Moti Gitik, Saharon Shelah
Answering two questions of D. Fremlin [Real-valued measurable cardinals, in Set Theory of the Reals, H. Judah ed. 1993, 151-305 ] we show the following: (1) If c is real-valued measurable then the Maharam type of (c,P(c),sigma) is 2^c. (2) It is consistent to have k real-valued measurable but for every submodel V_1 with k measurable in it there are no k reals which are random over V_1.
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Despite being an established notion in the large cardinal hierarchy, results about Woodin cardinals are sparse in the literature. Here we gather known results about the preservation of Woodin cardinals under certain forcing extensions, as well as giving a template for preserving Woodin cardinals through forcing. Using this template, we form an indestructibility result under certain Easton iterations.
We prove several consistency results concerning the notion of $ω$-strongly measurable cardinal in HOD. In particular, we show that is it consistent, relative to a large cardinal hypothesis weaker than $o(κ) = κ$, that every successor of a regular cardinal is $ω$-strongly measurable in HOD.
We give a detailed proof of the properties of the usual Prikry type forcing notion for turning a measurable cardinal into $\aleph_ω$.
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.
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.
We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof is by interpolation and uses the Mapping Reflection Principle.
Small forcing always ruins the indestructibility of an indestructible supercompact cardinal. In fact, after small forcing, any cardinal $κ$ becomes superdestructible---any further ${<}κ$-closed forcing which adds a subset to $κ$ will destroy the measurability, even the weak compactness, of $κ$. Nevertheless, after small forcing indestructible cardinals remain resurrectible, but never strongly resurrectible.
We show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $ω_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The combinatorial properties that we make use of are natural principles, and in particular for inaccessible cardinals, these principles are equivalent to their corresponding large cardinal properties. Our characterizations make use of the concepts of internal large cardinals introduced in this paper, and of the classical concept of generic elementary embeddings.
We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $κ$ with the property that the collection of all initial segments of the wellordering is definable by a $Σ_1$-formula with parameter $κ$. A short argument shows that the existence of a measurable cardinal $δ$ implies that such wellorderings do not exist at $δ$-inaccessible cardinals of cofinality not equal to $δ$ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a…
We build a supercompact version of the forcing defined in \cite{gitik2019}. For each singular cardinal in the ground model with any fixed cofinality, which is a limit of supercompact cardinals, it is possible to force so that the size of the powerset of the singular cardinal is arbitrarily large, while preserving the singular cardinal. An important feature of this forcing is that it is possible to define the forcing so that the successor of the singular cardinal is collapsed, but all the cardinals above it are preserved.
We prove that if there is an elementary embedding from the universe to itself, then there is a proper class of measurable successor cardinals.
Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer principles for chromatic numbers of graphs. In this paper, we study these in the case of the successors of singular cardinals. In particular, we show that Prikry forcing preserves the centeredness of ideals but kills the layeredness. We also study $\binom{μ^{++}}{μ^{+}}\to\binomκ{μ^{+}}_μ$ and $\mathrm{Tr}_{\mathrm{Chr}}(μ^{+++},μ^{+})$ in the extension by Prikr…
Small measurable cardinals
论文改进了j-可分解系统的定义，提升了对称扩展中初等嵌入的提升能力，展示了可测基数可作为弱临界基数或马洛基数，且在特定模型中具有至少2的Mitchell序。
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…
Between Ramsey and measurable cardinals
本文研究介于 $κ$-Ramsey 基数与可测基数之间的交织层级，通过 filter games 刻画 Bovykin 和 McKenzie 提出的可测性 baby 版本，并确定直至可测基数的宇宙理论。
Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles hold for $ω_2$. 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.
Cofinal types of ultrafilters over measurable cardinals
本文研究了可测基数上超滤子的 cofinal 类型理论，将其与 Galvin 属性联系起来，并展示了不同结构的超滤子模型。
A ccc-generically supercompact cardinal $κ$ can be smaller than or equal to the continuum. On the other hand, such a cardinal $κ$ still satisfies diverse largeness properties, like that it is a stationary limit of ccc-generically measurable cardinals (Theorem 4.1). This is in a strong contrast to $\cal P$-generically supercompact cardinals for the class $\cal P$ of all $σ$-closed posets, which can be $\aleph_n$ for any n>1.
This paper presents the main results in my Ph.D. thesis. In what follows several proofs of SCH are presented introducing a family of covering properties which implies both SCH and the failure of various forms of square. These covering properties are also applied to investigate models of strongly compact cardinals or of strong forcing axioms like MM or PFA.
