arXiv subjects

Sebastiano Thei

Publications and source records attributed to Sebastiano Thei.

6 recordsLinked to original sources

arXiv2026

On the Intermediate Models of Strongly Compact Prikry Forcing

We analyze the intermediate models of the strongly compact Prikry forcing. We exhibit a simple combinatorial property which, for a given supercompact cardinal $\kappa$, characterize the projections of all projections of the strongly compact Prikry forcing using $\kappa$-complete fine measures. Considering level-by-level results, if $\kappa$ is $2^\lambda$-strongly compact, we characterize the forcings of size $\leq\lambda$ which are projections of that $\lambda$-strongly compact Prikry forcing. Our characterization generalizes several known results, including those of Benhamou-Hayut-Gitik and folklore results regarding the class of $\kappa$-distributive forcing notions which are embedded into the supercompact Prikry forcing. Fixing a $\kappa$-complete fine measure $\mathcal{U}$ on $P_\kappa(\lambda)$, we also provide Rudin-Keisler like critiria for the existence projections from the strongly compact Prikry forcing with $\mathcal{U}$. Finally, we prove that among all projections of the $\lambda$-strongly compact Prikry forcing, the class of forcings of cardinality $\lambda$ are exactly those for which there is a projection map which depends only on the stem of the Prikry condition. We also give partial results regarding projections of arbitrary cardinality.

math.LO↗

arXiv2026

A Note on a Theorem of Apter

We show that the consistency of $\mathrm{ZF} + \mathrm{AD}_{\mathbb{R}} + ``\Theta$ is measurable$"$ implies the consistency of $\mathrm{ZF} +``\Theta$ is the least strongly regular cardinal and the least measurable cardinal$"$ + $``$all uncountable cardinals below $\Theta$ are of countable cofinality$"$.

math.LO↗

arXiv2025

Combinatorics in Higher Solovay Models

We construe the singular-cardinal analogue of the classical Solovay model. Starting with large cardinal assumptions in the realm of supercompactness, we show that the our inner model captures a substantial portion of the combinatorics of $L(\mathcal{P}(\kappa))$ that are typically implied by Woodin's axiom $I_0$. Among other things, we show that in our higher Solovay model there are no $\kappa^+$-sequences of distinct members of $\mathcal{P}(\kappa)$ and that Shelah's approachability property $\AP_\kappa$ fails. We prove that every set in our inner model satisfies a singular analogue of the complete Ramsey property and that the partition relation $\kappa\xrightarrow[]{\mathrm{OD}} (\omega)^\omega_{V_\mu}$ holds for all $\mu<\kappa$.

math.LO↗

arXiv2025

Higher Solovay Models

We introduce an axiomatisation of when a model of the form $L(V_{\kappa+1})^M$ can be considered a ``$\kappa$-Solovay model''; we show a characterisation of $\kappa$-Solovay models; and we prove elementary equivalences between $\kappa$-Solovay models.

math.LO↗

arXiv2024

No cardinal correct inner model elementarily embeds into the universe

An elementary embedding $j:M\rightarrow N$ between two inner models of ZFC is cardinal preserving if $M$ and $N$ correctly compute the class of cardinals. We look at the case $N=V$ and show that there is no nontrivial cardinal preserving elementary embedding from $M$ into $V$, answering a question of Caicedo.

math.LO↗

arXiv2024

The Baire and perfect set properties at singulars cardinals

We construct a model of ZFC with a singular cardinal $\kappa$ such that every subset of $\kappa$ in $L(V_{\kappa+1})$ has both the $\kappa$-Perfect Set Property and the $\mathcal{\vec{U}}$-Baire Property. This is a higher analogue of Solovay's result for $L(\mathbb{R})$. We obtain this configuration starting with large-cardinal assumptions in the realm of supercompactness, thus improving former theorems by Cramer, Shi and Woodin.

math.LO↗