A note on Löwenheim-Skolem cardinals

Magidor-Malitz Reflection

In this paper we investigate the consequences and consistency of the downward Löwenheim-Skolem theorem for extension of the first order logic by the Magidor-Malitz quantifier. We derive some combinatorial results and improve the known upper bound for the consistency of Chang's Conjecture at successor of singular cardinals.

Maximality of logic without identity

Lindström theorem obviously fails as a characterization of $\mathcal{L}_{ωω}^{-} $, first-order logic without identity. In this note we provide a fix: we show that $\mathcal{L}_{ωω}^{-} $ is \emph{maximal} among abstract logics satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in \cite{Casa}), the Löwenheim--Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs we use a form of strong upwards Löwenheim--Skolem theorem not avai…

A note on cardinal preserving embeddings

We prove the consistency of the theory ZFC + there is a strongly compact cardinal from the existence of a cardinal preserving embedding from the universe into an inner model. The proof almost shows that under SCH, every cardinal preserving embedding preserves the cofinality and continuum functions.

A note on "Another ordering of the ten cardinal characteristics in Cichoń's Diagram" and further remarks

In this note, we relax the hypothesis of the main results in Kellner-Shelah-Tǎnasie's "Another ordering of the ten cardinal characteristics in Cichoń's diagram".

A Note on Singular Cardinals in Set Theory Without Choice

We discuss how singular can cardinals be in absence of the axiom of choice. We show that, contrasting with known negative consistency results (of Gitik and others), certain positive results are provable. Then we pose some problems.

关于阿普特定理的注记

A Note on a Theorem of Apter

本文探讨了ZF与AD_R条件下Θ的可测性与最小强正则基数关系，揭示了其与可数余数的联系。

Notes on cardinals that are characterizable by a complete (Scott) sentence

This is part I of a study on cardinals that are characterizable by Scott sentences. Building on [3], [6] and [1] we study which cardinals are characterizable by a Scott sentence $ϕ$, in the sense that $ϕ$ characterizes $κ$, if $ϕ$ has a model of size $κ$, but no models of size $κ^+$. We show that the set of cardinals that are characterized by a Scott sentence is closed under successors, countable unions and countable products (cf. theorems 2.3, 3.4, and corollary 3.6). We also prove that if $\aleph_α$ is characterized by a Scott sentence, at least one of $\aleph_alpha$ and $\aleph_alpha^+$ is …

关于满射基数的注记

A note on surjective cardinals

本文证明ZF+DC_κ不能证明满射基数构成基数代数，但可构成满射基数代数，并由此给出消去律的更简洁证明。

About the review in Mathematical Reviews of my paper: The two-cardinal problem for languages of arbitrary cardinality The Journal of Symbolic Logic 75, Number 3, Sept., 2010, pp. 785-801

In this note I make some comments on the review MR2723767 (2011m:03064) appeared in Mathematical Reviews

Notes on countably generated complete Boolean algebras

We give a necessary and sufficient condition for an atomless Boolean algebra to be countably generated, and use it to give new proofs of some some know facts due to Gaifman-Hales and Solovay and also due to Jech, Kunen and Magidor. We also show that Jensen's coding theorem can be used to provide cardinal preserving countably generated complete Boolean algebras of arbitrary large cardinality. This answers a question of Jech [5] from 1976.

A note on $δ$-strongly compact cardinals

In this paper we investigate more characterizations and applications of $δ$-strongly compact cardinals. We show that, for a cardinal $κ$ the following are equivalent: (1) $κ$ is $δ$-strongly compact, (2) For every regular $λ\ge κ$ there is a $δ$-complete uniform ultrafilter over $λ$, and (3) Every product space of $δ$-Lindelöf spaces is $κ$-Lindelöf. We also prove that in the Cohen forcing extension, the least $ω_1$-strongly compact cardinal is a precise upper bound on the tightness of the products of two countably tight spaces.

