[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:
http://institucional.us.es/phum2594/
settheoryworkshop_Timetable&Abstracts.html
Attendance is free, but we kindly ask you to contact the organizers.
Best wishes,
José Ferreirós
Departamento de Filosofia y Logica
Universidad de Sevilla
http://personal.us.es/josef/
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:
"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
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.
José Ferreirós
Departamento de Filosofia y Logica
Universidad de Sevilla
http://personal.us.es/josef/
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