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Jason Zesheng Chen

Publications and source records attributed to Jason Zesheng Chen.

4 recordsLinked to original sources
arXiv2026

Two questions on $E_0$-like generic equivalence

We answer Problems 8.1 and 8.3 from \cite{Tianyuan2026}. We first note that the condition attributed there to ordinary Prikry forcing is not correct as written: the appropriate formulation uses finite symmetric difference of the ranges of the generic sequences, rather than eventual equality at the same coordinates. We record answers to the two problems for both formulations. A length-$ω$ Magidor forcing gives a genuinely Prikry-type example satisfying both conditions. Rigid real-adding forcing gives a broad source of further examples, and a forcing of Jech and Shelah shows that the condition does not carry any large cardinal strength. An $E_0$-invariant Jensen-type forcing of Kanovei and Lyubetsky gives a stronger nontrivial example in which the generic reals in a fixed extension form exactly one full $E_0$-class. Finally, a simple recoding turns the real-forcing previous examples into answers for the version of the problems with corrected condition.

math.LO↗
arXiv2026

Equivalence of Random Generics Is Not Essentially Free

Let $M$ be a countable transitive model of a sufficiently large finite fragment of ZFC, and let $E^{\mathbb B}_{M}$ be equivalence of random generics over $M$. Smythe proved that $E^{\mathbb B}_{M}$ is not Borel-reducible to the orbit relation of a free action of any countable group belonging to $M$, and asked whether the restriction on the target group can be removed. We prove that if $A$ is a positive-measure Borel set of random reals over $M$ and $F$ is an essentially free countable Borel equivalence relation, then every Borel homomorphism from $E^{\mathbb B}_{M}\upharpoonright A$ to $F$ maps a conull subset of $A$ into a single $F$-class. Hence no positive-measure restriction of $E^{\mathbb B}_{M}$ is essentially free, or even weakly Borel-reducible to an essentially free relation. The proof combines Thomas's consequence of Popa cocycle superrigidity with a perfect family of $2$-marked groups and a Fubini argument using two successive random reals.

math.LO↗
arXiv2026

Ideals, Well-Orderable Families, and Baire Category in Truss's Feferman-Type Model

Let $\mathsf{BCT}_{\mathsf{WO}}$ assert that every well-orderable family of dense open subsets of a perfect Polish space has dense intersection, and let $\mathsf{RS}_{\mathsf{WO}}$ assert that for every nonempty countable partial order, there is a filter meeting every member of any given well-orderable family of dense subsets. Over $\mathsf{ZF}$ they are shown to be equivalent. We show that in Truss's model, they characterize those $A\subseteq2^ω$ for which every $A$-indexed family of dense open sets has dense intersection. Assuming choice for well-orderable families of nonempty sets, we obtain an analogous characterization for total relations $R\subseteq X\times Y$ with meager vertical sections, whenever $Y$ is a surjective image of $2^ω$. The Feferman-type model $\mathfrak N_{\aleph_1}$ studied by Truss \cite{Truss1974} satisfies these hypotheses and $\mathsf{DC}$. We study the ideal of well-orderable subsets of $2^ω$ and its relations to other ideals in this model. We show, among other things: the well-orderable subsets of $2^ω$ are exactly the sets which are Rothberger in every finite power, have strong measure zero, are universally null, are Marczewski null, or contain no perfect subset. The ideal of well-orderable subsets of $2^ω$ is closed under well-ordered unions and is incomparable with the meager ideal. The equivalence with the Rothberger property does not extend to the Hurewicz property. Further more, in this model every set is Marczewski measurable, arbitrary maps into separable metric spaces have continuous perfect restrictions, and every set of positive outer measure contains a perfect subset.

math.LO↗