Updated on 2024/05/22

写真a

 
USUBA, Toshimichi
 
Affiliation
Faculty of Science and Engineering, School of Fundamental Science and Engineering
Job title
Professor
Degree
Ph. D. ( Nagoya University )
Profile

I am working in set theory, especially set theory with large cardinals.

Research Experience

  • 2021.04
    -
    Now

    Waseda University,   School of Fundamental Science and Engineering

  • 2016.04
    -
    2021.03

    Waseda University   School of Fundamental Science and Engineering   Associate Professor

  • 2013.04
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    2016.03

    Kobe University

  • 2010.09
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    2013.03

    Nagoya University   Nagoya University

  • 2009.05
    -
    2010.08

    University of Bonn   Hausdorff Center for Mathematics   Post-Doc Researcher

  • 2008.04
    -
    2009.03

    Yamagata University   Faculty of Science

  • 2008.04
    -
    2009.03

    Tohoku University   Graduate School of Science

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Professional Memberships

  •  
     
     

    JAPAN ASSOCIATION FOR PHILOSOPHY OF SCIENCE

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    European Set Theory Society

  •  
     
     

    Association for Symbolic Logic

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    THE MATHEMATICAL SOCIETY OF JAPAN

Research Areas

  • Applied mathematics and statistics   Axiomatic Set Theory / Basic mathematics   Axiomatic Set Theory / Applied mathematics and statistics   数学基礎論 / Basic mathematics   数学基礎論

Research Interests

  • Axiom of Choice

  • 集合論的位相空間論

  • 一般位相空間論

  • large cardinal axioms

  • Set theory

  • forcing

  • 集合論

  • 巨大基数

  • 数学基礎論

  • 公理的集合論

  • 巨大基数公理

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Awards

  • 2014年度日本数学会賞建部賢弘賞

    2014.09  

    Winner: 薄葉 季路

 

Papers

  • Geology of symmetric grounds

    Toshimichi Usuba

    To appear in Proceedings of Asian Logic Conference 2019    2021  [Refereed]

  • Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds

    Toshimichi Usuba

    Springer Proceedings in Mathematics & Statistics     161 - 179  2021  [Refereed]

    DOI

    Scopus

    2
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  • Extendible cardinals and the mantle

    Toshimichi Usuba

    Archive for Mathematical Logic   58 ( 1-2 ) 71 - 75  2019.02  [Refereed]

     View Summary

    © 2018, Springer-Verlag GmbH Germany, part of Springer Nature. The mantle is the intersection of all ground models of V. We show that if there exists an extendible cardinal then the mantle is the smallest ground model of V.

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  • G <inf>δ</inf> -topology and compact cardinals

    Toshimichi Usuba

    Fundamenta Mathematicae   246 ( 1 ) 71 - 87  2019  [Refereed]

     View Summary

    c Instytut Matematyczny PAN, 2019 For a topological space X, let X δ be the space X with the G δ -topology of X. For an uncountable cardinal κ, we prove that the following are equivalent: (1) κ is ω1-strongly compact. (2) For every compact Hausdorff space X, the Lindelöf degree of X δ is ≤ κ. (3) For every compact Hausdorff space X, the weak Lindelöf degree of X δ is ≤ κ. This shows that the least ω1-strongly compact cardinal is the supremum of the Lindelöf and the weak Lindelöf degrees of compact Hausdorff spaces with the G δ -topology. We also prove that the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with the G δ -topology. For the square of a Lindelöf space, using a weak G δ -topology, we prove that the following are consistent: (1) The least ω1-strongly compact cardinal is the supremum of the (weak) Lindelöf degrees of the squares of regular T1 Lindelöf spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular T1 Lindelöf spaces.

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  • The downward directed grounds hypothesis and very large cardinals

    Toshimichi Usuba

    Journal of Mathematical Logic   17 ( 2 )  2017.12  [Refereed]

     View Summary

    © 2017 World Scientific Publishing Company. A transitive model M of ZFC is called a ground if the universe V is a set forcing extension of M. We show that the grounds ofV are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) V has only set many grounds if and only if the mantle is a ground. We also show that if the universe has some very large cardinal, then the mantle must be a ground.

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  • Large regular Lindelöf spaces with points Gδ

    Toshimichi Usuba

    Fundamenta Mathematicae   237 ( 3 ) 249 - 260  2017  [Refereed]

     View Summary

    © Instytut Matematyczny PAN, 2017. By analyzing Dow's construction, we introduce a general construction of regular Lindelof spaces with points Gδ. Using this construction, we prove the following: Suppose that either (1) there exists a regular Lindel of P-space of pseudocharacter ≤ ω1 and of size > 2ω, (2) CH and (ω2) hold, or (3) CH holds and there exists a Kurepa tree. Then there exists a regular Lindel of space with points Gδ and of size > 2ω. This shows that, under CH, the non-existence of such a Lindel of space has a large cardinal strength. We also prove that every c.c.c. forcing adding a new real creates a regular Lindel of space with points Gδ and of size at least (2ω1 )V.

