同构性质的强度

ZFC论域非标准扩张的同构性质

Isomorphism property in nonstandard extensions of ZFC universe

我们研究模型的HST，一个非标准的集合论，其中包括，特别是，ZFC替换和分离图式的语言包含的成员资格和标准谓词，以及内部集的有序家庭的饱和度。这个理论承认同构性质IP的一个适当的公式：“任何两个基本等价的内部呈现结构的wellorderable语言是同构的。例如，IP意味着所有无限内部集合都是等数的，并且存在标准实数的唯一（模同构）内部初等扩张。我们证明了IP是独立的HST（使用类的所有集构造从内部集）和HST（使用通用扩展的$\HST$模型由足够数量的通用同构）一致。

When does elementary bi-embeddability imply isomorphism?

A first-order theory has the Schroder-Bernstein property if any two of its models that are elementarily bi-embeddable are isomorphic. We prove that if a countable theory T has the Schroder-Bernstein property then it is classifiable (it is superstable and has NDOP and NOTOP) and satisfies a slightly stronger condition than nonmultidimensionality, namely: there cannot be a model M of T, a type p over M, and an automorphism f of M such that for every two distinct natural numbers i and j, f^i(p) is orthogonal to f^j(p). We also make some conjectures about how the class of theories with the Schrode…

The Hrushovski property for hypertournaments and profinite topologies

We study the problem of extending partial isomorphisms for hypertournaments, which are relational structures generalizing tournaments. This is a generalized version of an old question of Herwig and Lascar. We show that the generalized problem has a negative answer, and we provide a positive answer in a special case. As a corollary, we show that the extension property holds for tournaments in case the partial isomorphisms have pairwise disjoint ranges and pairwise disjoint domains.

Definable tree property for successors of cardinals

It is proved that the consistency strength of having definable tree property for successors of all regular cardinals is the consistency strength of having proper class many small large cardinals which are defined very similar to indescribables but are much weaker in consistency strength. Also the consistency strength of definable tree property for successor of a singular cardinal is reduced to the existence of a supercompact cardinal and a measurable above it.

The consistency strength of the perfect set property for universally Baire sets of reals

We show that the statement "every universally Baire set of reals has the perfect set property" is equiconsistent modulo ZFC with the existence of a cardinal that we call a virtually Shelah cardinal. These cardinals resemble Shelah cardinals but are much weaker: if $0^\sharp$ exists then every Silver indiscernible is virtually Shelah in $L$. We also show that the statement $\text{uB} = {\bfΔ}^1_2$, where $\text{uB}$ is the pointclass of all universally Baire sets of reals, is equiconsistent modulo ZFC with the existence of a $Σ_2$-reflecting virtually Shelah cardinal.

关于共终Fraïssé极限的可计算性

On the computability of cofinal Fraïssé limits

本文研究了可计算age的共终Fraïssé极限的构造所需的计算资源，证明其需0'''且某些情况需0''，而Fraïssé极限则只需0'。