[FOM] Mini-Workshop "Maximalist and Minimalist Perspectives on Infinity" Seville, Spain, Nov. 7 & 8

José Ferreirós Domínguez josef at us.es
Thu Nov 3 07:13:28 EDT 2011

Mini-Workshop Maximalist and Minimalist Perspectives on Infinity: A  
meeting on set theory and its philosophy in Seville, Spain, Nov. 7 &  
8. Below we include information about the talks, including abstracts.  
You may also find the information in the webpage:


Attendance is free, but we kindly ask you to contact the organizers.

Best wishes,

José Ferreirós

Departamento de Filosofia y Logica
Universidad de Sevilla

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”

10:00             Talk by Sy D. Friedman:
"The Hyperuniverse"

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  

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  

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.

José Ferreirós

Departamento de Filosofia y Logica
Universidad de Sevilla

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