arXiv subjects

Chi Tat Chong

Publications and source records attributed to Chi Tat Chong.

5 recordsLinked to original sources

arXiv2025

Definability over $\mathrm BΣ^0_2$-models

Let $\mathfrak M=(M,\mathcal X)$ be a model of $\mathsf{RCA}_0+\text{$Σ^0_2$-bounding}$ in which $Σ^0_2(A)$-induction fails for some $A\in\mathcal X$. We show that (i) if $\mathfrak M$ is a model of the combinatorial principle Ramsey's Theorem for Pairs, the Cohesive Set Theorem or the Tree Theorem, then there is a $Δ^0_1(A)$-instance of the principle with no solution in $\mathfrak M$ that is arithmetically definable relative to $A$; and (ii) any set of minimal Turing degree in $\mathfrak M$ that is arithmetically definable relative to $A$ has Turing jump equivalent to $A'$.

math.LO↗

arXiv2025

Open Problems in Computability Theory and Descriptive Set Theory

These open problems were presented in the Problem Sessions held during the Tianyuan Workshop on Computability Theory and Descriptive Set Theory, June 16-20, 2025. The problems are organized into sections named after their contributors, in the order of their presentations during the workshop. Notes were taken and compiled by Wei Dai, Feng Li, Ruiwen Li, Ming Xiao, Xu Wang, Víctor Hugo Yañez Salazar, and Yang Zheng.

math.LO↗

arXiv2025

Independence and Induction in Reverse Mathematics

We continue the project of the study of reverse mathematics principles inspired by cardinal invariants. In this article in particular we focus on principles encapsulating the existence of large families of objects that are in some sense mutually independent. More precisely, we study the principle $\mathsf{MAD}$ stating that a maximal family of pairwise almost disjoint sets exists; and the principle $\mathsf{MED}$ expressing the existence of a maximal family of functions that are pairwise eventually different. We investigate characterisations of and relations between these principles and some of their variants. It turns out that induction strength at the levels of $\mathsf{B}\mathrmΣ_2^0$ or $\mathsf{I}\mathrmΣ_2^0$ is an essential parameter; for instance, over $\mathsf{B}\mathrmΣ_2^0$, we show that $\neg\mathsf{MAD}$ is equivalent to the principle $\mathsf{DOM}$ expressing that every weakly represented family of functions is dominated by some other function.

math.LO↗

arXiv2019

The Strength of Ramsey's Theorem For Pairs over trees: I. Weak König's Lemma

Let $\mathsf{TT}^2_k$ denote the combinatorial principle stating that every $k$-coloring of pairs of compatible nodes in the full binary tree has a homogeneous solution, i.e. an isomorphic subtree in which all pairs of compatible nodes have the same color. Let $\mathsf{WKL}_0$ be the subsystem of second order arithmetic consisting of the base system $\mathsf{RCA}_0$ together with the principle (called Weak König's Lemma) stating that every infinite subtree of the full binary tree has an infinite path. We show that over $\mathsf{RCA}_0$, $\mathsf{TT}^2_k$ doe not imply $\mathsf{WKL}_0$. This solves the open problem on the relative strength between the two major subsystems of second order arithmetic.

math.LO↗

arXiv1996

Ordinal Recursion Theory

In this article, intended for the Handbook of Recursion Theory, we survey recursion theory on the ordinal numbers, with sections devoted to $α$-recursion theory, $β$-recursion theory and the study of the admissibility spectrum.

math.LO↗