Ralf Schindler
We prove in the theory "ZFC + there is no inner model with a Woodin cardinal" that there is a universal weasel. This shows in particular that one doesn't have to assume the existence of large cardinals in V to prove the existence of some such weasel.
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On a Conjecture Regarding the Mouse Order for Weasels
研究Steel关于Ω+1-可迭代1-小型weasels的Mouse序猜想，证明其在存在满足KP且有Σ₁-Woodin基数的可迭代premouse时不成立，并解决相关开放问题。
Assuming the existence of a strong cardinal, we find a model of ZFC in which for each uncountable regular cardinal $λ,$ there is no universal graph of size $λ$.
From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal kappa is fully indestructible by kappa-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal.
Our theme is that not every interesting question in set theory is independent of $ZFC$. We give an example of a first order theory $T$ with countable $D(T)$ which cannot have a universal model at $\aleph_1$ without CH; we prove in $ZFC$ a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove --- again in ZFC --- that for a large class of cardinals there is no universal linear order (e.g. in every $\aleph_1<ł<2^{\aleph_0}$). In fact, what we show is that if there is a universal linear order at a regular $ł$ and its existence is not a result of …
The paper is concerned with the existence of a universal graph at the successor of a strong limit singular mu of cofinality aleph_0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such mu there are mu^{++} graphs on mu^+ that taken jointly are universal for the graphs on mu^+, while 2^{mu^+}>> mu^{++} . The paper also addresses the general problem of obtaining a framework for consistency results at the successor of a singular strong limit starting from the assumption that a supercompact cardinal kappa exists. The result on the exist…
Forcing and cardinal invariants of universally null sets
本文研究普遍空集上的基数不变量，证明了多个相关不变量之间的不等式关系，并探讨了在特定假设下的一致性结果。
No universal group in a cardinal
本文研究了在特定条件下，如群类，是否存在普遍成员的问题，并提出了新的充分条件，即橄榄性质，以解决传统方法无法覆盖的情况。
In this paper we investigate some properties of first order theories which prevent them from having universal models under certain cardinal arithmetic assumptions. Our results give a new syntactical condition, oak property, which is a sufficient condition for a theory not to have universal models in cardinality lambda when certain cardinal arithmetic assumptions implying the failure of GCH (and close to the failure of SCH) hold.
The paper settles the problem of the consistency of the existence of a single universal graph between a strong limit singular and its power. Assuming that in a model of $\mathbf{GCH}$ $κ$ is supercompact and the cardinals $θ< κ$, $λ> κ$ are regular, as an application of a more general method we obtain a forcing extension in which $\textrm{cf}(κ) = θ$, the Singular Cardinal Hypothesis fails at $κ$ and there exists a universal graph in cardinality $λ\in (κ,2^κ)$.
We deal with the existence of universal members in a given cardinality for several classes. First we deal with classes of Abelian groups, specifically with the existence of universal members in cardinalities which are strong limit singular of countable cofinality. Second, we then deal with (variants of) the oak property (from a work of Dzamonja and the author), a property of complete first order theories, sufficient for the non-existence of universal models under suitable cardinal assumptions. Third, we prove that the oak property holds for the class of groups (naturally interpreted, so for qu…
A model of the Axiom of Determinacy in which every set of reals is universally Baire
本文基于导出模型构造，结合泛型迭代方法，证明了满足“每个实数集都是泛Baire集”的确定性公理模型的一致性强度对应存在武丁基数与强基数的极限基数的ZFC系统。
We prove that for lambda = beta_omega or just lambda strong limit singular of cofinality aleph_0, if there is a universal member in the class K^lf_lambda of locally finite groups of cardinality lambda, then there is a canonical one (parallel to special models for elementary classes, which is the replacement of universal homogeneous ones and saturated ones in cardinals lambda = lambda^ < lambda). For this, we rely on the existence of enough indecomposable such groups, as proved in "Density of indecomposable locally finite groups". We also more generally deal with the existence of universal memb…
Starting from a supercompact cardinal we build a model in which $2^{\aleph_{ω_1}}=2^{\aleph_{ω_1+1}}=\aleph_{ω_1+3}$ but there is a jointly universal family of size $\aleph_{ω_1+2}$ of graphs on $\aleph_{ω_1+1}$. The same technique will work for any uncountable cardinal in place of $ω_1$.
