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Kenta Tsukuura

Publications and source records attributed to Kenta Tsukuura.

6 recordsLinked to original sources
arXiv2025

Walks along a weak square sequence and the non-semiproperness of Namba forcings

In this paper, we demonstrate that if, for every $κ$-complete fine filter $F$ over $\mathcal{P}_κλ$, the associated Namba forcing $\mathrm{Nm}(κ,λ,F)$ is semiproper, then $\square(μ,{<}\aleph_1)$ fails for all regular $μ\in [λ, 2^λ]$ under the certain cardinal arithmetic. In particular, this result establishes that the consistency strength of the semiproperness of $\mathrm{Nm}(\aleph_2,F)$ for every $\aleph_2$-complete filter $F$ over $\aleph_2$ exceeds the strength of infinitely many Woodin cardinals. Minimal walk methods associated with a square sequece play a central role in this paper. These observations introduce two-cardinal walks with naive $C$-sequences and show that the existence of non-reflecting stationary subsets implies $\mathcal{P}_κλ\not\to [I_{κλ}^{+}]^{3}_λ$.

math.LO↗
arXiv2023

On semiproperness of Namba forcings and Ideals in Prikry extensions

In this paper, we study some variations of Namba forcing $\mathrm{Nm}(κ,λ)$ over $\mathcal{P}_κλ$ and show that its semiproperness implies $\mathrm{SSR}([λ]^ω,{<}κ)$. In particular, Prikry forcing at $μ$ forces Namba forcing $\mathrm{Nm}(μ^{+})$ is not semiproper. This shows that there is no semiproper saturated ideal in the extensions by Prikry-type forcings. We also show that $[λ^{+}]^{μ^{+}}$ cannot carry a $λ^{+}$-saturated semiproper ideal if $μ$ is singular with countable cofinality.

math.LO↗
arXiv2022

On the centeredness of saturated ideals

We show that Kunen's saturated ideal over $\aleph_1$ is not centered. We also evaluate the extent of saturation of Laver's saturated ideal in terms of $(κ,λ,<ν)$-saturation.

math.LO↗
arXiv2022

Prikry-type forcings after collapsing a huge cardinal

Some models of combinatorial principles have been obtained by collapsing a huge cardinal in the case of the successors of regular cardinals. For example, saturated ideals, Chang's conjecture, polarized partition relations, and transfer principles for chromatic numbers of graphs. In this paper, we study these in the case of the successors of singular cardinals. In particular, we show that Prikry forcing preserves the centeredness of ideals but kills the layeredness. We also study $\binom{μ^{++}}{μ^{+}}\to\binomκ{μ^{+}}_μ$ and $\mathrm{Tr}_{\mathrm{Chr}}(μ^{+++},μ^{+})$ in the extension by Prikry forcing at $μ$.

math.LO↗
arXiv2022

Prikry-type forcing and the set of possible cofinalities

It is known that the set of possible cofinalities $\mathrm{pcf}(A)$ has good properties if $A$ is a progressive interval of regular cardinals. In this paper, we give an interval of regular cardinals $A$ such that $\mathrm{pcf}(A)$ has no good properties in the presense of a measurable cardinal, or in generic extensions by Prikry-type forcing.

math.LO↗