arXiv subjects

Vincenzo Dimonte

Publications and source records attributed to Vincenzo Dimonte.

11 recordsLinked to original sources

arXiv2026

The $λ$-PSP at $λ$-$Π^1_1$ sets

Given a strong limit cardinal $λ$ of countable cofinality, we show that if every (boldface) $λ\hyp\boldsymbolΠ^1_1$ subset of the generalised Cantor space ${}^λ2$ has the $λ$-$\mathsf{PSP}$, then $0^\dagger$ exists. We show too that if every (lightface) $λ\hypΠ^1_1$ subset of ${}^λ2$ has the $λ\hyp\mathsf{PSP}$, then there is an inner model with a measurable cardinal. The paper, a contribution to the ongoing research on generalised regularity properties in generalised descriptive set theory at singular cardinals of countable cofinality, is aimed at descriptive set theorists, and so it presents its results in as much detail as possible, particularly regarding the inner model-theoretic aspects. In doing so, we intend to provide the community with the tools needed to handle consistency strength arguments at the corresponding levels.

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arXiv2026

On the problem of generalized measures: an impossibility result

This paper investigates the problem of extending measure theory to non-separable structures, from generalized descriptive set theory to a broader class of spaces beyond this framework. While various notions, such as the ideal of measure zero sets, have been generalized, the question of whether a satisfactory notion of $λ^+$-measure could be defined in generalized descriptive set theory has remained open. We introduce a broad class of $λ^+$-measures as functions taking values in arbitrary positively totally ordered monoids equipped with an infinitary sum. This definition relies on minimal assumptions and captures most natural generalizations of measures to this context. We then prove that, under certain cardinal assumptions, no continuous $λ^+$-measure of this kind exists on ${}^κλ$, nor on any $λ^+$-Borel space or $T_0$ topological space of weight at most $λ$. We also show the optimality of these cardinal assumptions.

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arXiv2025

Generalized Descriptive Set Theory at Singular Cardinals of Countable Cofinality

We provide a comprehensive development of the basics of descriptive set theory for non-separable complete metric spaces whose weight is a singular cardinal $λ$ of countable confinality. Somewhat unexpectedly, the resulting theory is remarkably similar to the classical one, although the methods used are necessarily fairly different and combine ideas and results from general topology, infinite combinatorics, and set theory. More in detail, we study $λ$-Polish spaces and standard $λ$-Borel spaces (characterization of the generalized Cantor and Baire spaces, analogues of the Cantor-Bendixson theorem, classification up to $λ$-Borel isomorphism, etc.), their $λ$-Borel hierarchy (structural properties, changes of topologies, and so on), $λ$-analytic sets (including generalizations of the Lusin separation theorem and of the Souslin theorem), $λ$-coanalytic sets (including $λ$-$\boldsymbolΠ^1_1$-ranks and alike), and $λ$-projective sets. We also consider more advanced topics, and provide e.g. various uniformization results for $λ$-Borel set; these in turn lead to fundamental applications to the study of $λ$-Borel equivalence relations, such as a generalization of the celebrated Feldman-Moore theorem. Finally, we study a natural generalization of the classical Perfect Set Property, and develop tools to show that all definable sets enjoy such property under suitable large cardinal assumptions, most notably including Woodin's $\mathsf{I0}(λ)$.

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arXiv2024

Descriptive properties of I2-embeddings

We contribute to the study of generalizations of the Perfect Set Property and the Baire Property to subsets of spaces of higher cardinalities, like the power set $P(λ)$ of a singular cardinal $λ$ of countable cofinality or products $\prod_{i<ω}λ_i$ for a strictly increasing sequence $\langleλ_i ~ \vert ~ i<ω\rangle$ of cardinals. We consider the question under which large cardinal hypotheses classes of definable subsets of these spaces possess such regularity properties, focusing on rank-into-rank axioms and classes of sets definable by $Σ_1$-formulas with parameters from various collections of sets. We prove that $ω$-many measurable cardinals, while sufficient to prove the Perfect Set Property of all $Σ_1$-definable sets with parameters in $V_λ\cup\{V_λ\}$, are not enough to prove it if there is a cofinal sequence in $λ$ in the parameters. For this conclusion, the existence of an I2-embedding is enough, but there are parameters in $V_{λ+1}$ for which I2 is still not enough. The situation is similar for the Baire Property: under I2 all sets that are $Σ_1$-definable using elements of $V_λ$ and a cofinal sequence as parameters have the Baire property, but I2 is not enough for some parameter in $V_{λ+1}$. Finally, the existence of an I0-embedding implies that all sets that are $Σ^1_n$-definable with parameters in $V_{λ+1}$ have the Baire property.

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arXiv2024

The Baire and perfect set properties at singulars cardinals

We construct a model of ZFC with a singular cardinal $κ$ such that every subset of $κ$ in $L(V_{κ+1})$ has both the $κ$-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.

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arXiv2022

A Solovay-like model for singular generalized descriptive set theory

Kunen's proof of the non-existence of Reinhardt cardinals opened up the research on very large cardinals, i.e., hypotheses at the limit of inconsistency. One of these large cardinals, I0, proved to have descriptive-set-theoretical characteristics, similar to those implied by the Axiom of Determinacy: if $λ$ witnesses I0, then there is a topology for $V_{λ+1}$ that is completely metrizable and with weight $λ$ (i.e., it is a $λ$-Polish space), and it turns out that all the subsets of $V_{λ+1}$ in $L(V_{λ+1})$ have the $λ$-Perfect Set Property in such topology. In this paper, we find another generalized Polish space of singular weight $κ$ of cofinality $ω$ such that all its subsets have the $κ$-Perfect Set Property, and in doing this, we are lowering the consistency strength of such property from I0 to $κ$ $θ$-supercompact, with $θ>κ$ inaccessible.

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arXiv2018

The iterability hierarchy above I3

In this paper we introduce a new hierarchy of large cardinals between I3 and I2, the iterability hierarchy, and we prove that every step of it strongly implies the ones below.

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arXiv2017

Generic I0 at $\aleph_ω$

In this paper it is introduced a generic large cardinal akin to I0, and its consequences are analyzed in the case that $\aleph_ω$ is such a generic large cardinal. In this case $\aleph_ω$ is Jónsson, and in a choiceless inner model many properties hold that are in contrast with PCF in ZFC.

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arXiv2017

I0 and rank-into-rank axioms

Just a survey on I0: The basics, some things known but never published, some things published but not known.

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arXiv2017

LD-algebras beyond I0

The algebra of embeddings at the I3 level has been deeply analyzed, but nothing is known algebra-wise for embeddings above I3. In this paper it is introduced an operation for embeddings at the level of I0 and above, and it is proven that they generate an LD-algebra that can be quite different from the I3 one.

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arXiv2015

A general tool for consistency results related to I1

In this paper we provide a general tool to prove the consistency of $I1(λ)$ with various combinatorial properties at $λ$ typical at settings with $2^λ>λ^+$, that does not need a profound knowledge of the forcing notions involved. Examples of such properties are the first failure of GCH, a very good scale and the negation of the approachability property, or the tree property at $λ^+$ and $λ^{++}$.

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