arXiv subjects

Giorgio Venturi

Publications and source records attributed to Giorgio Venturi.

9 recordsLinked to original sources

arXiv2026

The Internal Modal Logic of Forcing

We connect modal set theory with Boolean-valued models by developing an \emph{internal} Kripke semantics for modal formulas whose atomic propositions are set-theoretic sentences. Given a complete Boolean algebra $B$, we view its elements as ``local perspectives on truth'' inside the Boolean-valued universe $V^{(B)}$ and interpret the modal operators using an accessibility relation $R$ on $B$ defined by \emph{co-consistency} (equivalently, Boolean compatibility): $aRb$ iff $a\wedge b\neq 0$. Our central conceptual point is that, for set-theoretic sentences $p$, the internal modality $\Diamond p$ holds at $b$ iff there is an ultrafilter $U$ of $B$ containing $b$ such that the classical quotient $V^{(B)}/U$ satisfies $p$. We compute several general and algebra-dependent modal validities, and analyze the special behavior of complete atomic Boolean algebras. Finally, adopting a translation-based semantics on the nonzero part $B^+=B\setminus\{0\}$, we prove a soundness-and-completeness theorem: the normal logic $\KTB$ is exactly the set of modal formulas valid in all translated co-consistency models with parameters.

math.LO↗

arXiv2024

Interpretations of ZF

In this paper, we unify the study of classical and non-classical algebra-valued models of set theory, by studying variations of the interpretation functions for identity and set-membership. Although, these variations coincide with the standard interpretation in Boolean-valued constructions, nonetheless they extend the scope of validity of ZF to new algebra-valued models.

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arXiv2023

What model companionship can say about the Continuum problem

We present recent results on the model companions of set theory, placing them in the context of the current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{κ^+}$, as $κ$ ranges among the infinite cardinals. We also single out $2^{\aleph_0}=\aleph_2$ as the unique solution of the Continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert's axiom of completeness.

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arXiv2021

The model companions of set theory

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin's generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $Π_2$-property formalized in an appropriate language for second order number theory is forcible from some $T\supseteq\mathsf{ZFC}+$large cardinals if and only if it is consistent with the universal fragment of $T$ if and only if it is realized in the model companion of $T$. In particular we show that the first order theory of $H_{ω_1}$ is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for $Δ_0$-properties and for all universally Baire sets of reals. We will extend these results also to the theory of $H_{\aleph_2}$ in a follow up of this paper.

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arXiv2019

Infinite forcing and the generic multiverse

In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing. This technique will be used to suggest a unified perspective on generic absoluteness principles.

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arXiv2016

A direct proof of the five element basis theorem

We present a direct proof of the consistency of the existence of a five element basis for the uncountable linear orders. Our argument is based on the approach of notion of saturation of Aronszajn trees considered by Koenig, Larson, Moore and Velickovic and simplifies the original proof of Moore.

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arXiv2015

Reflexive insensitive modal logics

We analyze a class of modal logics rendered insensitive to reflexivity by way of a modification to the semantic definition of the modal operator. We explore the extent to which these logics can be characterized, and prove a general completeness theorem on the basis of a translation between normal modal logics and their reflexive-insensitive counterparts. Lastly, we provide a sufficient semantic condition describing when a similarly general soundness result is also available.

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arXiv2014

Preservation of a Souslin tree and side conditions

We show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Souslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known by the standard countable support iteration, using a preservation theorem due to Miyamoto.

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arXiv2013

Proper forcing remastered

In these notes we present the method introduced by Neeman of generalized side conditions with two types of models. We then discuss some applications: the Friedman-Mitchell poset for adding a club in ω_2 with finite conditions, Koszmider's forcing construction of a strong chain of length ω_2 of functions from ω_1 to ω_1, and the Baumgartner-Shelah forcing construction of a thin very tall superatomic Boolean algebra.

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