Isomorphism property in nonstandard extensions of ZFC universe

非标准见证者与Π0_1句子在ZFC中的观察障碍：标准切口、统一反射失败与语义空洞

Nonstandard Witnesses and Observational Barriers for Π0_1 Sentences in ZFC: Standard Cuts, Uniform Reflection Failure, and the Semantic Void

本文研究了ZFC中Π0_1句子的不可判定性问题，揭示了统一反射失败导致的观察障碍及语义空洞现象。

A definable nonstandard model of the reals

We prove in ZFC the existence of a definable, countably saturated elementary extension of the reals. It seems that it has been taken for granted that there is no distinguished, definable nonstandard model of the reals. (This means a countably saturated elementary extension of the reals.) Of course if V=L then there is such an extension (just take the first one in the sense of the canonical well-ordering of L), but we mean the existence provably in ZFC. There were good reasons for this: without Choice we cannot prove the existence of any elementary extension of the reals containing an infinitel…

Barwise-Schlipf定理的集合论类似物

Set theoretical analogues of the Barwise-Schlipf theorem

该研究在集合论领域给出Barwise-Schlipf定理的类似结果，分别刻画了满足对应类理论扩张条件的任意基数ZF非标准模型与可数ZFC非标准模型。

The strength of the isomorphism property

In section 1 of this paper, we characterize the isomorphism property of nonstandard universes in terms of the realization of some second--order types in model theory. In section 2, several applications are given. One of the applications answers a question of D. Ross about infinite Loeb measure spaces

Computable quotient presentations of models of arithmetic and set theory

We prove various extensions of the Tennenbaum phenomenon to the case of computable quotient presentations of models of arithmetic and set theory. Specifically, no nonstandard model of arithmetic has a computable quotient presentation by a c.e. equivalence relation. No $Σ_1$-sound nonstandard model of arithmetic has a computable quotient presentation by a co-c.e. equivalence relation. No nonstandard model of arithmetic in the language $\{+,\cdot,\leq\}$ has a computably enumerable quotient presentation by any equivalence relation of any complexity. No model of ZFC or even much weaker set theori…

Uniform Roe coronas

A uniform Roe corona is the quotient of the uniform Roe algebra of a metric space by the ideal of compact operators. Among other results, we show that it is consistent with ZFC that isomorphism between uniform Roe coronas implies coarse equivalence between the underlying spaces, for the class of uniformly locally finite metric spaces which coarsely embed into a Hilbert space. Moreover, for uniformly locally finite metric spaces with property A, it is consistent with ZFC that isomorphism between the uniform Roe coronas is equivalent to bijective coarse equivalence between some of their cofinite…