arXiv subjects

Toshimasa Tanno

Publications and source records attributed to Toshimasa Tanno.

3 recordsLinked to original sources

arXiv2025

Perfect set dichotomy theorem in generalized Solovay model

We prove that the perfect set dichotomy theorem holds in the Solovay model $V ((ω^ω)^{V[G]})$. Namely, for every equivalence relation $E$ on $\mathbb{R}$, either $\mathbb{R}/E$ is well-orderable or there exists a perfect set consisting of $E$-inequivalent reals. Furthermore we consider a generalization of the Solovay model for an uncountable regular cardinal $μ$ and show the perfect set dichotomy theorem for $μ^μ$ also holds in that model. We establish the three element basis theorem for uncountable linear orders in the Solovay model for a weakly compact cardinal, in a general form covering the uncountable case.

math.LO↗

arXiv2025

Generalized Tukey reducibility between $σ$-directed sets

We introduce the pre-Tukey reducibility, a generalization of the Tukey reducibility between directed sets that works well in $\mathsf{ZF}$. We investigate the pre-Tukey reducibility between several $σ$-directed sets under assumptions on sets of reals, which hold in the Solovay model and in $L(\mathbb{R})$ satisfying $\mathsf{AD}$.

math.LO↗

arXiv2023

Stationary tower forcings and universally Baire sets

We investigate properties of stationary tower forcings and give conditions on stationary towers to derive the universally Baireness of sets of reals in $L(\mathbb{R})$.

math.LO↗