On Finitely Generated Models of Theories with at Most Countably Many Nonisomorphic Finitely Generated Models

Generic countably infinite groups

Countably infinite groups (with a fixed underlying set) constitute a Polish space $G$ with a suitable metric, hence the Baire category theorem holds in $G$. We study isomorphism invariant subsets of $G$, which we call group properties. We say that the generic countably infinite group is of property $P$ if $P$ is comeager in $G$. We prove that every group property with the Baire property is either meager or comeager. We show that there is a comeager elementary equivalence class in $G$ but every isomorphism class is meager. We prove that the generic group is algebraically closed, simple, not fin…

Some notes concerning the homogeneity of Boolean algebras and Boolean spaces

We consider homogeneity properties of Boolean algebras that have nonprincipal ultrafilters which are countably generated.It is shown that a Boolean algebra B is homogeneous if it is the union of countably generated nonprincipal ultrafilters and has a dense subset D such that for every a in D the relative algebra B restriction a:= {b in B:b <= a} is isomorphic to B. In particular, the free product of countably many copies of an atomic Boolean algebra is homogeneous. Moreover, a Boolean algebra B is homogeneous if it satisfies the following conditions: (i) B has a countably generated ultrafilter…

只存在可数多个co-trees的局部表格生物中间逻辑

There are only countably many locally tabular bi-intermediate logics of co-trees

本文证明只存在可数多个co-trees的局部表格生物中间逻辑，所有都是有限公理化的。

Cofinal Extensions and Coded Sets

Let M be a model of Peano Arithmetic that is countably generated over an exponentially closed cut I. We characterize those sets X of subsets of I for which there is a finitely (or countably) generated cofinal extension N of M such that I is the Greatest Common Initial Segment of M and N and X is the set of subsets of I coded by the extension. We also characterize such X for which the extension N can be, in addition, one of the following: non-filling; filling; n-filling.

无限帽子问题与大基数

Infinite Hat Problems and Large Cardinals

本文研究无限帽子问题，确定逻辑学家能保证至少一人猜对的颜色数κ，并推广到不同人数和猜测次数，发现最小失败κ与自由子集性质及大基数相关，且ZFC有时无法判定。

关于仅无限群的词问题

On the word problem for just infinite groups

研究了仅无限群的词问题，证明了有限生成群的词问题在可枚举关系集下可判定，并探讨了可数生成群在局部有限时的可判定性问题，构造了局部有限群的不可判定词问题示例。

具有不太快速的无标记增长的结构

Structures with not too fast unlabelled growth

研究一类增长速率受限的结构$\mathscr{S}$，通过自同构群对其进行完整分类，证明其直至双可定义性结构可数，在$(\mathbb{Q};<)$中一阶可解释且与有限有界齐次结构相互可定义，还证实其结构直至相互可定义性有有限多个一阶约简。

Reverse mathematics, Young diagrams, and the ascending chain condition

Let $S$ be the group of finitely supported permutations of a countably infinite set. Let $K[S]$ be the group algebra of $S$ over a field $K$ of characteristic $0$. According to a theorem of Formanek and Lawrence, $K[S]$ satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over RCA$_0$ (or even over RCA$_0^*$) to the statement that $ω^ω$ is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.