IUSS Philosophy - Visting Professors Seminars - Gabriel Uzquiano (USC) - The cardinal problem of absolute generality - July 10-12, 2017
Visiting Professors Seminars
Gabriel Uzquiano (University of Southern California)
The cardinal problem of absolute generality
July 10-12, 2017
1. Cantors theorem: Uses and abuses
Monday 10th: 16-18, Room 1.17
This chapter begins with a quick exposition of two different, though equivalent, versions of Cantors theorem in a set-theoretical framework: Not onto and Not one-one. I present further generalizations of each with an eye to at least two areas of application: Kaplans paradox (for Not onto) and Russells paradox of propositions (for Not one-one).
2. Class-forms of Cantors theorem
Tuesday 11th: 14-16, Room 1.17
This chapter sets out to generalize the usual Cantorian propositions to the language of classes where the target is the claim that a class including the universal class has strictly more subclasses than members. Unfortunately, the expressive limitations of the language of classes result in two different strategies for generalizing Cantors theorem. They result in class-theoretic generalizations of Not onto and Not one-one. But unlike their set-theoretic counterparts, they are not equivalent to each other. The generalization of Not onto is far weaker than the generalization of Not one-one. This results in questions to do with whether some of the usual applications of the former are in fact warranted. Some of these applications include the explanation of many set-theoretic antinomies, the claim that there are more pluralities than individuals or, in the philosophy of logic, more interpretations of a language than there are objects.
3. >From Cantorian propositions to cardinal inequalities
Wednesday 12th: 11-13, Room 1.17
This chapter begins with the observation that there is a significant gap between the class-theoretic generalization of Not onto and the further claim that a given class has more subclasses than members, which is what is required for the substantive uses to which it is generally put. The plan is, first, to distinguish the concern from two other sources of skepticism as to the import of Cantorian propositions in general and to check what is required to bridge the gap between one and the other.
NB: Additional related talk at 1st FilMat Graduate Conference, Trento:
Impredicativity and Paradox
First FilMat Graduate Conference, University of Trento, July 13th-14th.
The paper looks at the debate over the import of Russells paradox for Freges theory of extensions. One party in the debate claims that the real lesson of the paradox is that there are more concepts than objects whereas the other claims that impredicative second-order comprehension is what originates the problem. The paper confronts some arguments against the former and aims to put some pressure on the latter by noting that their preferred diagnosis of the inconsistency of Freges Law V fails to cover the inconsistency of a related proposition. This is not a problem for the former strategy since the two propositions are equivalent by their own lights.
Scuola Superiore Universitaria IUSS Pavia
Palazzo del Broletto, Room 1-17
Piazza della Vittoria 15, 27100 Pavia, Italy
How to reach the seminars venue: http://www.iusspavia.it/eng/index.php?id=40#.VAXkIGTV8mg
Everyone is invited. Info at: firstname.lastname@example.org