arXiv subjects

Takehiko Gappo

Publications and source records attributed to Takehiko Gappo.

8 recordsLinked to original sources

arXiv2026

Cardinal invariants of idealized Miller null sets

This paper provides an extensive study of the $\mathscr{I}$-Miller null ideals $M_\mathscr{I}$, $σ$-ideals on the Baire space parametrized by ideals $\mathscr{I}$ on countable sets. These $σ$-ideals are associated to the idealized versions of Miller forcing in the same way that the meager ideal is associated to Cohen forcing. We compute the cardinal invariants of $M_\mathscr{I}$ for typical examples of Borel ideals $\mathscr{I}$ and show that Cichoń's Maximum can be extended by adding the uniformity and covering numbers of $M_\mathscr{I}$ for different ideals $\mathscr{I}$.

math.LO↗

arXiv2026

Long games just beyond fixed countable length

We introduce a new type of game on natural numbers of variable countable length, which can be regarded as a diagonalization of all games of fixed countable length on natural numbers. Building on previous work by Trang and Woodin, we show that analytic determinacy of the game is equivalent to the existence of a sharp for a canonical inner model with a limit of Woodin cardinals $λ$ such that the order type of Woodin cardinals below $λ$ is $λ$.

math.LO↗

arXiv2025

Separating Maximality Principles

We investigate fragments of generic absoluteness principles known as Maximality Principles. We determine the consistency strength of $Σ_n$-$\mathsf{MP}(\mathbb R)$ and $Π_n$-$\mathsf{MP}(\mathbb R)$, the boldface Maximality Principle restricted respectively to $Σ_n$- and $Π_n$-formulas. Further, we show that no implication between $Σ_n$-$\mathsf{MP}(\mathbb R)$ and $Π_n$-$\mathsf{MP}(\mathbb R)$ is provable in $\mathsf{ZFC}$. We also establish the consistency, relative to a Woodin cardinal, of the Maximality Principle for $ω_1$-preserving posets with countable ordinal parameters and prove its consistency strength is bounded below by a Ramsey cardinal. Finally, we resolve questions of Ikegami-Trang and Goodman by separating the Maximality Principle for stationary set preserving posets restricted to $Σ_2$-formulas from $\mathsf{MM}^{++}$ in the presence of large cardinals.

math.LO↗

arXiv2025

Generic Absoluteness Revisited

The present paper is concerned with the relation between recurrence axioms and Laver-generic large cardinal axioms in light of principles of generic absoluteness and the Ground Axiom. M. Viale proved that Martin's Maximum$^{++}$ together with the assumption that there are class many Woodin cardinals implies $\mathcal{H}(\aleph_2)^{\mathsf{V}}\prec_{Σ_2}\mathcal{H}(\aleph_2)^{\mathsf{V}[\mathbb{G}]}$ for a generic $\mathbb{G}$ on any stationary preserving $\mathbb{P}$ which also preserves Bounded Martin's Maximum. We show that a similar but more general conclusion follows from each of $(\mathcal{P},\mathcal{H}(κ))_{Σ_2}$-${\sf RcA}^+$ (which is a fragment of a reformulation of the Maximality Principle for $\mathcal{P}$ and $\mathcal{H}(κ)$), and the existence of the tightly $\mathcal{P}$-Laver-generically huge cardinal. While under "$\mathcal{P}=$ all stationary preserving posets", our results are not very much more than Viale's Theorem, for other classes of posets, "$\mathcal{P}=$ all proper posets" or "$\mathcal{P}=$ all ccc posets", for example, our theorems are not at all covered by his theorem. The assumptions (and hence also the conclusion) of Viale's Theorem are compatible with the Ground Axiom. In contrast, we show that the assumptions of our theorems (for most of the common settings of $\mathcal{P}$ and with a modification of the large cardinal property involved) imply the negation of the Ground Axiom. This fact is used to show that fragments of Recurrence Axiom $(\mathcal{P},\mathcal{H}(κ))_Γ$-${\sf RcA}^+$ can be different from the corresponding fragments of Maximality Principle ${\sf MP}(\mathcal{P},\mathcal{H}(κ))_Γ$ for $Γ=Π_2$.

math.LO↗

arXiv2025

On $ω$-strongly measurable cardinals in $\mathbb{P}_{\max}$ extensions

We show that in the $\mathbb{P}_{\max}$ extension of a certain Chang-type model of determinacy, if $κ\in\{ω_1, ω_2, ω_3\}$, then the restriction of the club filter on $κ\cap\mathrm{Cof}(ω)$ to HOD is an ultrafilter in HOD. This answers Question 4.11 of [BNH23] raised by Ben-Neria and Hayut.

math.LO↗

arXiv2025

Chang models over derived models with supercompact measures

Based on earlier work of the third author, we construct a Chang-type model with supercompact measures extending a derived model of a given hod mouse with a regular cardinal $δ$ that is both a limit of Woodin cardinals and a limit of ${<}δ$-strong cardinals. The existence of such a hod mouse is consistent relative to a Woodin cardinal that is a limit of Woodin cardinals. We argue that our Chang-type model satisfies $\mathsf{AD}_{\mathbb{R}} + Θ$ is regular + $ω_1$ is ${<}δ_{\infty}$-supercompact for some regular cardinal $δ_{\infty}>Θ$. This complements Woodin's generalized Chang model, which satisfies $\mathsf{AD}_{\mathbb{R}}+ω_1$ is supercompact, assuming a proper class of Woodin cardinals that are limits of Woodin cardinals.

math.LO↗

arXiv2023

Determinacy in the Chang model

Assuming the existence of a certain hod pair with a Woodin cardinal that is a limit of Woodin cardinals, we show that the Chang model satisfies $\mathsf{AD}^+$ in any set generic extensions.

math.LO↗

arXiv2021

On the derived models of self iterable universes

We show that if the universe is self-iterable and $κ$ is an inaccessible limit of Woodin cardinal then $AD_R + "Θ$ is regular" holds in the derived model at $κ$. The proof is fine-structure free, and only assumes basic knowledge of iteration trees and iteration strategies. Our proof can be viewed as the fine-structure free version of the well-known fact that $AD_R + "Θ$ is regular" is true in the derived models of hod mice that have inaccessible limit of Woodin cardinals (see for example [6]). However, the proof uses a different set of ideas and is more general.

math.LO↗