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  • The list-chromatic number and the coloring number of uncountable graphs

    Toshimichi Usuba

    Israel Journal of Mathematics    2023.06  [Refereed]

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  • C-embedding and P-embedding in subspaces of products of ordinals

    Nobuyuki Kemoto, Toshimichi Usuba

    Topology and its Applications   318  2022.08  [Refereed]

     View Summary

    It is known that in X=A×B, where A and B are subspaces of ordinals, all closed C⁎-embedded subspaces of X are P-embedded. Also it is asked whether all closed C⁎-embedded subspaces of X are P-embedded whenever X is a subspace of products of two ordinals. In this paper, we prove that both of the following are consistent with ZFC: • there is a subspace X of (ω+1)×ω1 such that the closed subspace X∩({ω}×ω1) is C⁎-embedded in X but not P-embedded in X, • for every subspace X of (ω+1)×ω1, if the closed subspace X∩({ω}×ω1) is C⁎-embedded in X, then it is P-embedded in X.

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  • Equalities for the extent of infinite products and Σ-products

    Yasushi Hirata, Toshimichi Usuba, Yukinobu Yajima

    Topology and its Applications   307  2022.02  [Refereed]

     View Summary

    For a space X, let e(X)=ω⋅sup{|D|:D is a closed discrete subset in X}, which is called the extent of X. Here we deal with the following two questions: (1) For a product space X=∏λ∈ΛXλ, when is e(X)=|Λ|⋅sup{e(Xλ):λ∈Λ}? (2) For a Σ-product Σ of spaces Xλ,λ∈Λ, when is e(Σ)=sup{e(Xλ):λ∈Λ}? We show that the equalities in these questions hold if each Xλ is a strict p-space or a strong Σ-space and, in the case of the first question, if the cardinality of the index set Λ is less than the first weakly inaccessible. For semi-stratifiable spaces, we show that a slightly weaker form of these equalities holds.

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  • Strongly compact cardinals and the continuum function

    Arthur W. Apter, Stamatis Dimopoulos, Toshimichi Usuba

    Annals of Pure and Applied Logic   172 ( 9 ) 103013 - 103013  2021.10  [Refereed]

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  • A note on δ-strongly compact cardinals

    Toshimichi Usuba

    Topology and its Applications   301   107538 - 107538  2021.09  [Refereed]

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  • On Countable Stationary Towers

    Yo Matsubara, Toshimichi Usuba

    Springer Proceedings in Mathematics & Statistics     133 - 141  2021  [Refereed]

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  • A note on the tightness of G<inf>δ</inf>-modifications

    Toshimichi Usuba

    Topology and its Applications   265  2019.09  [Refereed]

     View Summary

    © 2019 Elsevier B.V. We construct a normal countably tight T1 space X with t(Xδ)>2ω. This is an answer to the question posed by Dow-Juhász-Soukup-Szentmiklóssy-Weiss [5]. We also show that if the continuum is not so large, then the tightness of Gδ-modifications of countably tight spaces can be arbitrary large up to the least ω1-strongly compact cardinal.

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  • On the existence of skinny stationary subsets

    Yo Matsubara, Hiroshi Sakai, Toshimichi Usuba

    Annals of Pure and Applied Logic   170 ( 5 ) 539 - 557  2019.05  [Refereed]

     View Summary

    © 2018 Elsevier B.V. Matsubara–Usuba [13] introduced the notion of skinniness and its variants for subsets of P κ λ and showed that the existence of skinny stationary subsets of P κ λ is related to large cardinal properties of ideals over P κ λ and to Jensen's diamond principle on λ. In this paper, we study the existence of skinny stationary sets in more detail.

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  • Products of Lindelöf spaces with points G<inf>δ</inf>

    Toshimichi Usuba

    Topology and its Applications   252   90 - 96  2019.02  [Refereed]

     View Summary

    © 2018 We show that if CH holds and either (i) there exists an ω1-Kurepa tree, or (ii) □(ω2) holds, then there are regular T1 Lindelöf spaces X0 and X1 with points Gδ such that the extent of X0×X1 is strictly greater than 2ω.

    DOI J-GLOBAL

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  • New combinatorial principle on singular cardinals and normal ideals

    Toshimichi Usuba

    Mathematical Logic Quarterly   64 ( 4-5 ) 395 - 408  2018.11  [Refereed]

     View Summary

    © 2018 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim We introduce a new combinatorial principle on singular cardinals. This principle allows us to take a kind of a diagonal intersection of more than λ many measure one sets of certain normal ideals over ℘(λ). Under the principle, we give various characterizations of the saturation property of normal ideals over ℘(λ). We also consider Chang's type transfer properties under the principle, and, when λ is Jónsson, we prove that every normal ideal over ℘(λ) with {x ⊆ λ : |x| = λ} measure one cannot have strong properties.

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  • Subtlety and partition relations

    Toshimichi Usuba

    Mathematical Logic Quarterly   62 ( 1-2 ) 59 - 71  2016.02  [Refereed]

     View Summary

    © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. We study the subtlety of a cardinal κ and of kΛ. We show that, under a certain large cardinal assumption, it is consistent that kΛ is subtle for some Λ > κ but κ is not subtle, and the consistency of such a situation is much stronger than the existence of a subtle cardinal. We show that the subtlety of PkΛ can be characterized by a certain partition relation on kΛ. We also study the property of faintness of κ, and the subtlety of kΛ with the strong inclusion.

