2024年度のセミナー

• Venue: Complex A 801 (and online via zoom).

Time: 15:00-16:30

Abstract: In [1], we introduced some characterizations of the set of Pi^1_2 sentences provable from Pi^1_1-CA_0 and a hierarchy dividing it. For a detailed study of the properties of this hierarchy, it was important to focus on the statements of the form [every coded beta-model satisfies sigma] for some specific sentences sigma.

       In this talk, we summarize a part of [1]. Then we give a characterization of the set of Sigma^1_2 or 

       Boole(Pi^1_2) sentences provable from Pi^1_1-CA_0 by statements of the form [every coded beta-model 

       satisfies sigma]. Here, Boole(Pi^1_2) is the class of formulas generated by disjunction, conjunction and 

       negation starting from Pi^1_2.

       This is joint work with J. Aguilera and K. Yokoyama.

[1] Suzuki, Y., & Yokoyama, K. (2024). On the $\Pi^ 1_2 $ consequences of $\Pi^ 1_1 $-$\mathsf {CA} _0$. arXiv preprint arXiv:2402.07136. 

• Venue: Complex A 801 (and online via zoom).

Time: 15:00-16:30

Abstract: ラムゼイの定理は長きにわたり逆数学研究の主要テーマのひとつであった。そのバリュエーションであるTree theoremの研究も近年進展を見せている。今回、Tree theoremの彩色のstable性に関する定理である0-S^3TT^2_2と帰納法公理の関係について、conservativityの手法により分析した。先行研究であるTT^1のconservationに関する結果を紹介するとともに研究の現状について報告する。 

• Venue: Complex A 801 (and online via zoom).

Time: 15:00-16:30

Abstract: 爆発律は推論規則として奇妙に見える。しかし、爆発律を除いた論理上での数学では不都合なことが起こる。このことから爆発律の数学での働きを調査すべく先ず爆発律を取り巻く環境を変えた。古典論理、直観主義論理も爆発律を持ちMinimal logicは爆発律を持たないが爆発律を加えると直観主義論理になってしまう。故に、Minimal logicより弱い論理(Subminimal Logics)で爆発律を観察した。その結果、新たな論理の構造が明らかになった。 この話では新たな論理上で数学に関する新たな事実を紹介する。 

• Venue: Complex A 801 (and online via zoom).

Time: 14:00-15:30

Abstract: In the study of reverse mathematics for analysis, it is observed that many theorems on minimal value principles for continuous functions are provable within the system of arithmetical comprehension (ACA_0) since they are usually depended on compactness. In contrast, we will see that Ekeland’s variational principle, which is a variation of minimization theorem in non-compact situation, is equivalent to Pi^1_1-comprehenstion.

In this talk, we will overview the reverse mathematics for analysis and then see the idea for the above result. 

• Venue: Complex A 801 (and online via zoom).

Time: 15:00-16:30

Abstract: Reverse mathematics has become one of the most important subjects of mathematical logic. It has a rich domain of problems arising from ordinary mathematics, as well as a programmatic orientation. To put it briefly, its results seem to matter not only for logic but also for ordinary mathematics. Yet there are several different accounts of why it matters, of what a reversal tells us. I want to discuss a few of these.