Toshimichi Usuba

activity

20132025

most citedChoiceless Löwenheim-Skolem property and uniform definability of grounds

1 citations · 3 across the 15 of their papers we have counts for

collaborators

19 papers

math.LO2025

Variants of Łoś's Theorem

Toshimichi Usuba

We study Łoś's theorem in a choiceless context. We introduce some variants of Łoś's theorem. These variants seem weaker than Łoś's theorem, but we prove that these are equivalent t…

math.LO2025

Monotonicity of the ultrafilter number function

Toshimichi Usuba

We investigate whether the ultrafilter number function $κ \mapsto u (κ)$ on the cardinals is monotone, that is, whether $u (λ) \leq u (κ)$ holds for all…

math.LO2024

On Recurrence Axioms

Sakaé Fuchino, Toshimichi Usuba

The Recurrence Axiom for a class $P$ of \pos\ and a set $A$ of parameters is an axiom scheme in the language of ZFC asserting that if a statement with parameters from $A$ …

math.LO2024

A note on uniform ultrafilters in a choiceless context

Toshimichi Usuba

In \cite{HK}, Hayut and Karagila asked some questions about uniform ultrafilters in a choiceless context. We provide several answers to their questions.

math.LO2023

A note on Łoś's Theorem without the Axiom of Choice

Toshimichi Usuba

We study some topics about Łoś's theorem without assuming the Axiom of Choice. We prove that Łoś's fundamental theorem of ultraproducts is equivalent to a weak form that every ultr…

math.LO2022

Generically extendible cardinals

Toshimichi Usuba

In this paper, we study the notion of a generically extendible cardinal, which is a generic version of an extendible cardinal. We prove that the generic extendibility of $ω_{1}$ or $…

math.LO2020

On Countable Stationary Towers

Yo Matsubara, Toshimichi Usuba

In this paper, we investigate properties of countable stationary towers. We derive the regularity properties of sets of reals in $L (R)$ from some properties of countable st…

math.LO2020

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

Toshimichi Usuba

We study the list-chromatic number and the coloring number of graphs, especially uncountable graphs. We show that the coloring number of a graph coincides with its list-chromatic n…

math.LO2020★ 1 cited

A note on Löwenheim-Skolem cardinals

Toshimichi Usuba

In this note we provide some applications of Löwenheim-Skolem cardinals introduced in \cite{U}.

math.LO2020

A note on $δ$ -strongly compact cardinals

Toshimichi Usuba

In this paper we investigate more characterizations and applications of $δ$ -strongly compact cardinals. We show that, for a cardinal $κ$ the following are equivalent: (1) $κ$ is $δ…

math.LO2019

Geology of symmetric grounds

Toshimichi Usuba

Let us say that a model of $ZF$ is a symmetric ground if $V$ is a symmetric extension of the model. In this paper, we investigate set-theoretic geology of symmetric ground…

math.LO2019★ 1 cited

Choiceless Löwenheim-Skolem property and uniform definability of grounds

Toshimichi Usuba

In this paper, without the axiom of choice, we show that if a certain downward Löwenheim-Skolem property holds then all grounds are uniformly definable. We also prove that the axio…

math.LO2019

A note on the tightness of $G_{δ}$ -modifications

Toshimichi Usuba

We construct a normal countably tight $T_{1}$ space $X$ with $t (X_{δ}) > 2^{ω}$ . This is an answer to the question posed by Dow-Juhász-Soukup-Szentmiklóssy-Weiss. We also show that if the…

math.LO2019

Strongly compact cardinals and the continuum function

Arthur W. Apter, Stamatis Dimopoulos, Toshimichi Usuba

We study the general problem of the behaviour of the continuum function in the presence of non-supercompact strongly compact cardinals.

math.LO2018

Products of Lindelöf spaces with points $G_{δ}$

Toshimichi Usuba

We show that if CH holds and either (i) there exists an $ω_{1}$ -Kurepa tree, or (ii) $□ (ω_{2})$ holds, then there are regular $T_{1}$ Lindelöf spaces $X_{0}$ and $X_{1}$ with points $G…

math.LO2018

Extendible cardinals and the mantle

Toshimichi Usuba

The mantle is the intersection of all ground models of $V$ . We show that if there exists an extendible cardinal then the mantle is a ground model of $V$ .

math.LO2017

$G_{δ}$ -topology and compact cardinals

Toshimichi Usuba

For a topological space $X$ , let $X_{δ}$ be the space $X$ with $G_{δ}$ -topology of $X$ . For an uncountable cardinal $κ$ , we prove that the following are equivalent: (1) $κ$ is $ω_{1}$ -st…

math.LO2017

The downward directed grounds hypothesis and very large cardinals

Toshimichi Usuba

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 of $V$ are downward set-directed. Consequently, we…

math.LO2013★ 1 cited

Superstrong and other large cardinals are never Laver indestructible

Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis +1

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-exten…

Variants of Łoś's Theorem

Monotonicity of the ultrafilter number function

On Recurrence Axioms

A note on uniform ultrafilters in a choiceless context

A note on Łoś's Theorem without the Axiom of Choice

Generically extendible cardinals

On Countable Stationary Towers

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

A note on Löwenheim-Skolem cardinals

A note on δ-strongly compact cardinals

Geology of symmetric grounds

Choiceless Löwenheim-Skolem property and uniform definability of grounds

A note on the tightness of Gδ​-modifications

Strongly compact cardinals and the continuum function

Products of Lindelöf spaces with points Gδ​

Extendible cardinals and the mantle

Gδ​-topology and compact cardinals

The downward directed grounds hypothesis and very large cardinals

Superstrong and other large cardinals are never Laver indestructible

A note on $δ$ -strongly compact cardinals

A note on the tightness of $G_{δ}$ -modifications

Products of Lindelöf spaces with points $G_{δ}$

$G_{δ}$ -topology and compact cardinals