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  • Superstrong and other large cardinals are never Laver indestructible

    Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis, Toshimichi Usuba

    Archive for Mathematical Logic   55 ( 1-2 ) 19 - 35  2016.02  [Refereed]

     View Summary

    © 2015, Springer-Verlag Berlin Heidelberg. Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σn-reflecting cardinals, Σn-correct cardinals and Σn-extendible cardinals (all for n ≥  3) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing (Formula presented.), the cardinal κ will exhibit none of the large cardinal properties with target θ or larger.

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  • Reflection principles for uω<inf>2</inf>and the semi-stationary reflection principle

    Toshimichi Usuba

    Journal of the Mathematical Society of Japan   68 ( 3 ) 1081 - 1098  2016  [Refereed]

     View Summary

    © 2016 The Mathematical Society of Japan. Starting from a model with a weakly compact cardinal, we construct a model in which the weak stationary reflection principle for uω2 holds but the Fodor-type reflection principle for uω2 fails. So the stationary reflection principle for uω2 fails in this model. We also construct a model in which the semi-stationary reflection principle holds but the Fodor-type reflection principle for uω2 fails.

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  • Two-cardinal versions of weak compactness: Partitions of triples

    Pierre Matet, Toshimichi Usuba

    Journal of the Mathematical Society of Japan   67 ( 1 ) 207 - 230  2015  [Refereed]

     View Summary

    © 2015 The Mathematical Society of Japan. Let k be a regular uncountable cardinal, and λ be a cardinal greater than k. Our main result asserts that if (λ<k)<(λ<k) = λ<k, then (pk,λ(NInk,λ<k))+ → + (NS[λ]<k k;λ )+; NSk;λs+)3 and (pk,λ(NAInk;λ<k))+ → (NSk;λs+)3, where NSk;λs (respectively, NS[λ]<k k;λ) denotes the smallest seminormal (respectively, strongly normal) ideal on Pk(λ), NInk,λ<k (respectively, NAInk;λ<k) denotes the ideal of non-ineffable (respectively, non-almost ineffable) subsets of Pk(λ<k), and pk,λ : Pk(λ<k) → Pk(λ) is defined by pk,λ(x) = x ∩ λ.

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  • The cardinality of compact spaces satisfying the countable chain condition

    Toshimichi Usuba

    Topology and its Applications   174 ( 1 ) 41 - 55  2014.09  [Refereed]

     View Summary

    We prove that for a compact Hausdorff space X, if λc(X)<w(X) for every infinite cardinal λ<w(X) and λc(X)<cf(w(X)) for every infinite cardinal λ<cf(w(X)), then Tikhonov cube [0,1]w(X) is a continuous image of X, in particular the cardinality of X is just 2w(X). As an application of this result, we consider elementary submodel spaces and improve Tall's result in [17]. © 2014 Elsevier B.V.

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  • Bounded dagger principles

    Toshimichi Usuba

    Mathematical Logic Quarterly   60 ( 4-5 ) 266 - 272  2014.08  [Refereed]

     View Summary

    For an uncountable cardinal κ, let (†)κ be the assertion that every ω1-stationary preserving poset of size ≤κ is semiproper. We prove that (†)ω2 is a strong principle which implies a strong form of Chang's conjecture. We also show that (†)2ω1 implies that NS ω1 is presaturated. © 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

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  • Characters of countably tight spaces and inaccessible cardinals

    Toshimichi Usuba

    Topology and its Applications   161 ( 1 ) 95 - 106  2014  [Refereed]

     View Summary

    In this paper, we study some connections between characters of countably tight spaces of size ω1 and inaccessible cardinals. A countable tight space is indestructible if every σ-closed forcing notion preserves countable tightness of the space. We show that, assuming the existence of an inaccessible cardinal, the following statements are consistent:(1)Every indestructibly countably tight space of size ω1 has character ≤ω1.(2)2ω1>ω2 and there is no countably tight space of size ω1 and character ω2. For the converse, we show that, if ω2 is not inaccessible in the constructible universe L, then there is an indestructibly countably tight space of size ω1 and character ω2. © 2013 Elsevier B.V.

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  • Lindelöf spaces with small pseudocharacter and an analog of Borel's conjecture for subsets of [0; 1]@1

    Franklin D. Tall, Toshimichi Usuba

    Houston Journal of Mathematics   40 ( 4 ) 1299 - 1309  2014  [Refereed]

     View Summary

    © 2014 University of Houston. e improve results of Shelah, Tall, and Scheepers concerning the cardinality of Lindelöf spaces with small pseudocharacter. We establish the consistency of an analog of Borel's Conjecture for subspaces of [0; 1]@1 .

