Joel David Hamkins, Mirna Džamonja
If kappa is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which Diamond_kappa(REG) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.
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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-->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 …
We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $θ$-supercompact, for any desired $θ$. In addition, we prove several global results showing how the entire class of weakly compact cardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly $θ_κ$-supercompact cardinals $κ$, for nearly any desired function $κ\mapstoθ_κ$. These results answer several questions that had been open in the literature and ex…
Patterns of Structural Reflection in the large-cardinal hierarchy
该研究揭示首个可测基数以下大基数层级的结构反射新模式，刻画强不可展开与微妙基数，引入$C^{(n)}$-强不可展开基数并证明其形成自然层级，证实大基数低区域与更高区域结构完全类似。
We introduce the strongly uplifting cardinals, which are equivalently characterized, we prove, as the superstrongly unfoldable cardinals and also as the almost hugely unfoldable cardinals, and we show that their existence is equiconsistent over ZFC with natural instances of the boldface resurrection axiom, such as the boldface resurrection axiom for proper forcing.
Large cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained that way (`Unfoldable cardinals') behave as a `boundary' between properties consistent with `V=L' and existence of indiscernibles. We also provide an `embedding characterisation' of the unfoldable cardinals and study their preservation and destruction by various different forcings.
We show that the notions of "strongly unfoldable cardinals", introduced by Villaveces in his model-theoretic studies of models of set theory, and "shrewd cardinals", introduced by Rathjen in a proof-theoretic context, coincide. We then proceed by using ideas from the proof of this equivalence to establish the existence of "ordinal anticipating Laver functions" for strong unfoldability. With the help of these functions, we show that the principle $\Diamond_κ(\mathrm{Reg})$ holds at every strongly unfoldable cardinal $κ$ with the property that there exists a subset $z$ of $κ$ such that every sub…
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 l:kappa-->V_kappa such that for any A in H(kappa+) there is an embedding j:V-->M having critical point kappa with j(l)(kappa)=A. This and similar principles formulated for many other large cardinal notions, including weakly compact, i…
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 isolate \emph{the approximating diamond principles}, which are consequences of the diamond principle at an inaccessible cardinal. We use these principles to find new methods for negating the diamond principle at large cardinals. Most notably, we demonstrate, using Gitik's overlapping extenders forcing, a new method to get the consistency of the failure of the diamond principle at a large cardinal $θ$ without changing cofinalities or adding fast clubs to $θ$. In addition, we show that the approximating diamond principles necessarily hold at a weakly compact cardinal. This result, combined wi…
Starting from suitable large cardinals, we force the failure of (weak) diamond at the least inaccessible cardinal. The result improves an unpublished theorem of Woodin and a recent result of Ben-Neria, Garti and Hayut.
Weak diamond and pcf theory
本文利用弱钻石原则推导出pcf(𝔞)的基数上界，并证明在温和假设下许多奇异基数ℵ_δ满足2^{ℵ_δ}<ℵ_{(|δ|^{+3})}。
We show that generalized eventually narrow sequences on a strongly inaccessible cardinal $κ$ are preserved under the Cummings-Shaleh non-linear iterations of the higher Hechler forcing on $κ$. Moreover assuming GCH, $κ^{<κ}=κ$, we show that: (1) if $κ$ is strongly unfoldable, $κ^+\leqβ=\hbox{cf}(β)\leq \hbox{cf}(δ)\leqδ\leqμ$ and $\hbox{cf}(μ)>κ$,then there is a cardinal preserving generic extension in which $$\mathfrak{s}(κ)=κ^+\leq\mathfrak{b}(κ)=β\leq\mathfrak{d}(κ)=δ\leq 2^κ=μ.$$ (2) if $κ$ is strongly inaccessible, $λ>κ^+$, then in the generic extension obtained as the $<κ$-support iterat…
Viale \cite{Viale_GuessingModel} introduced the notion of Generic Laver Diamond at $κ$---which we denote $\Diamond_{\text{Lav}}(κ)$---asserting the existence of a single function from $κ\to H_κ$ that behaves much like a supercompact Laver function, except with generic elementary embeddings rather than internal embeddings. Viale proved that the Proper Forcing Axiom (PFA) implies $\Diamond_{\text{Lav}}(ω_2)$. We strengthen his theorem by weakening the hypothesis to a statement strictly weaker than PFA. We also show that the principle $\Diamond_{\text{Lav}}(κ)$ provides a uniform, simple construc…
We consider a cardinal invariant closely related to Hindman's theorem. We prove that this cardinal invariant is small in the iterated Sacks perfect set forcing model, and that its corresponding parametrized diamond principle implies the existence of union ultrafilters. As a corollary, this establishes the existence of union ultrafilters in the iterated Sacks model of Set Theory.
