Title: A Introduction to Ramseys Theorem.
Venue: Complex A 202 (and online via zoom).
Time: 16:00-17:30.
Abstract: It will mainly be about Ramsey Numbers (with same examples) and generalizing Ramseys Theorem.
Title: Determinacy on the Edge of Second Order Arithmetic.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: An introduction of the first results proved on determinacy of simple games, without use of the power set axiom. We then present a proof of a paper of Montalbàn and Shore on the determincay of finite differences of \Pi^0_3 sets in second order arithmetic making use of high complexity comprehension scheme. It turns out this is the best result we can get without using power set axiom.
Title: Some comments about the mu-calculus' alternation hierarchy in modal logic and reverse mathematics.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: The mu-calculus is obtained by adding fixed point operators to modal logic. Its alternation hierarchy classified formulas by the entanglement of their fixed point operators. I explain the alternation hierarchy's collapse to modal logic over equivalence relations, in various non-standard semantics. I also sketch the relation between a formalized version of the mu-calculus and determinacy in second order arithmetic.
Title: Silver's theorem and its surroundings.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: TBA
Title: A formalized proof of a type of basis theorem.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: For a fixed problem type statement, we can consider various kinds of basis theorems, which require outputs to be bounded in some sense. In this talk, we observe one basis theorem, ll^2-basis theorem, and prove that for RT^2_2 in I Sigma^0_2.
Title: Silver's theorem and its surroundings.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: TBA
Title: On the complexity of fixed point theorem on the Cantor space. (Part 2)
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: In this talk, I will introduce two viewpoints of complexity of theorems. In Reverse Mathematics, we measure complexities by considering which axiom is needed to prove a theorem. On Weihrauch degrees, we consider how difficult it is to construct a solution for a problem.
I will compare them and consider the complexity of the fixed point theorem on the Cantor space from both viewpoints.
Title: Ramsey's Theorem and Recursion Theory.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: Jockuschの論文「Ramsey’s Theorem and Recursion Theory」の二つの主結果について解説する。この論文はラムゼーの定理と計算可能性理論の関係について論じている。主結果は、ある再帰的n分割で、いかなる均質な無限集合もSigma_n^0集合としてとれないこと、任意の再帰的n分割に対して、ある均質な無限集合をPi_n^0集合としてとれることである。
Title: On the complexity of fixed point theorem on the Cantor space.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: In this talk, I will introduce two viewpoints of complexity of theorems. In Reverse Mathematics, we measure complexities by considering which axiom is needed to prove a theorem. On Weihrauch degrees, we consider how difficult it is to construct a solution for a problem.
I will compare them and consider the complexity of the fixed point theorem on the Cantor space from both viewpoints.
Title: 全ての証明可能性論理の共通部分
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: 理論 T の証明可能性述語を様相演算子 □ とみたときに，T において証明可能な様相論理式全体の集合をその証明可能性述語の証明可能性論理という．導出可能性条件を満たす証明可能性述語の証明可能性論理が GL と一致するという Solovay の定理が知られているが，他方，例えば証明可能性論理が最小の正規様相論理 K になるような証明可能性述語が存在することなども分かっている．今回は証明可能性論理の分析対象を非正規様相論理に拡張する研究の一環として，全ての証明可能性論理の共通部分に注目し，それが Fitting, Marek, and Truszczyński の非正規様相論理 N に一致することを紹介する．
Title: 分離的微分閉体の理論の基礎的な性質
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: 分離閉体の微分的な類似として分離的微分閉体という新しいクラスの微分体を研究した。この理論は数理論理学の文脈において初等的であることが示され、これを SDCF と表記する。
本研究は 1970 年代の標数 p の微分閉体の理論 DCFp に関する Carol Wood の仕事を一般化する形で行われ、この理論 SDCFp,εの基礎的な性質;量化記号消去、安定性、素モデルの存在などを確立したので紹介したい。
Title: A very short introduction to modal logic and the modal mu-calculus.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: I will present the basic definitions of modal logic, two of its many interpretations. I will also sketch a few results on bisimulations and frame correspondence. After that I will define the modal mu-calculus and it’s alternation hierarchy. I will also discuss some situations where the alternation hierarchy collapses.
Title: Reverse mathematics from multiple points of view.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: The aim of the program of reverse mathematics is to classify mathematical theorems by comparing them with various types of axioms. The strength of axioms can be measured by several different viewpoints, such as consistency strength or computability strength, which are heavily connected to various fields of mathematical logic. Recently, the field of reverse mathematics has been widely expanding with newer perspectives and techniques. In this talk, we overview the recent developments of reverse mathematics and their connections to related fields.
Title: General topology on the second order arithmetic.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: Poset spaces were introduced by Carl Mummert in order to describe the general topology on the second order arithmetic. This talk introduces this concept and some recent results.
Title: 形式言語理論への測度論的アプローチ
Venue: Online via Zoom.
Time: 15:00-16:30.
Abstract: 形式言語理論は語の集合の種々の性質や「複雑さ」を調べる学問であり，その「複雑さ」の指標には様々なものがある．
伝統的には正規言語や文脈自由言語などの計算モデル・文法的な「複雑さ」によって定義された言語族がよく研究されたきた．
本講演ではこれら伝統的な言語族の紹介からはじめ，講演者が提案した測度論的な「複雑さ」の指標を導入し，その背景や最近の結果について解説を行う．
Title: Real objects and its expressions.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: In any formal system, we have to restrict expressions to ones which the system guarantees. We call the restricted expressions "codes". Of course we should be sensitive to the gap between objects and codes. In this talk, we will observe the gaps, working in systems for second order arithmetic and higher order arithmetic.
Title: Relativized leftmost path principle in Reverse mathematics and Weihrauch degrees.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: Any ill-founded tree T⊆ {N}^{<{N}} has the leftmost path w.r.t. the lexicographical ordering,
and this statement (called leftmost path principle, LPP) is equivalent to Pi^1_1\{CA}_0 over {RCA}_0 in reverse mathematics.
In [1], Towsner introduced weaker versions of this statement, Sigma^0_{alpha} relativized LPP,
stating that any ill-founded tree has a path f such that there is no f^{(alpha)} computable path located left of f.
Moreover, he showed that Sigma^0_{alpha} LPPs form a hierarchy between {ATR}_0 and Pi^1_1\{CA}_0.
In this talk, I will introduce Towsner's work and consider relativized LPPs from the viewpoint of Weihrauch degrees.
[1]Towsner, Henry.``Partial impredicativity in reverse mathematics." The Journal of Symbolic Logic 78.2 (2013): 459-488.
Title: Infinite games in reverse mathematics.
Venue: Complex A 202 (and online via zoom).
Time: 15:00-16:30.
Abstract: In this seminar I will survey infinite games in reverse mathematics. I will review basic definitions and theorems, and present a few recent results.