  • On skinny stationary subsets of Pkλ

    Yo Matsubara, Toshimichi Usuba

    Journal of Symbolic Logic   78 ( 2 ) 667 - 680  2013.06  [Refereed]

     View Summary

    We introduce the notion of skinniness for subsets ofP ë and its variants, namely skinnier and skinniest. We show that under some cardinal arithmetical assumptions, precipitousness or 2ë-saturation of NSë | X, where NSë denotes the non-stationary ideal over Pë, implies the existence of a skinny stationary subset of X. We also show that if ë is a singular cardinal, then there is no skinnier stationary subset of Pë. Furthermore, if ë is a strong limit singular cardinal, there is no skinny stationary subset of Pë. Combining these results, we show that if ë is a strong limit singular cardinal, then NSë | X can satisfy neither precipitousness nor 2ë-saturation for every stationary X Pë. We also indicate that ë(Eë <), where Eë < def = { < ë cf() < }, is equivalent to the existence of a skinnier (or skinniest) stationary subset of Pë under some cardinal arithmetical hypotheses.© 2013, Association for Symbolic Logic.

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  • Hierarchies of ineffabilities

    Toshimichi Usuba

    Mathematical Logic Quarterly   59 ( 3 ) 230 - 237  2013.05  [Refereed]

     View Summary

    We study combinatorial large cardinal properties on P κ λ, such as ineffability, almost ineffability, subtlety, and the Shelah property. We show that, even when λ > κ, the almost ineffability of P κ λ does not yield the ineffability of κ. We also show that the Shelah property and the partition property of P κ λ do not yield the subtlety of κ. © 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

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  • Notes on the partition property of Pκλ

    Yoshihiro Abe, Toshimichi Usuba

    Archive for Mathematical Logic   51 ( 5-6 ) 575 - 589  2012.08  [Refereed]

     View Summary

    We investigate the partition property of Pκλ. Main results of this paper are as follows: (1) If λ is the least cardinal greater than κ such that Pκλ carries a (λ <κ, 2)-distributive normal ideal without the partition property, then λ is Π n1-indescribable for all n < ω but not Π 12 -indescribable. (2) If cf(λ) ≥ κ, then every ineffable subset of Pκλ has the partition property. (3) If cf(λ) ≥ κ, then the completely ineffable ideal over Pκλ has the partition property. © 2012 Springer-Verlag.

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  • SPLITTING STATIONARY SETS IN p(lambda)

    Toshimichi Usuba

    JOURNAL OF SYMBOLIC LOGIC   77 ( 1 ) 49 - 62  2012.03  [Refereed]

     View Summary

    Let A be a non-empty set. A set S subset of p(A) is said to be stationary in p(A) if for every f: [A](&lt;omega) -&gt; A there exists x is an element of S such that x not equal A and f"[x](&lt;omega) subset of x. In this paper we prove the following: For an uncountable cardinal 1 and a stationary set S in p(lambda), if there is a regular uncountable cardinal kappa &lt;= lambda such that {x is an element of S : x boolean AND kappa is an element of kappa} is stationary, then S can be split into kappa disjoint stationary subsets.

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  • Two-cardinal versions of weak compactness: Partitions of pairs

    Pierre Matet, Toshimichi Usuba

    ANNALS OF PURE AND APPLIED LOGIC   163 ( 1 ) 1 - 22  2012.01  [Refereed]

     View Summary

    We study various partition properties on P(kappa)(lambda). Our main result asserts that lambda(&lt;lambda&lt;kappa) = lambda(&lt;kappa), then (p(NSh(kappa,lambda&lt;kappa)))(+) --&gt; (NSS(kappa,lambda)(+))(2), where p : P(kappa)(lambda(&lt;kappa)) --&gt; P(kappa)(lambda) is defined by p(x) = x boolean AND lambda. (C) 2011 Elsevier BM. All rights reserved.

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  • Roles of Large Cardinals in Set Theory(<Special Section>New Developments in Logic: Proof-Theoretic Ordinals and Set-Theoretic Ordinals)

    USUBA Toshimichi, FUJITA Hiroshi

    Journal of the Japan Association for Philosophy of Science   39 ( 2 ) 83 - 92  2012

     View Summary

    This article is to give a brief survey of roles of large cardinals (those ordinals which are in certain sense very big) in set theory of our time. In particular, close relationship of the structure of the continuum to large cardinals is emphasized. We also mention the inner model method which is a comparable approach to large cardinal axioms, so that we could make clear the reason why large cardinals are so important.

    DOI CiNii

  • Roles of large cardinals in set theory

    Hiroshi Fujita, Toshimichi Usuba

    Journal of the Japan Association for Philosophy of Science   39 ( 2 )  2012  [Refereed]

  • Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelofness

    Sakae Fuchino, Istvan Juhasz, Lajos Soukup, Zoltan Szentmiklossy, Toshimichi Usuba

    TOPOLOGY AND ITS APPLICATIONS   157 ( 8 ) 1415 - 1429  2010.06  [Refereed]

     View Summary

    We introduce a new reflection principle which we call "Fodor-type Reflection Principle" (FRP) This principle follows from but is strictly weaker than Fleissner&apos;s Axiom R For instance, FRP does not impose any restriction on the size of the continuum. while Axiom R implies that the continuum has size &lt;= aleph(2)
    We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelof if all of its subspaces of cardinality &lt;= aleph(1) are (Theorem 4 3) It follows that. under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality &lt;= aleph(1) are (Corollary 4 4) This improves a result of Balogh who proved the same assertion under Axiom R.
    We also give several other results in this vein, some in ZFC. others in some further extension of ZFC For example, we prove in ZFC that if X is a locally (countably) compact. space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5 2) (C) 2009 Elsevier B V All rights reserved

