldable (i.e. shrewd) cardinals, and strongly uplifting cardinals. Weak compactness of many other logics can be connected to certain variants of the notion of weakly extendible cardinals. We also show that, under V=L, a cardinal $κ$ is the weak compactness number of ${\cal L}^{\aleph_0,II}_{stat,κ,ω}$ if and only if it is th…
We consider compactness characterizations of large cardinals. Based on results of Benda \cite{b-sccomp}, we study compactness for omitting types in various logics. In $\bL_{κ, κ}$, this allows us to characterize any large cardinal defined in terms of normal ultrafilters, and we also analyze second-order and sort logic. In particular, we give a compactness for omitting types characterization of huge cardinals, which have consistency strength beyond Vopěnka's Principle.
If $κ$ is regular and $2^{<κ}\leqκ^+$, then the existence of a weakly presaturated ideal on $κ^+$ implies $\square^*_κ$. This partially answers a question of Foreman and Magidor about the approachability ideal on $ω_2$. As a corollary, we show that if there is a presaturated ideal $I$ on $ω_2$ such that $\mathcal{P}(ω_2)/I$ is semiproper, then CH holds. We also show some barriers to getting the tree property and a saturated ideal simultaneously on a successor cardinal from conventional forcing methods.
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we characterize strongly compact cardinals in terms of supercompactness: a strongly compact cardinal is either supercompact or a limit of supercompact cardinals. Assuming the Ultrapower Axiom and the GCH, we also prove a local result that roughly states that every countably complete ultrafilter factors as a finite iteration of supercompact ultrafilters.
The relationship between the large cardinal notions of strong compactness and supercompactness cannot be determined under the standard ZFC axioms of set theory. Under a hypothesis called the Ultrapower Axiom, we prove that the notions are equivalent except for a class of counterexamples identified by Menas. This is evidence that strongly compact and supercompact cardinals are equiconsistent.
We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on $κ$, if $κ$ is weakly compact, then $\diamondsuit(κ)$ holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently …
The weakly compact reflection principle $\text{Refl}_{\text{wc}}(κ)$ states that $κ$ is a weakly compact cardinal and every weakly compact subset of $κ$ has a weakly compact proper initial segment. The weakly compact reflection principle at $κ$ implies that $κ$ is an $ω$-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that $κ$ is $(ω+1)$-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at $κ$ then there is a forcing extension preserving this in which $κ$ is the least $ω$-weakly compact cardinal.…
We show that the tree property, stationary reflection and the failure of approachability at $κ^{++}$ are consistent with $\mathfrak{u}(κ) = κ^+ < 2^κ$, where $κ$ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if $λ$ is a regular cardinal, then stationary reflection at $λ^+$ is indestructible under all $λ$-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).
We establish the consistency of the failure of the diamond principle on a cardinal $κ$ which satisfies a strong simultaneous reflection property. The result is based on an analysis of Radin forcing, and further leads to a characterization of weak compactness of $κ$ in a Radin generic extension.
In the absence of the Axiom of Choice, the "small" cardinal $ω_1$ can exhibit properties more usually associated with large cardinals, such as strong compactness and supercompactness. For a local version of strong compactness, we say that $ω_1$ is $X$-strongly compact (where $X$ is any set) if there is a fine, countably complete measure on $\mathcal{P}_{ω_1}(X)$. Working in $\mathsf{ZF} + \mathsf{DC}$, we prove that the $\mathcal{P}(ω_1)$-strong compactness and $\mathcal{P}(\mathbb{R})$-strong compactness of $ω_1$ are equiconsistent with $\mathsf{AD}$ and $\mathsf{AD}_\mathbb{R} + \mathsf{DC}$…
We try to build, provably in ZFC, for a first order T a model in which any isomorphism between two Boolean algebras is definable. The problem, compared to [Sh:384], is with pseudo-finite Boolean algebras. A side benefit is that we do not use Skolem function (which do not matter for proving compactness of logics but still are of interest). Let lambda be 2^mu if regular and its successor otherwise. Model theoretically we investigate notions of bigness of types, usually those are ideals of the set of formulas in a model, definable in appropriate sense. We build a model of cardinality lambda^plus …
Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem
Tel Aviv University（特拉维夫大学）；Carnegie Mellon University（卡内基梅隆大学）；UC Berkeley（加州大学伯克利分校）；Hebrew University of Jerusalem（耶路撒冷希伯来大学）；CUNEF Universidad（CUNEF大学）；Universität Hamburg（汉堡大学）
Motivated by a characterization of weakly compact cardinals due to Todorcevic, we introduce a new cardinal characteristic, the C-sequence number, which can be seen as a measure of the compactness of a regular uncountable cardinal. We prove a number of ZFC and independence results about the C-sequence number and its relationship with large cardinals, stationary reflection, and square principles. We then introduce and study the more general C-sequence spectrum and uncover some tight connections between the C-sequence spectrum and the strong coloring principle U(...), introduced in Part I of this…
For an inaccessible cardinal $κ$, the super tree property (ITP) at $κ$ holds if and only if $κ$ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor of the limit of $ω$ many supercompact cardinals. Then we show that it can consistently hold at $\aleph_{ω+1}$. We also consider a stronger principle, ISP, and certain weaker variations of it. We determine which level of ISP can hold at a successor of a singular. These results fit in the broad program of testing how much compactness can exist in the universe, and obtaining…
The inner model problem for supercompact cardinals, one of the central open problems in modern set theory, asks whether there is a canonical model of set theory with a supercompact cardinal. The problem is closely related to the more precise question of the equiconsistency of strongly compact cardinals and supercompact cardinals. This dissertation approaches these two problems abstractly by introducing a principle called the Ultrapower Axiom which is expected to hold in all known canonical models of set theory. By investigating the consequences of the Ultrapower Axiom under the hypothesis that…
We discuss the rainbow Ramsey theorems at limit cardinals and successors of singular cardinals, addressing some questions in \cite{MR2354904} and \cite{MR2902230}. In particular, we show for inaccessible $κ$, $κ\to^{poly}(κ)^2_{2-bdd}$ does not characterize weak compactness and for singular $κ$, $\mathrm{GCH}+\square_κ$ implies $κ^+\not\to^{poly} (η)^2_{<κ-bdd}$ for any $η\geq cf(κ)^+$ and $κ^+\to^{poly} (ν)^2_{<κ-bdd}$ for any $ν<cf(κ)^+$. We also provide a simplified construction of a model for $ω_2\not\to^{poly} (ω_1)^2_{2-bdd}$ originally constructed in \cite{MR2902230} and show the witnes…