Diamonds in mice
该论文证明，在无最大基数的可迭代小鼠中，某些菱形原理变体成立，并给出相应条件。
I explore two separate topics: the concept of jointness for set-theoretic guessing principles, and the notion of grounded forcing axioms. A family of guessing sequences is said to be joint if all of its members can guess any given family of targets independently and simultaneously. I primarily investigate jointness in the case of various kinds of Laver diamonds. In the case of measurable cardinals I show that, while the assertions that there are joint families of Laver diamonds of a given length get strictly stronger with increasing length, they are all equiconsistent. This is contrasted with …
Forcing Diamond and Applications to Iterability
本文证明高阶Sacks力迫与club Miller力迫可添加diamond序列，并肯定回答了κ-Sacks力迫的κ-支撑迭代不坍缩κ+且为κ-proper的长期问题，结果适用于其他高阶树力迫。
Under some cardinal arithmetic assumptions, we prove that every stationary subset of lambda of a right cofinality has the weak diamond. This is a strong negation of uniformization. We then deal with a weaker version of the weak diamond- colouring restrictions. We then deal with semi- saturated (normal) filters.
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.
We prove the consistency of the failure of the weak diamond $Φ_λ$ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $Ψ_λ$ is equivalent to the statement $2^{<λ}<2^λ$ and hence holds at every strongly inaccessible cardinal.
We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinals axioms, ranging from supercompactness to rank-to-rank embeddings. The majority of these large cardinals properties can be defined in terms of suitable elementary embeddings j\colon V_γ\to V_λ. One key observation is that such embeddings are uniquely determined by the image structures j [ V_γ]\prec V_λ. These structures will be the prototypes guessing models. We shall show, usin…
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 …
We use a reverse Easton forcing iteration to obtain a universe with a definable well-ordering, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle diamond star at a proper class of successor cardinals.
We investigate the problem of when $\leqλ$--support iterations of $<λ$--complete notions of forcing preserve $λ^+$. We isolate a property -- {\em properness over diamonds} -- that implies $λ^+$ is preserved and show that this property is preserved by $λ$--support iterations. We close with an application of our technology by presenting a consistency result on uniformizing colorings of ladder systems on $\{δ<λ^+:\cf(δ)=λ\}$ that complements a theorem of Shelah.
We prove that, consistently, there exists a weakly but not strongly inaccessible cardinal $λ$ for which the sequence $\langle 2^θ:θ<λ\rangle$ is not eventually constant and the weak diamond fails at $λ$. We also prove that consistently diamond fails but a parametrized version of weak diamond holds at some strongly inaccessible $λ$.
Super black box (formerly: Middle diamond)
该研究在特定基数算术假设下，证明了足够大的正则基数下多类平稳子集具有super BB性质，是一致化的强否定，修正了旧作并统一了术语。
We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal $λ$, if $λ^{++}$ is not a Mahlo cardinal in Gödel's constructible universe, then $2^λ= λ^+$ entails the existence of a $λ^+$-complete $λ^{++}$-Souslin tree.
We prove two general results about the preservation of extendible and $C^{(n)}$-extendible cardinals under a wide class of forcing iterations (Theorems 5.4 and 7.5). As applications we give new proofs of the preservation of Vopěnka's Principle and $C^{(n)}$-extendible cardinals under Jensen's iteration for forcing the GCH, previously obtained by Brooke-Taylor and Tsaprounis, res\-pectively. We prove that $C^{(n)}$-extendible cardinals are preserved by forcing with standard Easton-support iterations for any possible $Δ_2$-definable behaviour of the power-set function on regular cardinals. We sh…
Diamond on ladder systems and countably metacompact topological spaces
本文证明在ZFC下存在梯子系统L使得对应空间X_L不是可数亚紧的，给出κ=cf(beth_{ω+1})的肯定答案，并在beth_ω=aleph_ω假设下得到更低基数的例子。
More minimal non-$σ$-scattered linear orders
本文在代理原则假设下构造大量极小非$\sigma$-分散线性序，扩展了先前结果，并在$\aleph_1$层次上以更弱假设回答了Baumgartner的问题。
Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should work for R-modules, etc. As in earlier cases part of the work is analyzing how to move between the set theory and the algebra. Set theoretically we try to force a universe which satisfies G.C.H. and diamond holds for many stationary sets but, for every regular uncountable lambda, in some sense anything which "may" hold for some stationary set, does hold for s…
We discuss the effect of adding a single real (for various forcing notions adding reals) on cardinal invariants associated with the continuum (like the unbounding or the dominating number or the cardinals related to measure and category on the real line). For random and Cohen forcing, this question was investigated by Cicho'n and Pawlikowski; for Hechler forcing, by Judah, Shelah and myself. We show here: (1) adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size omega_1; (2) Laver and Mathias forcing collapse the dominating number …
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