Time: March 02, 2024.

Vernue: 1105, Mengming Wei building, Nanjing University.

Speakers: 

Qi Feng (Chinese Academy of Sciences and Tsinghua University): Defining finiteness without infinity.

Abstract: We will give a definition of Finiteness without infinity using the formal ∈-language based on specifically finitely many concrete Frege’s Logic Rules and Zermelo’s Axioms of Sets and show that basic laws of Arithmetic can be established under a restricted form of Dedekind’s Principle. This will indicate clearly that the Axiom of Infinity only plays the rule of solving the Representation Problem of Finiteness as far as the basic Arithmetic is concerned. Of course, the methods of Mathematical Induction and Recursive Definition are tighten up with the Axiom of Infinity.

Jintao Luo (Sichuan University): Regularity Properties and Pathological Sets.

Abstract: We show that in ZF + DC, if all sets of reals have the Baire property, then there are no infinite maximal independent families. Also, if all sets are Ramsey, there are no Hamel bases of R over Q nor Vitali sets. Joint work with Jialiang He and Shuguo Zhang.

Xu Wang (Nankai University): A hierarchy on non-archimedean CLI Polish groups.

Abstract: We introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by α-CLI and L-α-CLI, where α is a countable ordinal. We establish three results: (1) G is 0-CLI iff G={1G}(2) G is 1-CLI iff G admits a compatible complete two-sided invariant metric; and(3) G is L-α-CLI iff G is locally α-CLI, i.e., G contains an open subgroup that is α-CLI. Next we prove this hierarchy is proper by constructing non-archimedean CLI Polish groups G_α and H_α for α<ω_1, such that (1) H_α is α-CLI but not L-β-CLI for β<α; and(2) G_α is (α+1)-CLI but not L-α-CLI.

Bokai Yao (Peking University): Boolean-Valued Models of Set Theory with Urelements.

Abstract: We explore Boolean-valued models of set theory with a class of urelements. In an existing construction, which we call $\UB$, every urelement is its own $\B$-name. We prove the fundamental theorem of $\UB$ in the context of $\ZFUR$ (i.e., ZF with urelements formulated with Replacement). In particular, $\UB$ is shown to preserve Replacement and hence $\ZFUR$. Moreover, $\UB$ can also destroy and recover certain axioms such as the Collection Principle. One drawback of $\UB$ is that it does not permit mixing names, resulting in a lack of fullness. To address this, we introduce a new construction, $\UBB$, which is closed under mixtures. We prove that there is an elementary embedding from $\UB$ to $\UBB$. Over $\ZFUR$ with the Axiom of Choice, $\UBB$ is full for every complete Boolean algebra $\B$ just in case the Collection Principle holds. This is joint work with Xinhe Wu.

Shuheng Zhang (Nanjing University): Differentiability of a function related to Chaitin's Omega.

Abstract: We modify Chaitin's Omega to a continuous function f(x)=Σ σ<x 2^-K(σ), and give a characterization of its differentiability.

Zhentao Zhang (Fudan University): Newelski's question on definably amenable groups over $p$-adics.

Abstract: Let $G$ be a definably amenable groups over $p$-adics. We study the definable topological dynamics $(G, S_G(\mathbb{Q}_p))$ of $G$ acting on its type space. We focus on the question of whether weakly generic types coincide with almost periodic types which was raised by Newelski and then restarted by Chernikov and Simon on NIP definably amenable groups. We show that the stationarity is a sufficient and necessary condition for the positive answer of the question when $G$ admits a dfg-fsg decomposition. This is  joint work with Ningyuan Yao. 

Schedule  :  

 9:30  Qi Feng: Defining finiteness without infinity.

10:15 Jintao Luo: Regularity Properties and Pathological Sets

11:00 Break

11:15 Xu Wang: A hierarchy on non-archimedean CLI Polish groups.

14:30 Bokai Yao: Boolean-Valued Models of Set Theory with Urelements.

15:15 Zhentao Zhang: Newelski's question on definably amenable groups over $p$-adics.

16:00 Break

16:15 Shuheng Zhang: Differentiability of a function related to Chaitin's Omega.

Organizers:

• Longyun Ding (Nankai University)

• Jialiang He (Sichuan University)

• Zhaokuan Hao (Fudan University)

• Liang Yu (Nanjing University)