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Toshimichi Usuba

Publications and source records attributed to Toshimichi Usuba.

At least 19 recordsLinked to original sources
arXiv2025

Monotonicity of the ultrafilter number function

We investigate whether the ultrafilter number function $κ\mapsto \mathfrak{u}(κ)$ on the cardinals is monotone, that is, whether $\mathfrak{u}(λ) \le \mathfrak{u}(κ)$ holds for all cardinals $λ< κ$ or not. We show that monotonicity can fail, but the failure has large cardinal strength. On the other hand, we prove that there are many restrictions of the failure of monotonicity. For instance, if $κ$ is a singular cardinal with countable cofinality or a strong limit singular cardinal, then $\mathfrak{u}(κ) \le \mathfrak{u}(κ^+)$ holds.

math.LO↗
arXiv2025

On Recurrence Axioms

The Recurrence Axiom for a class $\mathcal{P}$ 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 $A$ is forced by a poset in $\mathcal{P}$, then there is a ground containing the parameters and satisfying the statement. The tightly super-$C^{(\infty)}$-$\mathcal{P}$-Laver generic hyperhuge continuum implies the Recurrence Axiom for $\mathcal{P}$ and $\mathcal{H}(2^{\aleph_0})$. The consistency strength of this assumption can be decided thanks to our main theorems asserting that the minimal ground (bedrock) exists under a tightly $\mathcal{P}$-generic hyperhuge cardinal $κ$, and that $κ$ in the bedrock is genuinely hyperhuge, or even super $C^{(\infty)}$ hyperhuge if $κ$ is a tightly super-$C^{(\infty)}$-$\mathcal{P}$-Laver generic hyperhuge definable cardinal. The Laver Generic Maximum (LGM), one of the strongest combinations of axioms in our context, integrates practically all known set-theoretic principles and axioms in itself, either as its consequences or as theorems holding in (many) grounds of the universe. For example, double plus version of Martin's Maximum is a consequence of LGM while Cichoń's Maximum is a phenomenon in many grounds of the universe under LGM.

math.LO↗
arXiv2025

Variants of Łoś's Theorem

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 to Łoś's theorem.

math.LO↗
arXiv2024

Generically extendible cardinals

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 $ω_2$ has small consistency strength, but that of a cardinal $>ω_2$ does not. We also consider some results concerned with generically extendible cardinals, such as indestructibility, generic absoluteness of the reals, and Boolean valued second order logic.

math.LO↗
arXiv2024

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

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 ultrapower is elementary equivalent to its source structure. On the other hand, it is consistent that there is a structure $M$ and an ultrafilter $U$ such that the ultrapower of $M$ by $U$ is elementary equivalent to $M$, but the fundamental theorem for the ultrapower of $M$ by $U$ fails. We also show that weak fragments of the Axiom of Choice, such as the Countable Choice, do not follow from Łoś's theorem, even assuming the existence of non-principal ultrafilters.

math.LO↗
arXiv2021

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

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 number provided that the diamond principle holds. Under the GCH assumption, we prove the singular compactness theorem for the list-chromatic number. We also investigate reflection principles for the list-chromatic number and the coloring number of graphs.

math.LO↗
arXiv2021

Geology of symmetric grounds

Let us say that a model of $\mathsf{ZF}$ is a symmetric ground if $V$ is a symmetric extension of the model. In this paper, we investigate set-theoretic geology of symmetric grounds. Under a certain assumption, we show that all symmetric grounds of $V$ are uniformly definable. We also show that if $\mathsf{AC}$ is forceable over $V$, then the symmetric grounds are downward directed.

math.LO↗
arXiv2020

On Countable Stationary Towers

In this paper, we investigate properties of countable stationary towers. We derive the regularity properties of sets of reals in $L(\mathbf R)$ from some properties of countable stationary towers without explicit use of strong large cardinals such as Woodin cardinals. We also introduce the notion of semiprecipitousness and investigate its relation to precipitousness and presaturation of countable stationary towers. We show that precipitousness of countable stationary towers of weakly compact height implies the regularity properties of sets of reals in $L(\mathbf R)$.

math.LO↗
arXiv2020

A note on $δ$-strongly compact cardinals

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 $δ$-strongly compact, (2) For every regular $λ\ge κ$ there is a $δ$-complete uniform ultrafilter over $λ$, and (3) Every product space of $δ$-Lindelöf spaces is $κ$-Lindelöf. We also prove that in the Cohen forcing extension, the least $ω_1$-strongly compact cardinal is a precise upper bound on the tightness of the products of two countably tight spaces.

math.LO↗
arXiv2020

Choiceless Löwenheim-Skolem property and uniform definability of grounds

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 axiom of choice is forceable if and only if the universe is a small extension of some transitive model of $\mathsf{ZFC}$.

math.LO↗
arXiv2019

A note on the tightness of $G_δ$-modifications

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 continuum is not so large, then the tightness of $G_δ$-modifications of countably tight spaces can be arbitrary large up to the least $ω_1$-strongly compact cardinal.

math.LO↗
arXiv2018

Products of Lindelöf spaces with points $G_δ$

We show that if CH holds and either (i) there exists an $ω_1$-Kurepa tree, or (ii) $\square(ω_2)$ holds, then there are regular $T_1$ Lindelöf spaces $X_0$ and $X_1$ with points $G_δ$ such that $e(X_0 \times X_1)>2^ω$.

math.LO↗
arXiv2018

The downward directed grounds hypothesis and very large cardinals

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 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.

math.LO↗
arXiv2018

$G_δ$-topology and compact cardinals

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 $ω_1$-strongly compact. (2) For every compact Hausdorff space $X$, the Lindelöf degree of $X_δ$ is $\le κ$. (3) For every compact Hausdorff space $X$, the weak Lindelöf degree of $X_δ$ is $\le κ$. This shows that the least $ω_1$-strongly compact cardinal is the supremum of the Lindelöf and the weak Lindelöf degrees of compact Hausdorff spaces with $G_δ$-topology. We also prove the least measurable cardinal is the supremum of the extents of compact Hausdorff spaces with $G_δ$-topology. For the square of a Lindelöf space, using weak $G_δ$-topology, we prove that the following are consistent: (1) the least $ω_1$-strongly compact cardinal is the supremum of the (weak) Lindelöf degrees of the squares of regular $T_1$ Lindelöf spaces. (2) The least measurable cardinal is the supremum of the extents of the squares of regular $T_1$ Lindelöf spaces.

math.LO↗
arXiv2018

Extendible cardinals and the mantle

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.LO↗