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…
A stationary subset $S$ of a regular uncountable cardinal $κ$ {\it reflects fully} at regular cardinals if for every stationary set $T \subseteq κ$ of higher order consisting of regular cardinals there exists an $α\in T$ such that $S \cap α$ is a stationary subset of $α$. {\it Full Reflection} states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than $κ$ having Mitchell order $κ^{++}$ it is consistent that Full Reflection holds at every $λ\leq κ$ and $κ$ is measurable.
The purpose of this paper is to present some results which suggest that the Singular Cardinals Hypothesis follows from the Proper Forcing Axiom. What will be proved is that a form of simultaneous reflection follows from the Set Mapping Reflection Principle, a consequence of PFA. While the results fall short of showing that MRP implies SCH, it will be shown that MRP implies that if SCH fails first at kappa then every stationary subset of S_{kappa^+}^omega = {a < kappa^+ : cf(a) = omega} reflects. It will also be demonstrated that MRP always fails in a generic extension by Prikry forcing.
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 …
We define extender sequences, generelizing measure sequences from Radin Forcing. Using the extender sequences we combine Gitik-Magidor forcing for adding many Prikry sequences with Radin forcing. This forcing satisfies Prikry like condition, destroys no cardinals, and has a kind of properness. Depending on the large cardinal we start with, this forcing can blow the power of a cardinal together with changing its' cofinality to a prescribe value. If this prescribed value is the cardinal itself then it remains regular and even can remain measurable.
The Axiom of Full Reflection at a measurable cardinal has been conjectured to be equiconsitent with the existence of a coherent sequence of measures with a repeat point. However we prove that the Axiom of Full Reflection at a measurable cardinal is surprisingly equiconsistent just with the existence of a measurable cardinal. We also generalize the result to larger cardinals as strong or supercompact. Hence we can conclude that the Axiom of Full Reflection at large cardinals weaker than measurable, e.g. as n-Mahlo, does push the consistency strength up, but does not push the consistency strengt…
On the strength of ultrafilters above choiceless large cardinals and their Prikry forcings
本文研究无选择大基数上超滤子强度与普利克里力迫，证明超滤子的大基数强度界，拓展普利克里力迫理论并限制可坍缩/奇异化基数，提出张量普利克里系统以构造无选择集合论的一致性结果（如所有不可数基数奇异的模型）。
Let $D$ be an infinite discrete set of measurable cardinals. It is shown that generalized Prikry forcing to add a countable sequence to each cardinal in $D$ is subcomplete. To do this it is shown that a simplified version of generalized Prikry forcing which adds a point below each cardinal in $D$, called generalized diagonal Prikry forcing, is subcomplete. Moreover, the generalized diagonal Prikry forcing associated to $D$ is subcomplete above $μ$, where $μ$ is any regular cardinal below the first limit point of $D$.
Several variants of the Halpern-Läuchli Theorem for trees of uncountable height are investigated. For $κ$ weakly compact, we prove that the various statements are all equivalent. We show that the strong tree version holds for one tree on any infinite cardinal. For any finite $d \ge 2$, we prove the consistency of the Halpern-Läuchli Theorem on $d$ many $κ$-trees at a measurable cardinal $κ$, given the consistency of a $κ+d$-strong cardinal. This follows from a more general consistency result at measurable $κ$, which includes the possibility of infinitely many trees, assuming partition relation…
The first measurable can be the first inaccessible cardinal
该研究改进了此前关于最小不可达基数为最小可测基数时内模型的$o(κ)$下界结果，将其从$o(κ)\u003e=2$提升至$o(κ)\u003e=κ+1$，并证明$κ^{++}$-超紧基数可在对称扩张中成为最小的不可达与可测基数。
Let $\mathcal{E}$ be the $σ$-ideal generated by the closed measure zero sets of reals. We use an ultrafilter-extendable matrix iteration of ccc posets to force that, for $\mathcal{E}$, their associated cardinal characteristics (i.e.\ additivity, covering, uniformity and cofinality) are pairwise different.
The number of measures on very large measurable cardinals
研究在无内模型技术情况下，通过超幂公理探讨可测基数上正常测度数量的可能模式及扩展
We continue the work of [KlSh:362] and prove that for lambda successor, a lambda-categorical theory T in L_{kappa^*, omega} is mu-categorical for every mu, mu <= lambda which is above the (2^{LS(T)})^+-beth cardinal.