关于不可数基数的 Hindman 型定理的一则注记

A note on Hindman-type theorems for uncountable cardinals

本文将作者此前在可数情形下的结果推广至不可数基数，构造出一族自然的 Hindman 型定理，并给出部分变体的下界，为不可数交换群上的 Hindman 型性质提供正面结论。

为实现Cichoń最大值的强制技术：维也纳大学的研讨会讲义

Forcing techniques for Cichoń's Maximum: Lecture notes for the mini-course at the University of Vienna

本文探讨通过强制技术构建Cichoń最大值模型的方法，消除了对大基数的依赖，展示相关集合论特征之间的关系。

A Note on Edge Colorings and Trees

We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a cardinal $κ$ has a homogeneous set of size $κ$ provided that the number of colors, $μ$ satisfies $μ^+<κ$. Another result is that an uncountable cardinal $κ$ is weakly compact if and only if $κ$ is regular, has the tree property and for each $λ,μ<κ$ there exists $κ^*<κ$ such that every tree of height $μ$ with $λ$ nodes has less than $κ^*$ branches.

A note on the eightfold way

Assuming the existence of a Mahlo cardinal, we construct a model in which there exists an $ω_2$-Aronszajn tree, the $ω_1$-approachability property fails, and every stationary subset of $ω_2 \cap \mathrm{cof}(ω)$ reflects.

Monochromatic Sumset Without the use of large cardinals

We show in this note that in the forcing extension by $Add(ω,\beth_ω)$, the following Ramsey property holds: for any $r\in ω$ and any $f: \mathbb{R}\to r$, there exists an infinite $X\subset \mathbb{R}$ such that $X+X$ is monochromatic under $f$. We also show the Ramsey statement above is true in $\mathrm{ZFC}$ when $r=2$. This answers two questions by Komjáth, Leader, Russell, Shelah, Soukup and Vidnyánszky.

A note on infinite partitions of free products of Boolean algebras

If $A$ is an infinite Boolean algebra the cardinal invariant $\mathfrak{a}(A)$ is defined as the smallest size of an infinite partition of $A$. The cardinal $\mathfrak{a}(A\oplus B)$, where $A\oplus B$ is the free product of the Boolean algebras $A$ and $B$ (whose dual topological space is the product of the dual topological spaces of $A$ and $B$), is below both $\mathfrak{a}(A)$ and $\mathfrak{a}(B)$. The equality $\mathfrak{a}(A\oplus B)=\min\lbrace\mathfrak{a}(A),\mathfrak{a}(B)\rbrace$ is not known to hold for all infinite Boolean algebras $A$ and $B$. Here some lower bounds of $\mathfrak{…

A note on strong negative partition relations

We analyze a natural function definable from a scale at a singular cardinal, and using this function we are able to obtain quite strong negative square-brackets partition relations at successors of singular cardinals. The proof of our main result makes use of club-guessing, and as a corollary we obtain a fairly easy proof of a difficult result of Shelah connecting weak saturation of a certain club-guessing ideal with strong failures of square-brackets partition relations. We then investigate the strength of weak saturation of such ideals and obtain some results on stationary reflection.

Consistency of "the ideal of null restricted to some A is kappa-complete not kappa^+-complete, kappa weakly inaccessible and cov(meagre)=aleph_1"

In this note we answer the following question of Grinblat: Is it consistent that for some set X, cov(NULL restriction X)=lambda is a weakly inaccessible cardinal (so X not null of course) while cov(meagre) is small, say it is aleph_1.

On critical cardinalities related to $Q$-sets

In this note we collect some known information and prove new results about the small uncountable cardinal $\mathfrak q_0$. The cardinal $\mathfrak q_0$ is defined as the smallest cardinality $|A|$ of a subset $A\subset \mathbb R$ which is not a $Q$-set (a subspace $A\subset\mathbb R$ is called a $Q$-set if each subset $B\subset A$ is of type $F_σ$ in $A$). We present a simple proof of a folklore fact that $\mathfrak p\le\mathfrak q_0\le\min\{\mathfrak b,\mathrm{non}(\mathcal N),\log(\mathfrak c^+)\}$, and also establish the consistency of a number of strict inequalities between the cardinal $\…

On Harrington's model in which Separation holds but Reduction fails at the 3rd projective level, and on some related models of Sami

In a handwtitten note of 1975, Leo Harrington sketched a construction of a model of ZFC (no large cardinals or anything beyond ZFC!) in which $\mathbfΠ^1_3$-Separation holds but $\mathbfΣ^1_3$-Reduction fails. The result has never appeared in a journal or book publication except for a few of old references.