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Matteo Viale

Publications and source records attributed to Matteo Viale.

At least 19 recordsLinked to original sources
arXiv2025

The Boolean Compactness Theorem for $\mathrm{L}_{\infty\infty}$

We show that, contrary to the commonly held view, there is a natural and optimal compactness theorem for $\mathrm{L}_{\infty\infty}$ which generalizes the usual compactness theorem for first order logic. The key to this result is the switch from Tarski semantics to Boolean valued semantics. On the way to prove it, we also show that the latter is a (the?) natural semantics both for $\mathrm{L}_{\infty\infty}$ and for $\mathrm{L}_{\infty\omega}$.

math.LO↗
arXiv2024

Universally Baire sets in $2^{\kappa}$

We generalize the basic theory of universally Baire sets of $2^\omega$ to a theory of universally Baire subsets of $2^\kappa$. We show that the fundamental characterizations of the property of being universally Baire have natural generalizations that can be formulated also for subsets of $2^\kappa$, in particular we provide four equivalent uniform definitions in the parameter $\kappa$ (for $\kappa$ an infinite cardinal) characterizing for each such $\kappa$ the class of universally Baire subsets of $2^\kappa$. For $\kappa=\omega$, these definitions bring us back to the original notion of universally Baire sets of reals given by Feng, Magidor and Woodin [2].

math.LO↗
arXiv2024

Reflecting compact $T_1$-spaces into bounded distributive lattices

We present a contravariant reflection of the compact $T_1$-spaces with arrows given by closed continuous functions into the category of bounded distributive lattices with arrows given by closed subfit morphisms. This reflection extends both Stone duality and Isbell's duality between frames and sober spaces for those compact $T_1$-spaces that fall within each of these dualities, that is, respectively, zero-dimensional compact Hausdorff spaces, and compact sober $T_1$-spaces. On the topological side, we allow all compact $T_1$-spaces rather than just sober ones and we identify points in these with minimal prime filters on some base. On the lattice side, the shift goes from the notion of frame homomorphism to that of closed subfit morphism between bounded distributive lattices (closed subfit morphisms are defined by a natural and first order expressible constraint). The reflection becomes a duality when one restricts on the algebraic side to the complete and compact subfit lattices (i.e. compact subfit frames). Furthermore, restricting our duality on the topological side to the subcategory of compact $T_2$-spaces with all continuous maps, we obtain a duality for these with the category of complete, compact and normal lattices, thus recovering a classical result of Cornish. We also relate our adjunction to the duality introduced by Maruyama between $T_1$-spaces with continuous maps and a category having as objects a particular type of subfit frames and as arrows a certain type of morphism, of which we give an alternative and explicit algebraic characterization.

math.GN↗
arXiv2023

Universality properties of forcing

The purpose of this paper is to investigate forcing as a tool to construct universal models. In particular, we look at theories of initial segments of the universe and show that any model of a sufficiently rich fragment of those theories can be embedded into a model constructed by forcing. Our results rely on the model-theoretic properties of good ultrafilters, for which we provide a new existence proof on non-necessarily complete Boolean algebras.

math.LO↗
arXiv2022

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 $H_{\kappa^+}$, as $\kappa$ 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.

math.LO↗
arXiv2021

Boolean valued semantics for infinitary logics

It is well known that the completeness theorem for $\mathrm{L}_{\omega_1\omega}$ fails with respect to Tarski semantics. Mansfield showed that it holds for $\mathrm{L}_{\infty\infty}$ if one replaces Tarski semantics with boolean valued semantics. We use forcing to improve his result in order to obtain a stronger form of boolean completeness (but only for $\mathrm{L}_{\infty\omega}$). Leveraging on our completeness result, we establish the Craig interpolation property and a strong version of the omitting types theorem for $\mathrm{L}_{\infty\omega}$ with respect to boolean valued semantics. We also show that a weak version of these results holds for $\mathrm{L}_{\infty\infty}$ (if one leverages instead on Mansfield's completeness theorem). Furthermore we bring to light (or in some cases just revive) several connections between the infinitary logic $\mathrm{L}_{\infty\omega}$ and the forcing method in set theory.

math.LO↗
arXiv2021

Another proof that $\mathsf{MM}^{++}$ implies Woodin's axiom $(*)$

Let $\mathsf{MM}^{++}(\kappa)$ state that the forcing axiom $\mathsf{MM}^{++}$ can be instantiated only for stationary set preserving posets of size at most $\kappa$. We give a detailed account of Asper\`o and Schindler's proof that $\mathsf{MM}^{++}(\kappa)+$there are class many Woodin cardinals implies Woodin's axiom $(*)$ if $\Diamond_\kappa$ holds and $\kappa>\aleph_2$. Our presentation takes advantage of the notion of consistency property: specifically we rephrase Asper\`o and Schindler's forcing as a specific instantiation of the notion of ``consistency property'' used by Makkai, Keisler, Mansfield and others in the study of infinitary logics. We also reorganize the order of presentation of the various parts of the proof. Taken aside these variations, our account is quite close to the original proof of Asper\`o and Schindler.