The effects of forcing on exacting and ultraexacting cardinals
本文研究集合论强制对exacting与ultraexacting基数的影响，证明I2嵌入产生具有真类exacting基数的ZFC模型，并探讨HOD猜想下相关性质及强制扩张。
W.H. Woodin showed that if $κ_1 < \cdots < κ_n$ are strong cardinals then two-step ${\bfΣ}^1_{n+3}$ generic absoluteness holds after collapsing $2^{2^{κ_n}}$ to be countable. We show that this number can be reduced to $2^{κ_n}$, and to $κ_n^+$ in the case $n = 1$, but cannot be further reduced to $κ_n$.
Adding cofinal countable sequences through multiple regular cardinals by ssp forcing
本研究提出一种基于ZFC的初等平稳集保持力迫构造方法，可使指定的大于ω₁的正则基数集元素都具有ω共尾性，同时保证基模其他不可数正则基数不会在力迫扩张中获得可数共尾性。
The Necessary Maximality Principle for c.c.c. forcing asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already true in the minimal extension containing the real. We show that this principle is equiconsistent with the existence of a weakly compact cardinal.
We isolate a new large cardinal concept, "remarkability." Consistencywise, remarkable cardinals are between ineffable and omega-Erdos cardinals. They are characterized by the existence of "0^sharp-like" embeddings; however, they relativize down to L. It turns out that the existence of a remarkable cardinal is equiconsistent with L(R) absoluteness under proper forcings. In particular, said absoluteness does not imply Pi^1_1 determinacy.
A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing notion for~$λ$" we shall say "for every such family of forcing notions, depending on stationary $S\subseteq λ$, for some such stationary set we have\dots". Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.
Generalizing random real forcing for inaccessible cardinals
本文针对不可达基数$\u03bb$对应的$\u03bb^\u03bb$空间，在附加假设下构造了具备随机实数力迫性质的推广力迫，证明其保持基数与共尾性且不添加无支配实数，填补了随机实数力迫在高阶无穷空间的推广空白。
Assume $λ$ is a singular limit of $η$ supercompact cardinals, where $η\leq λ$ is a limit ordinal. We present two forcing methods for making $λ^+$ the successor of the limit of the first $η$ measurable cardinals while the tree property holding at $λ^+.$ The first method is then used to get, from the same assumptions, tree property at $\aleph_{η^2+1}$ with the failure of $SCH$ at $\aleph_{η^2}$. This extends results of Neeman and Sinapova. The second method is also used to get tree property at successor of an arbitrary singular cardinal, which extends some results of Magidor-Shelah, Neeman and S…
Suppose $κ$ is a singular strong limit cardinal of countable cofinality and let $\langle κ_{n}: n<ω\rangle$ be an incrasing sequence of regular cardinals cofinal in $κ$. We show that if $cf(2^κ)= κ^+$, then forcing with the full product $\prod_{n<ω}Add(κ_n,1)$ collapses $2^κ$ into $κ^+$. This result gives a consistent positive answer to a question of Sy Friedman. We also give a new proof of a result due to Shelah by showing that if the sequence carries a scale of length $κ^+,$ then forcing with $\prod_{n<ω}Add(κ_n,1)$ adds a generic filter for $Add(κ^+, 1)$, and indeed \[ \prod_{n<ω}Add(κ_n,1)…
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.
Answering a question of Usuba, we show that an extendible cardinal can be preserved by a set forcing that is not a small forcing.
We define a weak iterability notion that is sufficient for a number of arguments concerning $Σ_1$-definability at uncountable regular cardinals. In particular we give its exact consistency strength firstly in terms of the second uniform indiscernible for bounded subsets of $κ$: $u_2(κ)$, and secondly to give the consistency strength of a property of Lücke's. Theorem: The following are equiconsistent: (i) There exists $κ$ which is stably measurable; (ii) for some cardinal $κ$, $u_2(κ)=σ(κ)$; (iii) The {\boldmath $Σ_1$}-club property holds at a cardinal $κ$. Here $σ(κ)$ is the height of the smal…
Existence of a Model of $o(κ)=κ^{++}$ from Failure of GCH at a Measurable Cardinal
本文针对 GCH 在可测基数处失效与 o(κ)=κ^{++} 存在性之间的等一致性关系，补全了文献中缺失的下界证明细节。
This dissertation includes many theorems which show how to change large cardinal properties with forcing. I consider in detail the degrees of inaccessible cardinals (an analogue of the classical degrees of Mahlo cardinals) and provide new large cardinal definitions for degrees of inaccessible cardinals extending the hyper-inaccessible hierarchy. I showed that for every cardinal $κ$, and ordinal $α$, if $κ$ is $α$-inaccerssible, then there is a $\mathbb{P}$ forcing that $κ$ which preserves that $α$-inaccessible but destorys that $κ$ is $(α+1)$-inaccessible. I also consider Mahlo cardinals and d…
We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κfor which 2^κ=κ^+, another for which 2^κ=κ^++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable c…
On forcing axioms and weakenings of the Axiom of Choice
本文证明H_κ与DC_κ等选择公理弱化形式的力迫公理等价物，并借助针对特定稠密集族的新模板给出选择公理的新表述。
Suppose that there is a measurable cardinal. If \aleph_ωis a strong limit cardinal, but the power of \aleph_ωis bigger than \aleph_{ω_1}, then there is an inner model with a Woodin cardinal. Modulo the need of the measurable cardinal this answers a question of Gitik and Mitchell.
