Matteo Viale

E-mail: matteo dot viale at unito dot it

Phone: + 39 - 011 - 670 28 91

I'm currently an associate professor in the math department of the University of Torino.

My research interests lie in the area of mathematical logic, more specifically, in set theory. For those interested, here is a very short propaganda to the subject and to myself. A more detailed survey paper (with Giorgio Venturi) on these topics can be found here.

Here is a link to the official website of the Turin logic group of which I'm one of the members.



In english

In italian


Winner of the Kurt Gödel Research Prize Fellowship 2010, awarded in 2010 to support original research in the foundations of mathematics. Here you can find the slides of the conference I held in occasion of the ceremony award.

Winner of the premio Fubini 2010, awarded in 2010 to a young mathematician working in Italy in the fields of logic, geometry and algebra.

Sacks Prize for the best Ph.D. thesis in Mathematical Logic in 2006 attributed by the Association for Symbolic Logic for the Ph.d thesis Applications of the proper forcing axiom to cardinal arithmetic.

Papers and reviews

Absoluteness via resurrection, 33 pages (with Giorgio Audrito), Journal of Mathematical Logic17 (2017), no. 2.

This paper introduces the forcing axioms of iterated resurrection with the aim to prove the generic absoluteness of the theory of H(c) with parameteres with respect to various classes of forcings modulo the preservation of these axioms. We are inspired by the works on resurrection axioms of Hamkins Johnstone on one side and Tsaprounis on the other. Compared to the generic absoluteness results obtained in the paper below on category forcings, the proofs are much simpler, but the results obtained are less informative and slightly weaker. On the other hand the machinery we develop can be applied uniformly to all classes of forcings for which an iteration theorem is known (CCC, proper, semiproper, axiom A,....). In the case of the iterated resurrection axioms for SSP forcings we show that this axiom sits in between MM^{+++} and BMM^{++}.

Forcing the truth of a weak form of Schanuel's conjecture, Confluentes Mathematici. 8 (2016), no. 2, 59–83.

This paper contains a proof using generic absoluteness of the existence of a countable field K such that Schanuel's conjecture holds if we repalce rational numbers with K in its statement.

Generic absoluteness and boolean names for elements of a Polish space (with Andrea Vaccaro) Boll. Unione Mat. Ital. 10 (2017), no. 3, 293–319.

"Category forcings, MM+++, and generic absoluteness for the theory of strong forcing axioms." J. Amer. Math. Soc. 29 (2016), no. 3, 675-728 Arxiv

The paper studies the category whose objects are stationary set preserving complete boolean algebras and whose arrows are complete homomorphisms between them with a stationary set preserving quotient. It first introduces the concept of total rigidity: An object B of this category is totally rigid if for all other objects Q of this category there is at most one homomorphism i with domain B and range Q which is an arrow of this category. It is first shown that in the presence of large cardinals many initial segments of this category seen as a class partial orders are themselves stationary set preserving and totally rigid and that the class of totally rigid partial orders is dense in this class forcing. Next it is argued that MM++ can be stated as the property that the class of presaturated normal towers is dense in this class partial order. Following this characterization of MM++ we introduce a new forcing axiom (MM+++) which is slightly stronger than the assertion that the class of presaturated normal towers which are also totally rigid is dense in this class forcing. Finally it is shown that the theory of the Chang model L(ORD^{\omega_1}) is generically invariant with respect to stationary set preserving forcings which preserve MM+++, and that MM+++ is consistent relative to large cardinal axioms.


"Martin's maximum revisited" Arch. Math. Logic 55 (2016), no. 1-2, 295-317. 03E35 (03E40 03E57) Arxiv

This paper contains preliminary results on generic absoluteness with respect to stationary set preserving forcings. In particular it is shown among other things that the \Sigma_2-theory of H(\omega_2) over models of MM^{++} is generically invariant with respect to SSP forcings which preserve BMM in the presence of class many Woodin cardinals.


"Martin's maximum and tower forcing", Israel Journal of Mathematics, volume 197 (2013), no. 1, 347-376: Arxiv

"Guessing models and generalized Laver diamond", Annals of Pure and Applied Logic, Annals of Pure and Applied Logic, volume 163 (2012), no. 11, 1660-1678: viale.pdf

"On the consistency strength of the proper forcing axioms" (with Christoph Weiss), Advances in Mathematics, Volume 228 (2011), no. 5, 2672-2687: viale_weiss.pdf

"On the mapping reflection principle MRP" The Bulletin of Symbolic Logic 15(3), 322-325, 2009: BSLMRPrevMV.pdf

"Some consequences of reflection on the approachability ideal" (with Assaf Sharon) Transactions of the American Mathematical Society 362, 4201-4212, 2009 : RAMVAS.pdf

"Forcing axioms, supercompact cardinals, singular cardinals combinatorics" The Bulletin of Symbolic Logic 14 (2008), no. 1, 99--113: BSLMV.pdf

"A family of covering properties" Mathematical Research Letters, 2008, volume 15, no. 2, 221--238: CPMV.pdf

"The proper forcing axiom and the singular cardinal hypothesis." The Journal of Symbolic Logic, 2006, volume 71, no. 2, 473--479: proper-singularMV.pdf

"The cumulative hierarchy and the constructible universe of ZFA." Mathematical Logic Quarterly, 2004, volume 50, no. 1, 99--103: zfaMV.pdf

"A binary modal logic for the intersection types of lambdacalculus." (with Silvio Valentini), Information and Computation, 2003, volume 185, no. 2, 211--232: lambdaMV.pdf




Useful axioms

A survey of my results on generic absoluteness, also containing a way to interpret generic absoluteness results as strong forms of Los theorem for boolean ultrapowers.