math.LO↗
arXiv2021

Absolute model companionship, forcibility, and the continuum problem

Absolute model companionship (AMC) is a strict strengthening of model companionship defined as follows: For a theory $T$, $T_{\exists\vee\forall}$ denotes the logical consequences of $T$ which are boolean combinations of universal sentences. $T^*$ is the AMC of $T$ if it is model complete and $T_{\exists\vee\forall}=T^*_{\exists\vee\forall}$. We use AMC to study the continuum problem and to gauge the expressive power of forcing. We show that (a definable version of) $2^{\aleph_0}=\aleph_2$ is the unique solution to the continuum problem which can be in the AMC of a "partial Morleyization" of the $\in$-theory $\mathsf{ZFC}+$"there are class many supercompact cardinals". We also show that (assuming large cardinals) forcibility overlaps with the apparently weaker notion of consistency for any mathematical problem $\psi$ expressible as a $\Pi_2$-sentence of a (very large fragment of) third order arithmetic ($\mathsf{CH}$, the Suslin hypothesis, the Whitehead conjecture for free groups are a small sample of such problems $\psi$). Partial Morleyizations can be described as follows: let $\mathsf{Form}_{\tau}$ be the set of first order $\tau$-formulae; for $A\subseteq \mathsf{Form}_\tau$, $\tau_A$ is the expansion of $\tau$ adding atomic relation symbols $R_\phi$ for all formulae $\phi$ in $A$ and $T_{\tau,A}$ is the $\tau_A$-theory asserting that each $\tau$-formula $\phi(\vec{x})\in A$ is logically equivalent to the corresponding atomic formula $R_\phi(\vec{x})$. For a $\tau$-theory $T$ $T+T_{\tau,A}$ is the partial Morleyization of $T$ induced by $A\subseteq \mathsf{Form}_\tau$. Finally we characterize a strong form of Woodin's axiom $(*)$ as the assertion that the first order theory of $H_{\aleph_2}$ as formalized in a certain natural signature is model complete.

math.LO↗
arXiv2021

The model-companionship spectrum of set theory, generic absoluteness, and the Continuum problem

We show that for $\Pi_2$-properties of second or third order arithmetic as formalized in appropriate natural signatures the apparently weaker notion of forcibility overlaps with the standard notion of consistency (assuming large cardinal axioms). Among such $\Pi_2$-properties we mention: the negation of the Continuum hypothesis, Souslin Hypothesis, the negation of Whitehead's conjecture on free groups, the non-existence of outer automorphisms for the Calkin algebra, etc... In particular this gives an a posteriori explanation of the success forcing (and forcing axioms) met in producing models of such properties. Our main results relate generic absoluteness theorems for second order arithmetic, Woodin's axiom $(*)$ and forcing axioms to Robinson's notion of model companionship (as applied to set theory). We also briefly outline in which ways these results provide an argument to refute the Continuum hypothesis.

math.LO↗
arXiv2021

Incompatible bounded category forcing axioms

We introduce bounded category forcing axioms for well-behaved classes $\Gamma$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{\lambda_\Gamma^+}$ modulo forcing in $\Gamma$, for some cardinal $\lambda_\Gamma$ naturally associated to $\Gamma$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation $\lambda_\Gamma=\omega$--to classes $\Gamma$ with $\lambda_\Gamma>\omega$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions on $V$. We also show the existence of many classes $\Gamma$ with $\lambda_\Gamma=\omega_1$, and giving rise to pairwise incompatible theories for $H_{\omega_2}$.

math.LO↗
arXiv2020

Boolean valued models, presheaves, and \'etal\'e spaces

Boolean valued models for a signature $\mathcal{L}$ are generalizations of $\mathcal{L}$-structures in which we allow the $\mathcal{L}$-relation symbols to be interpreted by boolean truth values. For example, for elements $a,b\in\mathcal{M}$ with $\mathcal{M}$ a $\mathsf{B}$-valued $\mathcal{L}$-structure for some boolean algebra $\mathsf{B}$, $(a=b)$ may be neither true nor false, but get an intermediate truth value in $\mathsf{B}$. In this paper we introduce a topological characterization of the sheafification process for presheaves on topological spaces induced by the dense Grothendieck topology. On the way to produce our characterization, we also relate the notion of open continuous mapping between topological spaces to that of complete homomorphism between complete boolean algebras, and to that of adjoint homomorphism between boolean algebras (e.g. an homomorphism which has a left adjoint, if seen as a functor between partial orders/categories). Next we link these topological/category theoretic results to the theory of boolean valued models. We give a different proof of a result by Monro identifying topological presheaves on Stone spaces with boolean valued models, and sheaves (according to the dense Grothendieck topology) with boolean valued models having the mixing property. We also give an exact topological characterization (the so called fullness property) of which boolean valued models satisfy Lo\'s Theorem (i.e. the general form of the Forcing Theorem which Cohen -- Scott, Solovay, Vopenka -- established for the special case given by the forcing method in set theory). Then we separate the fullness property from the mixing property, by showing that the latter is strictly stronger. Finally we give an exact categorical characterization of which presheaves correspond to full boolean valued models in terms of the structure of global sections of their associated \'etal\'e space