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  • Splitting stationary sets in P-kappa lambda for lambda with small cofinality

    Toshimichi Usuba

    FUNDAMENTA MATHEMATICAE   205 ( 3 ) 265 - 287  2009  [Refereed]

     View Summary

    For a regular uncountable cardinal kappa and a cardinal lambda with cf(lambda) &lt; kappa &lt; lambda, we investigate the consistency strength of the existence of a stationary set in P-kappa lambda which cannot be split into lambda(+) many pairwise disjoint stationary subsets. To do this, we introduce a new notion for ideals, which is a variation of normality of ideals. We also prove that there is a stationary set S in P-kappa lambda such that every stationary subset of S can be split into lambda(+) many pairwise disjoint stationary subsets.

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  • Ineffability of P-kappa lambda for lambda with small cofinality

    Toshimichi Usuba

    JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN   60 ( 3 ) 935 - 954  2008.07  [Refereed]

     View Summary

    We study ineffability, the Shelah property, and indescribability of P-kappa lambda when cf(lambda) &lt; kappa. We prove that if lambda is a strong limit cardinal with cf(lambda) &lt; kappa then the ineffable ideal, the Shelah ideal, and the completely ineffable ideal over P-kappa(lambda) are the same, and that it can be precipitous. Furthermore we show that Pi(1)(1)-indescribability of P-kappa(lambda) is much stronger than ineffability if 2(lambda) = lambda(&lt;kappa).

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  • Local saturation of the non-stationary ideal over P-k lambda

    Toshimichi Usuba

    ANNALS OF PURE AND APPLIED LOGIC   149 ( 1-3 ) 100 - 123  2007.11  [Refereed]

     View Summary

    Starting with lambda-supercompact cardinal K, where lambda is a regular cardinal greater than or equal to K, we produce a model with a stationary subset S of P-k lambda such that NSk lambda vertical bar S, the ideal generated by the non-stationary ideal NSk lambda over P-k lambda together with P-k lambda\S, is; lambda(+)-saturated. Using this model we prove the consistency of the existence of such a stationary set together with the Generalized Continuum Hypothesis (GCH).
    We also show that in our model we can make NSk lambda vertical bar S(k, lambda) lambda(+)-saturated, where S(k, lambda) is the set of all x epsilon P-k lambda such that ot(x), the order type of x, is a regular cardinal and x is stationary in sup(x). Furthermore we construct a model where NSk lambda vertical bar S(k, lambda) is k(+)-saturated but GCH fails. We show that if S\S(k, lambda) is stationary in P-k lambda, then S can be split into lambda many disjoint stationary subsets. (c) 2007 Elsevier B.V. All rights reserved.

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Books and Other Publications

Presentations

  • Löwenheim-Skolem-Tarski property of rings of continuous functions

    Toshimichi Usuba  [Invited]

    Symposium on Advances in Mathematical Logic 2022 

    Presentation date: 2022.06

  • Generic setwise large cardinals

    Toshimichi Usuba  [Invited]

    Kobe Set Theory Workshop 2021 

    Presentation date: 2021.03

    Event date:
    2021.03
    -
     
  • Large cardinals as natural upper bounds on cardinal functions

    Toshimichi Usuba  [Invited]

    3rd Pan-Pacific International Conference on Topology and Applications  (Chengdu) 

    Presentation date: 2019.11

  • Choiceless set-theoretic geology

    Toshimichi Usuba  [Invited]

    Asian Logic Conference 2019 

    Presentation date: 2019.06

  • Syntactical and semantical approach to generic multiverse

    Toshimichi Usuba  [Invited]

    Sendai Logic School 2018 

    Presentation date: 2018.12

  • Set-theoretic geologies

    Toshimichi Usuba  [Invited]

    6th European Set Theory Conference 

    Presentation date: 2017.07

  • Lindeloef spaces and large cardinals

    Toshimichi Usuba  [Invited]

    Toposym 2016 

    Presentation date: 2016.07

  • Uniform ultrafilters in a choiceless context

    Toshimichi Usuba

    Presentation date: 2024.03

  • 空集合について

    薄葉季路

    第9回山陰数学と基礎論研究集会 

    Presentation date: 2024.01

  • Monotonicity of the ultra lter number

    Toshimichi Usuba

    Large cardinals and the continuum 

    Presentation date: 2023.10

  • Los's theorem without the axiom of choice

    Toshimichi Usuba

    Presentation date: 2023.09

  • C∗-embedding and P-embedding in subspaces of products of ordinals

    Presentation date: 2023.06

  • Monotonicity of ultrafilter numbers

    Toshimichi Usuba

    Presentation date: 2023.03

  • 可算集合の可算和について(ただし選択公理は仮定しない)