If it is consistent that there is a measurable cardinal, then it is consistent that all points g-delta Rothberger spaces have "small" cardinality.
In this paper we present a few properties of $K$-partitions, which are partitions of Baire spaces such that all subfamilies of such a partition sum to a set with the Baire property. Among the result proven we have general existence result that state that the existence of any $K$-partition implies the existence of $K$-partition of a metrizable space as well as existence of $K$-partition of a compact space implies the existence of $K$-partition of a completely metrizable space. We also prove some connections between existence of $K$-partitions and existence of precipitous ideals as well as measu…
It is known that the set of possible cofinalities $\mathrm{pcf}(A)$ has good properties if $A$ is a progressive interval of regular cardinals. In this paper, we give an interval of regular cardinals $A$ such that $\mathrm{pcf}(A)$ has no good properties in the presense of a measurable cardinal, or in generic extensions by Prikry-type forcing.
We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal of uncountable cofinality, while its successor enjoys various combinatorial properties. As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $κ$ of uncountable cofinality where SCH fails and for which there is a collection of graphs on $κ^+$ whose size…
Many of the most common reverse Easton iterations found in the large cardinal context, such as the Laver preparation, admit a gap at some small delta in the sense that they factor as P*Q, where P has size less than delta and Q is forced to be delta-strategically closed. In this paper, generalizing the Levy-Solovay theorem, I show that after such forcing, every embedding j:V[G]-->M[j(G)] in the extension which satisfies a mild closure condition is the lift of an embedding j:V-->M in the ground model. In particular, every ultrapower embedding in the extension lifts an embedding from the ground m…
Eventual Capture on a Measurable Cardinal
本文研究正则不可数基数上的局部化基数，证明可测基数使其平凡化，并通过力迫法证明结果最优，同时回答了一个关于不同函数对应局部化基数严格不等的一致性公开问题。
We present a forcing for blowing up 2^lambda and making ``many positive polarized partition relations'' (in a sense made precise in (c) of our main theorem) hold in the interval [lambda, 2^lambda]. This generalizes results of [276], Section 1, and the forcing is a ``many cardinals'' version of the forcing there.
Forcing "$\mathrm{NS}_{ω_1}$ is $ω_1$-dense" From Large Cardinals
本文回答 Woodin 的问题，证明在特定大基数假设下存在保持静态集的迫近使 NS_{ω_1} 为 ω_1-稠密，并引入新公理 QM 蕴含 Q_max-(*)，从而也蕴含该结论。
Supercompact extender based forcings are used to construct models with HOD cardinal structure different from those of V. In particular, a model with all regular uncountable cardinals measurable in HOD is constructed.
In this paper we investigate some properties of forcing which can be considered "nice" in the context of singularizing regular cardinals to have an uncountable cofinality. We show that such forcing which changes cofinality of a regular cardinal, cannot be too nice and must cause some "damage" to the structure of cardinals and stationary sets. As a consequence there is no analogue to the Prikry forcing, in terms of "nice" properties, when changing cofinalities to be uncountable.
We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals $\langle κ_n: n< ω\rangle$, Prikry defined the forcing $\mathbb{P}$ all perfect subtrees of $\prod_{n<ω}κ_n$, and proved that for $κ=\sup_{n<ω}κ_n$, assuming the necessary cardinal arithmetic, the Boolean completion $\mathbb{B}$ of $\mathbb{P}$ is $(ω,μ)$-distributive for all $μ<κ$ but $(ω,κ,δ)$-distributivity fails for all $δ<κ$, implying failure of the $(ω,κ)$-d.l. …
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