[FOM] IUSS Philosophy - Visting Professors Seminars - Gabriel Uzquiano (USC) - The cardinal problem of absolute generality - July 10-12, 2017

[Andrea Sereni]

*IUSS Philosophy
Visiting Professors Seminars**
*
/IUSS/UniSR PhD program Cognitive Neurosciences and Philosophy of Mind////
//NEtS Centre @ IUSS Pavia//
/www.iusspavia.it - www.nets.iusspavia.it

*
Gabriel Uzquiano (University of Southern California)*
/*The cardinal problem of absolute generality**
*/
July 10-12, 2017



*1. **Cantor’s 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 Cantor’s 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: Kaplan’s paradox (for Not 
onto) and Russell’s paradox of propositions (for Not one-one).

*2. Class-forms of Cantor’s 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 Cantor’s 
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*
/Friday 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.
/https://filmatnetwork.com/activities/gradconf2017/
/
The paper looks at the debate over the import of Russell’s paradox for 
Frege’s 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 Frege’s 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.

***


*Seminars Venue*
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*: andrea.sereni at iusspavia.it
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