    薄葉季路

    第8回山陰基礎論と数学およびその周辺の研究集会 

    Presentation date: 2023.01

  • The continuum function on the countable unions of countable sets

    Rin Miyauchi, Toshimichi Usuba

    Presentation date: 2022.09

  • Equalities for the extent of infinite products and Σ-products

    矢島幸信, 平田康史, 薄葉季路

    日本数学会2021 年度秋季総合分科会 

    Presentation date: 2021.09

  • Generically extendible cardinals

    Toshimichi Usuba

    Presentation date: 2021.09

  • On Reinhardt cardinals

    薄葉季路

    日本数学会2020年度秋季総合分科会 

    Presentation date: 2020.09

  • Variants of Strong Chang's Conjecure

    Presentation date: 2020.03

  • Geology of symmetric grounds

    Toshimichi Usuba

    Set Theory and Infinity 

    Presentation date: 2019.11

  • Standard and choiceless set-theoretic geology

    薄葉季路

    Summer School 2019 

    Presentation date: 2019.08

  • Choiceless set-theoretic geology

    Toshimichi Usuba

    Higher recursion theory and set theory 

    Presentation date: 2019.06

  • GCH at strongly compact cardinals

    薄葉季路

    日本数学会2019年度年会 

    Presentation date: 2019.03

  • 多元宇宙の理論

    薄葉季路

    山陰 基礎論・解析学セミナー 2019 

    Presentation date: 2019.02

  • Axiomatization of Generic Multiverse

    薄葉季路

    証明論研究集会 

    Presentation date: 2018.12

  • 数学基礎論と位相空間論のコンパクト

    薄葉季路  [Invited]

    筑波大学数学域談話会 

    Presentation date: 2018.11

  • Extendible cardinals and the mantle

    薄葉季路

    日本数学会2018年度秋季総合分科会 

    Presentation date: 2018.09

  • The generic multiverse and large cardinals

    Toshimichi Usuba

    Symposium on Advances in Mathematical Logic 2018 

    Presentation date: 2018.09

  • omega_1-strongly compact cardinals and cardinal functions in topology

    Toshimichi Usuba

    Reflections on Set Theoretic Reflection 

    Presentation date: 2018.09

  • G_delta-topology and compact cardinals

    薄葉季路

    一般位相幾何学の進展と諸問題 

    Presentation date: 2018.06

  • Products of Lindeloef spaces

    薄葉季路

    日本数学会2018年度年会 

    Presentation date: 2018.03

  • Compactness cardinals and covering properties of topological spaces

    薄葉季路  [Invited]

    手形L4研究集会 

    Presentation date: 2018.03

  • 多元宇宙論入門

    薄葉季路

    山陰 基礎論・解析学セミナー 2018 

    Presentation date: 2018.01

  • 集合論の宇宙と強制法

    薄葉季路  [Invited]

    東北大学理学部数学科談話会 

    Presentation date: 2017.12

  • G_delta-modification and large cardinals

    Toshimichi Usuba

    Iterated Forcing Theory and Cardinal Invariants 

    Presentation date: 2017.11

  • Set-theoretic geoligies

    Toshimichi Usuba  [Invited]

    Workshop on Computability Theory and Foundations of Mathematics 

    Presentation date: 2017.09

  • パラコンパクト空間と強制法公理

    薄葉季路  [Invited]

    第64回トポロジーシンポジウム 

    Presentation date: 2017.08

  • 集合論の宇宙 -univreseとmultiverse-

    薄葉季路  [Invited]

    日本数学会2017年度年会 

    Presentation date: 2017.03

  • The Downward Directed Ground Hypothesis

    Toshimichi Usuba

    Academy of Mathematics and Systems Science Colloquium 

    Presentation date: 2017.02

  • 大域的と局所的: パラコンパクトを題材に

    薄葉季路

    山陰 基礎論・解析学セミナー 2017 

    Presentation date: 2017.01

  • Paracompactness of Locally Lindeloef spaces

    薄葉季路

    2016年度ジェネラルトポロジーシンポジウム 

    Presentation date: 2016.12

  • Definability of ground models in ZF

    Toshimichi Usuba

    Infinite Combinatorics and Forcing Theory 

    Presentation date: 2016.12

  • Set-theoretic geology without AC

    薄葉季路

    日本数学会2016年度秋季総合分科会 

    Presentation date: 2016.09

  • The universe and the multiverse

    薄葉季路

    科学基礎論学会2016年度総会・講演会 

    Presentation date: 2016.06

  • 有向集合の分類

    薄葉季路  [Invited]

    山陰 基礎論・解析学セミナー 2016 

    Presentation date: 2016.01

  • The Downward Directed Grounds hypothesis

    Toshimichi Usuba

    IMS-JSPS Joint Workshop in Mathematical Logic and Foundations of Mathematics 

    Presentation date: 2016.01

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Research Projects

  • Set theoretic multiverse and large cardinal axioms

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research

    Project Year :

    2018.04
    -
    2021.03
     

    Usuba Toshimichi

     View Summary

    We studied set-theoretic multiverse, in particular the structure of the entire multiverse under the large cardinal axiom, and obtained the following results: 1. If there exists a large cardinal such as an extendible cardinal, there exists an inner model that is invariant with respect to the forcing method. 2. Even in the absence of the Axiom of Choice, all ground models are uniformly definable if there are enough large cardinals, and thus it is possible to develop a multiverse theory without the Axiom of Choice. 3. Symmetric extension is a generalization of the forcing extension. We showed that all symmetric grounds are uniformly definable, and thus it is possible to develop a more extensive multiverse theory including the symmetric extensions.

