Department of Mathematical Informatics | Graduate School of Informatics | Nagoya University

## Regular Members

 Takayuki Kihara (Nagoya Univ.), Yo Matsubara (Nagoya Univ.), Hiroaki Minami (Aichi Gakuin Univ.), Tadatoshi Miyamoto (Nanzan Univ.), Yasuo Yoshinobu (Nagoya Univ.)

## Upcoming Seminars

 Date 2020.3.2. 15:30〜17:00, Room 314 Speaker Toshimichi Usuba (Waseda Univ.) Title Choiceless Löwenheim-Skolem property This seminar was cancelled due to the coronavirus outbreak
 Date 2020.2.20. 15:30〜17:00, Room 322 Speaker Souji Shizuma (Osaka Prefecture Univ.) Title Infinite versions of "puzzles of prisoners and hats" and set theory

## Past Seminars: Fall Semester 2019

 Date 2020.1.24. 15:30〜17:00, Room 206 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title Basic instances of Y-proper forcing notions
 Date 2019.12.13. 15:30〜17:00, Room 206 Speaker Takayuki Kihara (Nagoya Univ.) Title From Weihrauch reducibility to Lifschitz-like realizability for intuitionistic Zermelo-Fraenkel set theory
 Date 2019.11.29. 15:30〜17:00, Room 206 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title A note on a countable fast function, its forcing axiom, and a weak tail club guessing, Part 3
 Date 2019.11.1. 15:30〜17:00, Room 206 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title A note on a countable fast function, its forcing axiom, and a weak tail club guessing, Part 2
 Date 2019.10.25. 15:30〜17:00, Room 206 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title A note on a countable fast function, its forcing axiom, and a weak tail club guessing Abstract The countable fast function poset is strongly sigma-closed but does not have a usual chain condition. It forces a closed and cofinal subset of the second uncountable cardinal. Therefore its forcing axiom that takes care of appropriate number of dense sets kills a weak tail club guessing. This weak club guessing is a consequence of a tiny fragment of GCH. We discuss a consistency of the forcing axiom together with CH. We use an Aspero-Mota type iterated forcing with countable symmetric systems of sigma-closed uncountabe elementary substructres of relevant expanding relational structures on a fixed trasitive set universe of a large fragment of set theory. We also dicuss a class of posets that includes this poset and posts with stronger forms of relevant chain condition.

## Past Seminars: Spring Semester 2019

 Date 2019.6.14. 15:30〜17:00, Room 206 Speaker Takayuki Kihara (Nagoya Univ.) Title On Martin's conjecture, Part 2
 Date 2019.6.7. 15:30〜17:00, Room 206 Speaker Takayuki Kihara (Nagoya Univ.) Title On Martin's conjecture, Part 1
 Date 2019.5.31. 15:30〜17:00, Room 122 (Lecture Room 2) Speaker Sakaé Fuchino (Kobe Univ.) Title The trichotomy of the possible size of the continuum under the existence of a Laver-generic large cardinal Abstract A cardinal $$\kappa$$ is said to be Laver-generically supercompact for a class of posets $$\mathcal{P}$$ if, for any $$\mathbb{P}\in\mathcal{P}$$ and $$\delta\geq\kappa$$, there are $$\mathbb{Q}\in\mathcal{P}$$, $$({\sf V},\mathbb{P})$$-generic $$\mathbb{H}$$ and $$j$$, $$M\subseteq{\sf V}[\mathbb{H}]$$ such that $$\mathbb{P}\mathrel{{\leqslant}\hspace{-0.86ex}{\lower-0.53ex\hbox{\(\scriptscriptstyle\circ$$}}}\mathbb{Q}\), $$M$$ is an inner model of $${\sf V}[\mathbb{H}]$$, $${j:{\sf V}\stackrel{\preccurlyeq\hspace{0.8ex}}{\rightarrow}M}$$, $$crit(j)=\kappa$$, $$j(\kappa)>\delta$$ and $$\mathbb{P}$$, $$\mathbb{H}$$, $$j{}^{\,{\prime}{\prime}}\delta\in M$$ . Laver-generic super almost-hugeness and Laver-generic superhugeness are defined similarly. For the three “iterable” classes of proper posets $$\mathcal{P}_0=$$ $$\sigma$$-closed posets, $$\mathcal{P}_1=$$ proper posets, and $$\mathcal{P}_2=$$ c.c.c. posets, the existence of a Laver-generically supercompact cardinal $$\kappa$$ for $$\mathcal{P}_i$$ ($$i=0$$, $$1$$, $$2$$) implies: $$\kappa=\aleph_2$$ and $${\sf CH}$$ holds (in case of $$i=0$$); $$\kappa=2^{\aleph_0}=\aleph_2$$ (for $$i=1$$), or $$\kappa\leq 2^{\aleph_0}$$ and $$\kappa$$ is very large (e.g. $$\kappa$$-weakly Mahlo and more, if $$i=2$$). In each of these cases we also obtain a very strong reflection principle (reflection down to $$<\aleph_2$$ in case of $$i=0$$ and reflection down to $$<2^{\aleph_0}$$ in case of $$i=1$$ or $$2$$) and $${\sf MA}^+(\mathcal{P}_i)$$. In this rather technical seminar talk, I will presend the proof (sketch) of the statement above and also the consistecy proof of the existence of Laver-generically supercompact cardinals for $$\mathcal{P}_i$$ starting from a supercompact cardinal (or superhuge cardinal --- actually super almost-huge should be enough --- in case of $$i=1$$). The key to the consistency proofs is the use of a Laver function from which the nomenclature of the “Laver-genericity” is originated.
 Date 2019.5.17. 15:30〜17:00, Room 206 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title Forcing a continuous ∈-chain of length ω1 by finite side conditions, Part 3
 Date 2019.5.10. 15:30〜17:00, Room 206 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title Forcing a continuous ∈-chain of length ω1 by finite side conditions, Part 2
 Date 2019.4.26. 15:30〜17:00, Room 206 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title Forcing a continuous ∈-chain of length ω1 by finite side conditions, Part 1

