7-8 November, 2011
Maximalist and Minimalist Perspectives on Infinity: A meeting on set theory and its philosophy
Speakers:Tatiana Arrigoni (Fondazione Bruno Kessler, Trento & Kurt Gödel Centre, Vienna); Joan Bagaría (Universitat de Barcelona); Laura Crosilla (University of Leeds); Sy Friedman (Kurt Gödel Centre for Mathematical Logic, Vienna); Ignasi Jané (Universitat de Barcelona)
==========================MONDAY 7 Nov.
16:00 Opening words by J. Ferreirós and Luis Narvaez
16:15 Talk by Tatiana Arrigoni:
“On intuitive plausibility in set theory. The case of V = L.”
17:45 Talk by Joan Bagaria:
“Maximality vs. structural richness in the universe of all sets”
TUESDAY 8 Nov.
10:00 Talk by Sy D. Friedman:
11:45 Talk by Ignasi Jané:
“Philosophical concerns regarding the power-set operation”
16:00 Talk by Laura Crosilla:
“Constructive set theory and the foundations of constructive mathematics”
17:30 General discussion
ARRIGONI -- On intuitive plausibility in Set Theory. The case of V = L.
Abstract: In this talk I will consider whether some kind of intuitive plausibility can be legitimately ascribed to set theoretic axioms and methodological principles that apparently conflict with the recommendation to maximize. The case of the axiom of constructibility (V = L) will be especially focussed on, as well as the arguments that have been given, in particular by R. B. Jensen, in defense of the view that it is "a very attractive axiom". As a result a novel proposal will be advanced as to how matters of intuitive plausibility in contemporary set theory could be suitably understood.
CROSILLA -- Constructive set theory and the foundations of constructive mathematics
Abstract: Constructive Zermelo Fraenkel set theory is one of a number of systems introduced as foundations for constructive mathematics Bishop style. From a classical perspective it can be seen as a double restriction of Zermelo Fraenkel set theory: firstly the logic is intuitionistic, and secondly the notion of set is crafted in such a way to comply with a certain notion of predicativity.
In this talk I shall first of all recall the system CZF and how it differs from classical ZF. Then I shall hint at some questions which emerge when looking at CZF as a foundational system for constructive mathematics. For example, the notion of predicativity is prone to different interpretations and constructive foundational systems are usually bound to the notion of generalised predicativity. In addition, constructive mathematicians usually see their practice as fully compatible with classical mathematics. This, however, rises some natural questions on the justification of constructive mathematics.
BAGARÍA -- Maximality vs. structural richness in the universe of all sets
Abstract: We will present several notions of structural richness for the set-theoretic universe, and we shall argue that the axioms of large cardinals in set theory are better justified in terms of structural richness, rather than in terms of maximality. We shall also discuss the relationship between structural richness and reflection in the universe of all sets.
FRIEDMAN -- The Hyperuniverse
Abstract: I discuss the Hyperuniverse approach to discovering desirable properties of the universe V of all sets. In this approach, one considers what properties a countable transitive model of ZFC will have in order to give it a "privileged status" within the Hyperuniverse of all models of its height, and then transfers these properties back to V. Natural sources of "privileged status" are maximility principles. Surprisingly, a compelling principle of "logical maximality", the Inner Model Hypothesis (IMH), leads to a refutation of large cardinals, contradicting the common claim that large cardinals are essential for maximality (see my paper with Arrigoni, "Foundational Implications of the Inner Model Hypothesis"). On the other hand, maximality principles expressed through reflection do lead to large cardinals. I will propose a possible solution to this dilemma by formulating a new maximality principle which embodies both logical maximality and strong reflection.
JANÉ -- Philosophical concerns regarding the power-set operation
Abstract: The power set of any given set A is easy to describe: it consists of all sets that are included in A. The inclusion relation being a very simple one, the power set of A is fully determined as soon as the universe of all sets is fixed. This, however, (even disregarding the difficulty to fix the extent of the set-theoretical universe, as witnessed by the huge variety of allowable models of set theory) is hardly a satisfactory way to understand the power-set operation. This is so partly because of the notion (embedded in the iterative conception of sets) that the universe of all sets is built from the power set operation, which entails that the power-set operation is prior to the whole set-theoretical universe and should be explained without recourse to it. In my talk, I will deal with the difficulties of accounting for the power set of any given infinite set and I will advance and argue for a somewhat unorthodox proposal for meeting them.