  • On combinatiral problems using P_kappa lambda structures

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Young Scientists (B)

    Project Year :

    2015.04
    -
    2018.03
     

    Usuba Toshimichi, Bagaria Joan, Hamkins Joel David, Tsaprounis Konstantinos, MATSUBARA Yo, SAKAI Hiroshi, ISHII Hiromi, YAMAURA Naoki

     View Summary

    On infinitely combinatorics, we obtained various results such as: applications of the unbranching principle to the ideal theory, comparison between cardinals and P_kappa lambda via combinatorial properties, applications of large cardinals to Lindeloef spaces. For the reflection principle, we separated the strong reflection principle from the weak one, and got the characterization of paracompact spaces.
    In addition, for the set-theoretic multiverse and the set-theoretic geology, we showed the fundamental theorem that the downward directedness of all ground models. Furthermore, we proved that there exists the minimum ground model if there exists a large cardinal.

  • Pκλ上のイデアルの構造的性質と無限組合せ論

    日本学術振興会  科学研究費助成事業 基盤研究(C)

    Project Year :

    2018.04
    -
    2021.03
     

    阿部 吉弘, 薄葉 季路, 南 裕明

     View Summary

    Local P-pointの次のような特徴付けを与えた:WをPκλのunbounded setによる分割とするとき、bounded イデアルとWの元から生成されるイデアルをJ(W)とする。Iがlocal P-pointではないことと、あるWに対してJ(W)をIが含んでいることは同値である。
    イデアルのカテトフ順序に関連して、ED_fin イデアルについての有限分岐するtree type強制法がどの程度の基数不変量を保存するかを調べた。
    無限帽子パズルとイデアルの組み合わせ論:(1)自然数人囚人がいて一方通行の視界しかない場合、囚人の視界は自然数上のグラフとして表現できる。双対フィルターI*-many 正解になるような戦略が存在するならば、グラフの意味でI positive-manyの独立集合は存在しない。この逆が成り立つイデアルをNISイデアルとよぶ。極大イデアル I に関しては、I*がRamsey ultrafilterであることとIがNISイデアルであることは同値である。Definable(Borel,analytic,co-analytic)なイデアルでNISイデアルとなるようなものがないか調べている。(2)「視界」は十分あるが,色の見分けがつかない場合について、必勝法や必敗法のあるようなゲームの枠組みを自然数の集合の分割を用いて調べた。
    強制法への波及効果:適当な巨大基数の仮定の下では選択公理を使わずに集合論的地質学展開できることを示した。また、集合論的地質学をsymmetric extensionを含む形に拡張した。
    集合論的位相幾何学への応用:ω1-strongly compact cardinalのgeneral topologyへの新たな応用をいくつか得た。

  • Structural properties of ideals over Pk\lambda

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2015.04
    -
    2018.03
     

    Abe Yoshihiro

     View Summary

    We developed the theory of structural properties of ideals over Pκλ.
    First it was shown that, at the most cases, the ideal isomorphic to the bounded ideal,that is the smallest idal,does not contain the nonstationary ideal, the smallest normal ideal. Second, we define Ulam ideals similar to the case of κ, and show the bounded ideal is not Ulam, and give the characterization of Ulam ideals using the coherence of its extensions.
    Last, we study the rigidity of ideals. The relation between the rigid ideals and Ulam ideals have been turned out.

  • Deepening of the research in set-theoretic topology with forcing and large cardinal properties

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2013.04
    -
    2018.03
     

    Kada Masaru, Yoshinobu Yasuo, Tomoyasu Kazuo, Fuchino Sakaé, Usuba Toshimichi, Iwasa Akira, Kamo Shizuo, Kato Takuto, Shizuma Souji

     View Summary

    After the start of this project, several non-academic difficulties for research activities happened, which caused the delay of the project and brought less results than I expected. In particular, I could not get a significant progress in the field of general topology using the set-theoretic method of large cardinal properties, which was one of the main topic I mentioned in the proposal. However, I got several meaningful results in the field of set theory and general topology, including the following topics: (i) Convergence of a sequence to a set after forcing extensions, (ii) Interplay among variants of the axiom of choice when we discard the axiom of union, and (iii) Generalizations of hat-guessing games into infinitary combinatorics.

  • Invariant structures on medium-size infinite cardinals

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2012.04
    -
    2017.03
     

    YOSHINOBU YASUO, FUCHINO SAKAE, MIYAMOTO TADATOSHI, KADA MASARU, TOMOYASU KAZUO, SAKAI HIROSHI, USUBA TOSHIMICHI, MATSUBARA YO, König Bernhard

     View Summary

    In the study of axiomatic set theory, combinatorial properties of infinite cardinals greater than or equal to aleph 2 are less understood than those of smaller cardinals. In this research, we studied invariance of combinatorial properties of relatively small infinite cardinals greater than or equal to aleph 2 under various forcing extensions. As a result, we obtained several significant consequences, mainly about problems to what extent and with what sort of partially ordered sets well-known axioms of set theory called forcing axioms are preserved under forcing extensions.