The consistency strength of determinacy when all sets are universally Baire
本文针对所有实数集合均为普遍 Baire 集时决定性公理的大基数强度，提出混合老鼠的新翻译程序，证明其恰等价于存在 Woodin 基数与强基数的共同极限。
The notion of commutation of operations in universal algebra leads to the concept of centralizer clone and gives rise to a well-known class of problems that we call centralizer problems, in which one seeks to determine whether a given set of operations arises as a centralizer or, equivalently, coincides with its own double centralizer. Centralizer clones and centralizer problems in universal algebra have been studied by several authors, with early contributions by Cohn, Kuznecov, Danil'čenko, and Harnau. In this paper, we work within a generalized setting of infinitary universal algebra relati…
We prove that a universal class categorical in a high-enough cardinal is categorical on a tail of cardinals. As opposed to other results in the literature, we work in ZFC, do not require the categoricity cardinal to be a successor, do not assume amalgamation, and do not use large cardinals. Moreover we give an explicit bound on the "high-enough" threshold: $\mathbf{Theorem}$ Let $ψ$ be a universal $\mathbb{L}_{ω_1, ω}$ sentence. If $ψ$ is categorical in some $λ\ge \beth_{\beth_{ω_1}}$, then $ψ$ is categorical in all $λ' \ge \beth_{\beth_{ω_1}}$. As a byproduct of the proof, we show that a conj…
The exact strength of generic absoluteness for the universally Baire sets
本文研究了泛Baire集的泛绝对性条件$\sf{Sealing}$的精确一致性强度，利用Woodin的$\sf{Core\\ Model\\ Induction}$技术证明其与$\sf{LSA-over-uB}$等一致，并分离出对应的精确大基数理论。
We investigate, in ZFC, the behavior of abstract elementary classes (AECs) categorical in many successive small cardinals. We prove for example that a universal $\mathbb{L}_{ω_1, ω}$ sentence categorical on an end segment of cardinals below $\beth_ω$ must be categorical also everywhere above $\beth_ω$. This is done without any additional model-theoretic hypotheses (such as amalgamation or arbitrarily large models) and generalizes to the much broader framework of tame AECs with weak amalgamation and coherent sequences.
Shelah has provided sufficient conditions for an $L_{ω_1, ω}$-sentence $ψ$ to have arbitrarily large models and for a Morley-like theorem to hold of $ψ$. These conditions involve structural and set-theoretic assumptions on all the $\aleph_n$'s. Using tools of Boney, Shelah, and the second author, we give assumptions on $\aleph_0$ and $\aleph_1$ which suffice when $ψ$ is restricted to be universal: $\mathbf{Theorem}$ Assume $2^{\aleph_{0}} < 2 ^{\aleph_{1}}$. Let $ψ$ be a universal $L_{ω_{1}, ω}$-sentence. - If $ψ$ is categorical in $\aleph_{0}$ and $1 \leq I(ψ, \aleph_{1}) < 2 ^{\aleph_{1}}$, …
If M is a proper class inner model of ZFC and omega_2^M=omega_2, then every sound mouse projecting to omega and not past 0-pistol belongs to M. In fact, under the assumption that 0-pistol does not belong to M, K^M \| omega_2 is universal for all countable mice in V. Similarly, if M is a proper class inner model of ZFC, delta>omega_1 is regular, (delta^+)^M = delta^+, and in V there is no proper class inner model with a Woodin cardinal, then K^M \| delta is universal for all mice in V of cardinality less than delta.
We prove: $\mathbf{Theorem}$ Let $K$ be a universal class. If $K$ is categorical in cardinals of arbitrarily high cofinality, then $K$ is categorical on a tail of cardinals. The proof stems from ideas of Adi Jarden and Will Boney, and also relies on a deep result of Shelah. As opposed to previous works, the argument is in ZFC and does not use the assumption of categoricity in a successor cardinal. The argument generalizes to abstract elementary classes (AECs) that satisfy a locality property and where certain prime models exist. Moreover assuming amalgamation we can give an explicit bound on t…
We prove that for each universal algebra $(A,\mathcal A)$ of cardinality $|A|\ge 2$ and an infinite set $X$ of cardinality $|X|\ge|\mathcal A|$, the $X$-th power $(A^X,\mathcal A^X)$ of the algebra $(A,\mathcal A)$ contains a free subset $\mathcal F\subset A^X$ of cardinality $|\mathcal F|=2^{|X|}$. This generalizes the classical Fichtenholtz-Kantorovitch-Hausdorff result on the existence of an independent family $\mathcal I\subset\mathcal P(X)$ of cardinality $|\mathcal I|=|\mathcal P(X)|$ in the Boolean algebra $\mathcal P(X)$ of subsets of an infinite set $X$.
Universal countably chromatic graph
证明存在大小为ℵ₁的通用可数染色图且连续统假设不成立的一致性，通过强真ccc偏序集的力迫迭代完成，构造适用于任何正则基数κ对应的不可数后继基数κ⁺。
Let T be an abelian group and lambda an uncountable regular cardinal. We consider the question of whether there is a lambda-universal group G^* among all torsion-free abelian groups G of cardinality less than or equal to lambda satisfying Ext(G,T)=0. Here G^* is said to be lambda-universal for T if, whenever a torsion-free abelian group G of cardinality less than or equal to lambda satisfies Ext(G,T)=0, then there is an embedding of G into G^*. For large classes of abelian groups T and cardinals lambda it is shown that the answer is consistently no. In particular, for T torsion, this solves a …
For every uncountable cardinal $λ$, suitable negations of the Generalized Continuum Hypothesis imply: - For all infinite $α$ and $β$, there is no universal $K_{α,β}$-free graphs in $λ$ - For all $α\ge 3$, there is no universal $K_α$-free graph in $λ$ The instance $K_{ω,ω_1}$ for $λ=\aleph_1$ was settled by Komjath and Pach from the principle $\diamondsuit(ω_1)$.
