Saharon Shelah
We succeed to say something on the identities of (mu^+, mu) when mu>theta>cf(mu), mu strong limit theta--compact. This hopefully will help to prove the consistency of ``some pair (mu^+,mu) is not compact'', however, this has not been proved.
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In [Bon20], model theoretic characterizations of several established large cardinal notions were given. We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and subtle cardinals.
This is the second combinatorial proof of the compactness theorem for singular from 1977. In fact it gives a somewhat stronger theorem.
In a paper from 1997, Shelah asked whether $Pr_1(λ^+,λ^+,λ^+,λ)$ holds for every inaccessible cardinal $λ$. Here, we prove that an affirmative answer follows from $\square(λ^+)$. Furthermore, we establish that for every pair $χ<κ$ of regular uncountable cardinals, $\square(κ)$ implies $Pr_1(κ,κ,κ,χ)$.
We study uniqueness of limit models in abstract elementary classes (AECs) with no maximal models. We prove (assuming instances of diamonds) that categoricity in a cardinal of the form $μ^{+(n + 1)}$ implies the uniqueness of limit models of cardinality $μ^{+}, μ^{++}, \ldots, μ^{+n}$. This sheds light on a paper of Shelah and Villaveces, who were the first to consider uniqueness of limit models in this context. We also prove that (again assuming instances of diamonds) in an AEC with no maximal models, tameness (a locality property for types) together with categoricity in a proper class of card…
We present a modification to the Prikry on Extenders forcing notion allowing the blow up of the power set of a large cardinal, change its cofinality to omega without adding bounded subsets, working directly from arbitrary extender (e.g., n-huge extender). Using this forcing, starting from a superstrong cardinal, we construct a model in which the added Prikry sequences are a scale in the normal Prikry sequence.
We argue that we solved Hilbert's first problem positively (after reformulating it just to avoid the known consistency results) and give some applications. Let lambda to the revised power of kappa, denoted lambda^{[kappa]}, be the minimal cardinality of a family of subsets of lambda each of cardinality kappa such that any other subset of lambda of cardinality kappa is included in the union of <kappa members of the family. The main theorem says that almost always this revised power is equal to lambda. Our main result is The Revised GCH Theorem: Assume we fix an uncountable strong limit cardinal…
Narrow systems revisited
本文引入并研究广义窄系统性质，证明了其足以蕴含$κ$以上的奇异基数假设，并在所有正则基数$κ\geqω_2$处一致成立，同时揭示了其与强无界次可加着色的联系。
We extend and improve the result of Makkai and Paré that the powerful image of any accessible functor F is accessible, assuming there exists a sufficiently large strongly compact cardinal. We reduce the required large cardinal assumption to the existence of $L_{μ,ω}$-compact cardinals for sufficiently large μ, and also show that under this assumption the λ-pure powerful image of F is accessible. From the first of these statements, we obtain that the tameness of every Abstract Elementary Class follows from a weaker large cardinal assumption than was previously known. We provide two ways of empl…
This is a revised version (of late 2020) of [Sh:700], which is arXiv:math/0012170 . First point is noting that the proof of Theorem 4.3 in [Sh:700], which says that the proof giving the consistency $ \mathfrak{b} = \mathfrak{d} = \mathfrak{u} < \mathfrak{a} $ also gives $ \mathfrak{s} = \mathfrak{d} $. The proof uses a measurable cardinal and a c.c.c. forcing so it gives large $ \mathfrak{d} $ and assumes a large cardinal. Second point is adding to the results of \S2,\S3 which say that (in \S3 with no large cardinals) we can force $ {\aleph_1} < \mathfrak{b} = \mathfrak{d} < \mathfrak{a}$. We …
Assuming the existence of a strong cardinal $κ$, a weakly compact cardinal $λ$ above it and $γ> λ,$ we force a generic extension in which $κ$ is a singular strong limit cardinal of any given cofinality $δ$, $2^κ\geq γ$ and such that the tree property holds at $κ^{++}$.
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.