Slides of some of my talks

Here are the slides of my thesis defense (september 2006): A self contained proof of the singular cardinal hypothesis from the P-ideal dichotomy is presented.

Here are the slides of my talk in Luminy (october 2010) concerning the consistency strength of PFA.

Here are the the slides of my talk as plenary speaker in the annual north american metting of the ASL held in Waterloo - Canada (may 2013) where I first introduce the category of stationary set preseving forcings (with arrows given by complete embeddings with a stationary set preserving quotient), then present MM^{+++}, a strong form of Martin's maximum, as a density property of this category, and finally sketch a proof the generic absoluteness of the theory of the Chang model L(Ord^{\omega_1}) for models of ZFC+MM^{+++}+large cardinals.

Here is a link to the slides of the winter school in abstract analysis: section set theory and topology (Czech Republic January 2014), among which those of my tutorial on a boolean algebraic presentation of iterated forcing and on category forcings.

Here are the slides of my talk in the Luminy workshop in set theory held in Luminy (Marseille) in September 2014. I give an overview of the variety of forcing axioms which produce generic absoluteness and present forcing axioms as a strengthening of the axiom of choice which can be introduced either by topological means (the standard presentation of forcing axioms) or by algebraic means. The algebraic language allows to formulate forcing axioms which are even stronger then the one which can be usually formulated in topological terms.

Here are the slides of my talk in the XXth UMI congress held in Siena in September 2015. It is aimed at a general audience of mathematicians and gives a sketchy presentation of what are and how to recognize undecidable problems in mathematics, what is forcing and how to use it to prove undecidability results and how to use it to prove theorems.

Here are slides on model companionship results for set theory + large cardinals and their implications for CH.

Many inaccuracies may occur in the slides, caveat lector!


Giorgio Audrito: "Generic large cardinals and absoluteness", PhD thesis (2016): audrito-phd.pdf


Giorgio Audrito: "Characterizations of set generic extensions", master thesis (2011): audrito.pdf

Vincenzo Giambrone: "Boolean valued models for set theory and Grothendieck topoi", master thesis (2017): giambrone.pdf

Fiorella Guichardaz: "Limits of boolean algebras and boolean valud models", master thesis (2013): guichardaz.pdf

Giuseppe Moranarocca: "Ultraproducts of finite partial orders and some of their applications in model theory and set theory", master thesis (2013): moranarocca.pdf

Francesco Parente: "Boolean valued models, saturation, forcing axioms", master thesis (2015): parente.pdf

Moreno Pierobon: "Saturated structures constructed using forcing and applications", master thesis (2019): pierobon.pdf

Daniele Truzzi: "Continuous logic as a tool to transfer model-theoretic dichotomies to functional analysis and conversely", master thesis (2020): truzzi.pdf

Andrea Vaccaro: "C*-algebras and B-names for complex numbers", master thesis (2015): vaccaro.pdf

Giorgio Venturi: "Forcing axioms and cardinality of the continuum", master thesis (2009): venturi.pdf


Morena Porzio: "The rank of Ext^1(A,Z) and Whitehead's problem", bachelor thesis (2018): Porzio.pdf


Set Theory

Notes on forcing: A complete presentation of the forcing method via an approach based on boolean valued models, and with detailed proofs (obtained by means of forcing) of the independence of the continuum hypothesis with respect to the ZFC axioms.

Logic for undergraduates

Notes on propositionale and first order logic: a short introduction to propositional and first order logic for undergraduates.

Notes on model companionship

Notes on model companionship: a self contained account of the main results on model companionship and model completeness.

Lecture notes on advanced topics in Set Theory

A boolean algebraic approach to semiproper iterations (with Giorgio Audrito and Silvia Steila), 60 pages: a self contained development of iterated forcing by means of directed systems of boolean algebras. It proves the main results on semiproper iterations expanding and developing on the work of Donder and Fuchs. This material expands on a Ph.D course I gave in summer and fall 2013 on this topic.

FORCING AXIOMS AND REFLECTION PRINCIPLES lecture notes taken by Giorgio Audrito on a Ph.D course I gave in spring 2012 on this topic.

THE OPEN COLORING AXIOM lecture notes taken by Gemma Carotenuto on a Ph.D course I gave in spring 2012 on this topic.

Valid HTML 4.01!