We consider a class K of structures e.g. trees with omega +1 levels, metric spaces and mainly, classes of Abelian groups like the one mentioned in the title and the class of reduced separable (Abelian) p-groups. We say M in K is universal for K if any member N of K of cardinality not bigger than the cardinality of M can be embedded into M . This is a natural, often raised, problem. We try to draw consequences of cardinal arithmetic to non-existence of universal members for such natural classes.
The $\mathsf{HOD}$ Hypothesis and a supercompact cardinal
本文在超紧基数与HOD假设的前提下，证明V_κ中存在真类多在HOD中可测的正则基数，推导出Woodin局部普遍性定理，揭示了HOD假设下两类大基数的差异及V中大基数向HOD的局部反映特性。
Universal wide Aronszajn tree
本文在弱紧致基数一致性假设下，证明存在一棵包含所有宽 Aronszajn 树同构副本的通用宽 Aronszajn 树。
Problem 5.1 in page 181 of [Fuc15] asks to find the cardinals $λ$ such that there is a universal abelian $p$-group for purity of cardinality $λ$, i.e., an abelian $p$-group $U_λ$ of cardinality $λ$ such that every abelian $p$-group of cardinality $\leq λ$ purely embeds in $U_λ$. In this paper we use ideas from the theory of abstract elementary classes to show: $\textbf{Theorem.}$ Let $p$ be a prime number. If $λ^{\aleph_0}=λ$ or $\forall μ< λ( μ^{\aleph_0} < λ)$, then there is a universal abelian $p$-group for purity of cardinality $λ$. Moreover for $n\geq 2$, there is a universal abelian $p$-…
We introduce reflection properties of cardinals in which the attributes that reflect are expressible by infinitary formulas whose lengths can be strictly larger than the cardinal under consideration. This kind of generalized reflection principle leads to the definitions of $L_{κ^+,κ^+}$-indescribability and $Π^1_ξ$-indescribability of a cardinal $κ$ for all $ξ<κ^+$. In this context, universal $Π^1_ξ$ formulas exist, there is a normal ideal associated to $Π^1_ξ$-indescribability and the notions of $Π^1_ξ$-indescribability yield a strict hierarchy below a measurable cardinal. Additionally, given…
A learning problem whose consistency is equivalent to the non-existence of real-valued measurable cardinals
本文证明 k-最近邻学习规则的普遍一致性与实值可测基数的不存在性等价，揭示了机器学习一致性问题与集合论大基数假设之间的深刻联系。
We find new "reasons" for a class of models for not having a universal model in a cardinal $λ$. This work, though it has consequences in model theory, is really in combinatorial set theory. We concentrate on a prototypical class which is a simply defined class of models, of combinatorial character - models of $T_{\rm ceq}$ (essentially another representation of $T_{\rm feq}$ which was already considered but the proof with $T_{\rm ceq}$ is more transparent). Models of $T_{\rm ceq}$ consist essentially of an equivalence relation on one set and a family of choice functions for it. This class is n…
On $(Σ^2_1)^{uB}$ Absoluteness Between V and HOD
本文结合 Woodin 的 Σ²₁ 基定理与 Vopěnka 定理，证明在存在真类多 Woodin 基数时，V 中为真的 (Σ²₁)^uB 命题在 HOD 中也为真，并探讨了反向绝对性及其与 Ultimate L 公理的关系。
Free Left Distributive Algebras and a Canonical Extension
在大基数假设下，作者证明了有限生成自由左分配代数在不同生成元数量下Σ₁-等价但非Σ₂-等价，并构造了扩展的单生成自由左分配代数，其中任一固定元素的应用是左分配代数的初等嵌入。
For k being the first uncountable cardinal w_1 or k being the cardinality of the continuum c, we prove that it is consistent that there is no Banach space of density k in which it is possible to isomorphically embed every Banach space of the same density which has a uniformly Gâteaux differentiable renorming or, equivalently, whose dual unit ball with the weak* topology is a subspace of a Hilbert space (a uniform Eberlein compact space). This complements a consequence of results of M. Bell and of M. Fabian, G. Godefroy, V. Zizler that assuming the continuum hypothesis, there is a universal spa…
We prove that for any constant $K>0$ there exists a separable group equipped with a complete bi-invariant metric bounded by $K$, isometric to the Urysohn sphere of diameter $K$, that is of `almost-universal disposition'. It is thus an object in the category of separable groups with bi-invariant metric analogous in its properties to the Gurarij space from the category of separable Banach spaces. We show that this group contains an isometric copy of any separable group equipped with bi-invariant metric bounded by $K$. As a consequence, we get that it is a universal Polish group admitting compati…
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