Assuming the existence of a Mahlo cardinal, we produce a generic extension of Gödel's constructible universe $L$, in which the transfer principles $(\aleph_2, \aleph_0) \to (\aleph_3, \aleph_1)$ and $(\aleph_3, \aleph_1) \to (\aleph_2, \aleph_0)$ fail simultaneously. The result answers a question of Silver from 1971. We also extend our result to higher gaps.
Generic Absoluteness Revisited
本文研究递归公理与拉弗泛型大基数公理的关系，证明从特定递归公理片段和紧拉弗泛型巨大基数存在性可推出更一般的泛型绝对性结论，且相关假设在多数常见偏序集类下蕴含地基公理的否定，区分了两类公理的对应片段。
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…
Let $\mathrm{cof}(μ)=μ$ and $κ$ be a supercompact cardinal with $μ<κ$. Assume that there is an increasing and continuous sequence of cardinals $\langleκ_ξ\mid ξ<μ\rangle$ with $κ_0:=κ$ and such that, for each $ξ<μ$, $κ_{ξ+1}$ is supercompact. Besides, assume that $λ$ is a weakly compact cardinal with $\sup_{ξ<μ}κ_ξ<λ$. Let $Θ\geqλ$ be a cardinal with $\mathrm{cof}(Θ)>κ$. Assuming the $\mathrm{GCH}_{\geqκ}$, we construct a generic extension where $κ$ is strong limit, $\mathrm{cof}(κ)=μ$, $2^κ= Θ$ and both $\mathrm{TP}(κ^+)$ and $\mathrm{TP}(κ^{++})$ hold. Further, in this model there is a very …
Infinite Combinatorics revisited in the absence of Axiom of Choice
该研究在ZF公理系统下考察无穷组合定理的可证性，通过组合论证与绝对性方法得到多个基数相关的组合结论，证明了若干陈述的等价性与不可证性，并明确了一致可数枚举存在性相关命题的等一致性强度。
We present a new partial order for directly forcing morasses to exist that enjoys a significant homogeneity property. We then use this forcing in a reverse Easton iteration to obtain an extension universe with morasses at every regular uncountable cardinal, while preserving all n-superstrong (0<n<omega+1), hyperstrong and 1-extendible cardinals. In the latter case, a preliminary forcing to make the GCH hold is required. Our forcing yields morasses that satisfy an extra property related to the homogeneity of the partial order; we refer to them as mangroves and prove that their existence is equi…
We give an application of our extender based Radin forcing to cardinal arithmetic. Using a preparation forcing and interleaving of Cohen and Levy forcings in the normal Radin sequence we get a model with a power function having a fixed finite gap on all cardinals.
On a relation between $λ$-full well-ordered sets and weakly compact cardinals
本文通过超限递归证明了在适当正则基数 $λ$ 的线性序集中存在 $λ$-满良序子集，并以此证明若正则基数集 $H$ 的下确界为弱紧基数，则对任意LOTS $X$，$H$-紧性等价于不存在 $H$ 中类型的间隙。
For an infinite cardinal $κ$, let $dedκ$ denote the supremum of the number of Dedekind cuts in linear orders of size $κ$. It is known that $κ<dedκ\leq 2^κ$ for all $κ$ and that $dedκ<2^κ$ is consistent for any $κ$ of uncountable cofinality. We prove however that $2^κ\leq ded ( ded ( ded ( dedκ)))$ always holds. Using this result we calculate the Hanf numbers for the existence of two-cardinal models with arbitrarily large gaps and for the existence of arbitrarily large models omitting a type in the class of countable dependent first-order theories. Specifically, we show that these bounds are as…
We consider a question: Can a given AD-family be ADR for two orthogonal uncountable towers? If $b > ω_1$, then we rebuilt any AD-family of the cardinality $ω_1$ onto a Hausdorff pre-gap. Moreover, if a such AD-family is a Luzin gap, then we obtain a Hausdorff gap. Under $b = ω_1$, a similar rebuilding is impossible.
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