## Past Seminars: Fall Semester 2018

 Date 2019.3.19. 15:30〜17:00, Room 314 Speaker Sabrina Ouazzani (École Polytechnique) Title How to compute with an infinite time Turing machine? [Slides] Abstract In this talk, we present infinite time Turing machines (ITTM), from the original definition of the model to some new infinite time algorithms. We will present algorithmic techniques that allow to highlight some properties of the ITTM-computable ordinals. In particular, we will study gaps in ordinal computation times, that is to say, ordinal times at which no infinite time program halts. We will explain some properties of the gaps in the ordinal computation times.
 Date 2019.3.8. 15:30〜17:00, Room 314 Speaker Sakaé Fuchino (Kobe Univ.) Title Reflection axioms which imply the continuum is very large
 Date 2019.2.22. 15:30〜17:00, Room 314 Speaker Toshimichi Usuba (Waseda Univ.) Title Prikry forcing with non-normal filters
 Date 2019.1.25. 15:30〜17:00, Room 206 Speaker Yo Matsubara (Nagoya Univ.) Title Reflection Principles and Stationary Towers
 Date 2018.11.16. 15:30〜17:00, Room 314 Speaker Satoru Kuroda (Gunma Prefectural Women’s Univ.) Title Bounded Arithmetic and Forcing Method
 Date 2018.11.5 -- 2018.11.8, RIMS, Kyoto Workshop RIMS Workshop on Axiomatic Set Theory and its Applications
 Date 2018.10.26. 15:30〜17:00, Room 206 Speaker Yo Matsubara (Nagoya Univ.) Title On Countable Stationary Towers
 Date 2018.9.18 -- 2018.9.20, Takigawa Memorial Hall, Kobe Workshop Symposium on Advances in Mathematical Logic 2018 Dedicated to the Memory of Professor Gaisi Takeuti (1926-2017)
 Date 2018.9.14. 15:30〜17:00, Room 206 Speaker Yo Matsubara (Nagoya Univ.) Title Precipitousness of Contable Stationary Towers