math.LO↗
arXiv2020

Tameness for set theory $I$

The paper is a first of two and aims to show that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship. Specifically we develop a general framework linking generic absoluteness results to model companionship and show that (with the required care in details) a $\Pi_2$-property formalized in an appropriate language for second or third 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 $T$. The paper is accessible to any person who has a fair acquaintance with set theory and first order logic at the level of an under-graduate course in both topics; however bizarre this may appear (given the results we aim to prove) no knowledge of forcing or large cardinals is required to get the proofs of its main results (if one accepts as black-boxes the relevant generic absoluteness results). On the other hand familiarity with the notions of model completeness and model companionship is essential. All the necessary model-theoretic background will be given in full detail. The present work expands and systematize previous results obtained with Venturi.

math.LO↗
arXiv2020

Tameness for set theory $II$

The paper is the second of two and shows that (assuming large cardinals) set theory is a tractable (and we dare to say tame) first order theory when formalized in a first order signature with natural predicate symbols for the basic definable concepts of second and third order arithmetic, and appealing to the model-theoretic notions of model completeness and model companionship. Specifically we use the general framework linking generic absoluteness results to model companionship introduced in the first paper to show that strong forms of Woodin's axiom $(*)$ entail that any theory $T$ extending $\mathsf{ZFC}$ by suitable large cardinal axioms has a model companion $T^*$ with respect to certain signatures $\tau$ containing symbols for $\Delta_0$-relations and functions, constant symbols for $\omega$ and $\omega_1$, a predicate symbol for the nonstationary ideal on $\omega_1$, symbols for certain lightface definable universally Baire sets. Moreover $T^*$ is axiomatized by the $\Pi_2$-sentences $\psi$ for $\tau$ such that $T$ proves that $$ L(\mathsf{UB})\models(\mathbb{P}_\max\Vdash\psi^{H_{\omega_2}}), $$ where $L(\mathsf{UB})$ denotes the smallest transitive model containing the universally Baire sets. Key to our results is the recent breakthrough of Asper\`o and Schindler establishing that a strong form of Woodin's axiom $(*)$ follows from $\mathsf{MM}^{++}$.

math.LO↗
arXiv2019

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 $\Pi_2$-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 $H_{\omega_1}$ 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 $\Delta_0$-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.

math.LO↗
arXiv2018

Incompatible category forcing axioms

Given a cardinal $\lambda$, category forcing axioms for $\lambda$-suitable classes $\Gamma$ are strong forcing axioms which completely decide the theory of the Chang model $\mathcal C_\lambda$, modulo generic extensions via forcing notions from $\Gamma$. $\mathsf{MM}^{+++}$ was the first category forcing axiom to be isolated (by the second author). In this paper we present, without proofs, a general theory of category forcings, and prove the existence of $\aleph_1$-many pairwise incompatible category forcing axioms for $\omega_1$-suitable classes.

math.LO↗
arXiv2016

Useful axioms

We give a brief survey on the interplay between forcing axioms and various other non-constructive principles widely used in many fields of abstract mathematics, such as the axiom of choice and Baire's category theorem. First of all we outline how, using basic partial order theory, it is possible to reformulate the axiom of choice, Baire's category theorem, and many large cardinal axioms as specific instances of forcing axioms. We then address forcing axioms with a model-theoretic perspective and outline a deep analogy existing between the standard {\L}o\'s Theorem for ultraproducts of first order structures and Shoenfield's absoluteness for $\Sigma^1_2$-properties. Finally we address the question of whether and to what extent forcing axioms can provide a "complete" semantics for set theory. We argue that to a large extent this is possible for certain initial fragments of the universe of sets: The pioneering work of Woodin on generic absoluteness show that this is the case for the Chang model $L(\text{Ord}^\omega)$ in the presence of large cardinals, and recent works by the author show that this can also be the case for the Chang model $L(\text{Ord}^{\omega_1})$ in the presence of large cardinals and maximal strengthenings of Martin's maximum or of the proper forcing axiom. The major open question we leave open is whether this situation is peculiar to these Chang models or can be lifted up also to $L(\text{Ord}^\kappa)$ for cardinals $\kappa>\omega_1$.

math.LO↗
arXiv2016

Generic absoluteness and boolean names for elements of a Polish space

It is common knowledge in the set theory community that there exists a duality relating the commutative $C^*$-algebras with the family of $B$-names for complex numbers in a boolean valued model for set theory $V^B$. Several aspects of this correlation have been considered in works of the late $1970$'s and early $1980$'s, for example by Takeuti, and by Jech. Generalizing Jech's results, we extend this duality so as to be able to describe the family of boolean names for elements of any given Polish space $Y$ (such as the complex numbers) in a boolean valued model for set theory $V^B$ as a space $C^+(X,Y)$ consisting of functions $f$ whose domain $X$ is the Stone space of $B$, and whose range is contained in $Y$ modulo a meager set. We also outline how this duality can be combined with generic absoluteness results in order to analyze, by means of forcing arguments, the theory of $C^+(X,Y)$.

math.LO↗