Axiomatizing some small classes of set functions

对周期小的l-预群变种进行公理化

Axiomatizing small varieties of periodic l-pregroups

本文研究了周期小的l-预群变种的公理化，揭示了有限连接的格结构，并characterize了有限子直接不可约代数的性质。

公理化 Rumsfeld 无知

Axiomatizing Rumsfeld Ignorance

针对 Rumsfeld 无知可由无知定义导致公理化平凡的问题，本文区分两个隐式知识可达关系并令其一为另一个的任意子集，给出多种真双框架类上的公理化并用于分析 Fine 的结果。

Measurability and Perfect Set Theorems for Equivalence Relations with Small Classes

We ask whether $\mathbf{Δ^1_2}$ or $\mathbf{Σ^1_2}$ equivalence relations with $I$-small classes for $I$ a $σ$-ideal must have perfectly many classes. We show that for a wide class of ccc $σ$-ideals, a positive answer for $\mathbf{Δ^1_2}$ equivalence relations is equivalent to the $I$-measurability of $\mathbf{Δ^1_2}$ sets. However, the analogous statement for $\mathbf{Σ^1_2}$ equivalence relations is false: $\mathbf{Σ^1_2}$ equivalence relations with meager classes have a perfect set of pairwise inequivalent elements if and only if $\mathbf{Δ^1_2}$ sets have the Baire property.

Perfect Set Theorems for Equivalence Relations with $I$ - small classes

A classical theorem due to Mycielski states that an equivalence relation $E$ having the Baire property and meager equivalence classes must have a perfect set of pairwise inequivalent elements. We consider equivalence relations with $I$-small equivalence classes, where $I$ is a proper $σ$-ideal, and ask whether they have a perfect set of pairwise inequivalent elements. We give a positive answer for $E$ universally Baire. We show that the answer for $E$ $\mathbf{Δ_{2}^{1}}$ is independent of $ZFC$, and find set theoretic assumptions equivalent to it when $I$ is the countable ideal. For equivalen…

The small index property for homogeneous models in AECs

We prove a version of a small index property theorem for strong amalgamation classes. Our result builds on an earlier theorem by Lascar and Shelah (in their case, for saturated models of uncountable first-order theories). We then study versions of the small index property for various non-elementary classes. In particular, we obtain the small index property for quasiminimal pregeometry structures.

在局部抽象初等类中的小基数下构建模型

Building models in small cardinals in local abstract elementary classes

本文证明了在局部抽象初等类中，假设Galois类型满足温和局部性条件，稳定性足以在小基数下构建更大模型，并显著改进了已有定理使其适用于部分不稳定类。