## Past Seminars: Spring Semester 2018

 Date 2018.7.20. 15:30-17:00, Room 206 Speaker Yasuo Yoshinobu (Nagoya Univ.) Title Properness under closed forcing, Part 3
 Date 2018.7.13. 15:30-17:00, Room 206 Speaker Yasuo Yoshinobu (Nagoya Univ.) Title Properness under closed forcing, Part 2
 Date 2018.7.6. 15:30-17:00, Room 206 Speaker Yasuo Yoshinobu (Nagoya Univ.) Title Properness under closed forcing, Part 1
 Date 2018.6.29. 15:30-17:00, Room 314 Speaker Paul-Elliot Anglès D'Auriac (Université Paris-Est Créteil) Title On Infinite Time Turing Machine and its related ordinals Abstract In 1998, Hamkins and Lewis introduced Infinite Time Turing Machines (ITTMs), a version of Turing Machines where time is allowed to run through the ordinals instead of the integers. This model of computation revealed itself to have interesting connections with set theory and in particular Godel's constructible hierarchy. In this talk, we will be interested in the properties of the ordinals that naturally arises in the study of ITTMs, such as those that correspond to halting time, or that have a code that can be written on the tape of an ITTM.
 Date 2018.6.22. 15:30-17:00, Room 206 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title An (ω0,1)-morass and a partition for ω1, Part 2
 Date 2018.6.15. 15:30-17:00, Room 206 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title An (ω0,1)-morass and a partition for ω1, Part 1
 Date 2018.6.8. 15:30-17:00, Room 206 Speaker Yo Matsubara (Nagoya Univ.) Title Stationary tower revisited, Part 3
 Date 2018.5.25. 15:30-17:00, Room 206 Speaker Yo Matsubara (Nagoya Univ.) Title Stationary tower revisited, Part 2
 Date 2018.5.11. 15:30-17:00, Room 206 Speaker Yo Matsubara (Nagoya Univ.) Title Stationary tower revisited, Part 1
 Date 2018.4.27. 15:30-17:00, Room 206 Speaker Yasuo Yoshinobu (Nagoya Univ.) Title Preserving forcing axioms, Part 4
 Date 2018.4.20. 15:30-17:00, Room 206 Speaker Yasuo Yoshinobu (Nagoya Univ.) Title Preserving forcing axioms, Part 3

## Past Seminars: Fall Semester 2017

 Date 2018.3.12. 15:30-17:00, Room 314 Speaker Vassilios Gregoriades (Univ. of Turin) Title The Preiss Separation Theorem uniformly Abstract The typical example of a uniformity-type result in descriptive set theory is the Souslin-Kleene Theorem, which says that the separation property of the class of analytic sets can be witnessed by a recursive function in the codes. An important consequence of the latter is the extension of the result HYP = effectively bi-analytic, in all recursive Polish spaces. In this talk we present the uniform version of a separation result by Preiss that deals with the convex analytic subsets of the Euclidean space. We show that the separation can be realized by a HYP function in the codes. Similarly to the case of the Souslin-Kleene Theorem, we conclude that every HYP convex subset of the Euclidean space can be obtained from the class of HYP compact convex sets by taking HYP increasing unions and HYP intersections.
 Date 2018.3.9. 15:30-17:00, Room 314 Speaker Toshimichi Usuba (Waseda Univ.) Title Extendible cardinals and the mantle
 Date 2018.1.26. 15:30-17:00, Room 206 Speaker Yasuo Yoshinobu (Nagoya Univ.) Title Preserving forcing axioms, Part 2
 Date 2017.12.15. 15:30-17:00, Room 206 Speaker Takayuki Kihara (Nagoya Univ.) Title On Semmes' proof, Part 5 Report A priority argument in descriptive set theory: A very detailed exposition of Semmes’ proof.
 Date 2017.12.8. 15:30-17:00, Room 206 Speaker Takayuki Kihara (Nagoya Univ.) Title On Semmes' proof, Part 4
 Date 2017.12.1. 15:30-17:00, Room 206 Speaker Takayuki Kihara (Nagoya Univ.) Title On Semmes' proof, Part 3
 Date 2017.11.24. 15:30-17:00, Room 206 Speaker Takayuki Kihara (Nagoya Univ.) Title On Semmes' proof, Part 2
 Date 2017.11.17. 15:30-17:00, Room 206 Speaker Takayuki Kihara (Nagoya Univ.) Title On Semmes' proof, Part 1
 Date 2017.11.6 -- 2017.11.9, RIMS, Kyoto Workshop RIMS Workshop on Iterated Forcing Theory and Cardinal Invariants
 Date 2017.10.27. 15:30-17:00, Room 206 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title On iterated forcing with side conditions, Part 4
 Date 2017.10.20. 15:30-17:00, Room 206 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title On iterated forcing with side conditions, Part 3
 Date 2017.10.13. 15:30-17:00, Room 206 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title On iterated forcing with side conditions, Part 2
 Date 2017.10.6. 15:30-17:00, Room 314 Speaker Tadatoshi Miyamoto (Nanzan Univ.) Title On iterated forcing with side conditions Abstract Aspero-Mota introduced an iterated forcing that used symmetric systems of elementary substructures with the markers. We reproduce it. Our construction features the following: We stick to a single transitive set universe to form various clubs. We use a pre-forced stationary set to manage amalgamations. We use what we call signed coordinates rather than the markers. We consider these features by iteratively forcing the following examples. Posets that force what we call fast functions. Posets that kill weak club guessings.