  • Research toward a solution of Galvin's conjecture

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Challenging Exploratory Research

    Project Year :

    2014.04
    -
    2016.03
     

    Fuchino Sakae, SAKAI Hiroshi, USUBA Toshimichi

     View Summary

    Galvin's Conjecture is the assertion "any partial ordering X such that any subordering of X of size ω_1 is a union of countably many chains is by itself a union of countably many chains". Its consistency is still open. In our research, we introduced the reflection numbers corresponding to Rado's conjecture, Galvin's conjecture etc. and studied the relationships of between these cardinal numbers. Galvin's Conjecture is characterized by the corresponding reflection number being ω_2.
    We showed the consistency of a restricted form of Galvin's conjecture claiming that the Galvin type reflection number can be ω_2 for the class of partial ordering for which the property that the partial ordering is not a union of countably many chains is preserved by σ-closed forcing and that for this class it is also consistent that the reflection number is less than or equal to the continuum while the continuum is fairly large.

  • Ideals on P\kappa\lambda which does not depend on large cardinals

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2011
    -
    2013
     

    ABE Yoshihiro, USUBA Toshimichi, KAMO Shizuo, SHIOYA Masahiro

     View Summary

    The basic concepts of the structural theory of ideals on P_{\kappa}\lambda, P-points, Q-points, and selective ideals are defined, and several theorems are proved, some of which hold similarly for ideals on \kappa, whereas the others are very different from those on \kappa. We get several facts which says that some theorems on \kappa result from the special situation that \kappa=\lambda.
    It has been long known that the stationary sets of P_{\kappa}\lambda reflects when \kappa is supercompact. We construct a forcing model in which \kappa is strongly compact and P_{\kappa}\lambda has a stationary set without such reflection.

  • Reflection principles on stationarity and possible extensions of the axiom system of set thoery

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2009
    -
    2011
     

    FUCHINO Sakae, LAJOS Soukup, FRIEDMAN Sy-david, SAKAI Hiroshi, USUBA Toshimichi

     View Summary

    In this research project, we studied the so-called Fodor-type Reflection Principle(FRP) which is situated between stationarity reflection of sets of ordinals and that of sets of countable sets of ordinals. We proved that FRP is equivalent to many of the "mathematical" reflection principles which have been known as consequences of Fleissner's Axiom R. We also proved that FRP implies Shelah's Strong Hypothesis and obtained an almost complete picture of the implications and non implications among reflection principles including FRP.

  • Combinatorics on medium-sized infinite cardinals

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2008
    -
    2011
     

    YASUO Yoshinobu, FUCHINO Sakae, MATSUBARA Yo, MIYAMOTO Tadatoshi, KADA Masaru, TOMOYASU Kazuo, SAKAI Hiroshi, USUBA Toshimichi, KONIG Bernhard, B.LARSON Paul

     View Summary

    Combinatorics of infinite cardinals greater than or equal to aleph 2 take on a different aspect from that of aleph 1, that is, the least uncountable cardinal. In this research, we studied inherent combinatorial nature of relatively small infinite cardinals greater than or equal to aleph 2, from the aspect of its interaction with several well-known axioms of set theory and that of its influence on set-theoretic topology, and obtained several significant results in this area.

  • Infinite combinatorial principles and compact cardinals

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2008
    -
    2010
     

    ABE Yoshihiro, HIRATAYA Sushi, USUBA Toshimichi, KAMO Shizuo, SHIOYAMA Sahiro

     View Summary

    We proved several facts in the combinatorial set theory on Pκλ={x⊂λ:|x|<κ}, the set of all subsets of λ with cardinality less than κ. For instance :
    (1) If the cofinality ofλis smaller thanκ, then there exists a stationary subset S of Pκλsuch that every stationary subset of S can be splitted intoλ^+many disjoint stationary sets.
    (2) If the cofinality ofλis not smaller thanκand there is a weakly normal ideal onPκλ, then the cardinality of PκλisMax(2^<κ,λ).
    (3) Suppose that the cofinality ofλis not smaller thanκand X is a subset ofPκλ. Then, X is ineffable if and only if it has the partition property.

  • Search of extensions of the axiom system of set-theory form a global view point

    Japan Society for the Promotion of Science  Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)

    Project Year :

    2007
    -
    2008
     

    FUCHINO Sakae, BRENDLE Jorg, SAKAI Hiroshi, USUBA Toshimichi, JOAN Bagaria, SOUKUP Lajos, JUHASZ Istvan, SZENTMIKLOSSY Soltan

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Misc

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Syllabus

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Sub-affiliation

  • Faculty of Science and Engineering   Graduate School of Fundamental Science and Engineering

Research Institute

  • 2022
    -
    2024

    Waseda Research Institute for Science and Engineering   Concurrent Researcher