## Past Seminars: Spring Semester 2017

 Date 2017.8.24, 15:30-17:00, Room 314 Speaker Linda Brown Westrick (University of Connecticut) Title Uncountable free abelian groups via admissible computability [Slides] Abstract One way to study structures of uncountable cardinality κ is to generalize the notion of computation. Saying that a subset of κ is κ-c.e. if it is Σ01 definable (with parameters, in the language of set theory) over Lκ provides the notion of κ-computability. We may also quantify over subsets of Lκ, providing a notion of a κ-analytic set (here we assume V=L). In this setting, we consider the difficulty of recognizing free groups and the complexity of their bases. For example, if κ is a successor cardinal, the set of free abelian groups of size κ is Σ11-complete. The resolution of questions of this type is more complex for other κ, and a few questions remain open. This is joint work with Greenberg and Turetsky.
 Date 2017.7.21. 15:30-17:00, Room 314 Speaker Arno Pauly (Université libre de Bruxelles) Title Computability: From ωω to κκ Abstract Recently Galeotti and Nobrega [1,2] have suggested to generalize computable analysis and the theory of Weihrauch degrees to higher cardinalities. The central role taken by Baire space ωω in the classic theories is then filled by κκ for a cardinal κ with κ<κ, the reals are replaced by initial segments of the surreal numbers and Turing machines are generalized to ordinal time Turing machines. Initial investigations have reveiled that some results carry over directly, whereas other core questions can become inpendent of ZFC. I will outline both the classic theory and its the generalization to higher cardinalities. In particular, I will highlight some open questions and challenges. [1] Lorenzo Galeotti: A candidate for the generalised real line, CiE 2016. [2] Lorenzo Galeotti and Hugo Nobrega: Towards computable analysis on the generalised real line, CiE 2017.
 Date 2017.7.14. 15:30-17:00, Room 206 Speaker Yasuo Yoshinobu (Nagoya Univ.) Title Preserving forcing axioms
 Date 2017.6.30. 15:30-17:00, Room 206 Speaker Hiroaki Minami (Aichi Gakuin Univ.) Title Many simple cardinal invariants, Part 8
 Date 2017.6.23. 15:30-17:00, Room 206 Speaker Hiroaki Minami (Aichi Gakuin Univ.) Title Many simple cardinal invariants, Part 7
 Date 2017.6.12. 15:00-, Room 322 Speaker Antonio Montalbán (UC Berkeley) Title A classification of the natural many-one degrees [Slides] Abstract A common phenomenon in mathematics is that naturally-occurring objects behave better than general objects. This is definitely the case of the many-one degrees within Computability Theory. Our theorem, in a sense, completely classifies the natural many-one degrees and sets them apart from the non-natural ones. The theorem is a version of the uniform Martin's conjecture, but for the case of the many-one degrees.
 Date 2017.6.9. 15:30-17:00, Room 206 Speaker Hiroaki Minami (Aichi Gakuin Univ.) Title Many simple cardinal invariants, Part 6
 Date 2017.5.26. 15:30-17:00, Room 206 Speaker Hiroaki Minami (Aichi Gakuin Univ.) Title Many simple cardinal invariants, Part 5
 Date 2017.5.12. 15:30-17:00, Room 206 Speaker Hiroaki Minami (Aichi Gakuin Univ.) Title Many simple cardinal invariants, Part 4
 Date 2017.4.28. 15:30-17:00, Room 206 Speaker Hiroaki Minami (Aichi Gakuin Univ.) Title Many simple cardinal invariants, Part 3
 Date 2017.4.21. 15:30-17:00, Room 206 Speaker Hiroaki Minami (Aichi Gakuin Univ.) Title Many simple cardinal invariants, Part 2
 Date 2017.3.31. 15:30-17:00, Room 206 Speaker Hiroaki Minami (Aichi Gakuin Univ.) Title Many simple cardinal invariants, Part 1