Week 6, 17 February 2014 (16:00-18:00, 52 Pritchatts Road LT1)

Sam Sanders (Ghent, Munich)

Title: On the contingency of predicativism

Abstract:

Following his discovery of the paradoxes present in naive set theory, Russell proposed to ban the vicious circle principle, nowadays called impredicative definition. A set is impredicative if its definition quantifies over the totality of sets it belongs to, like the definition of supremum as the least upper bound. Russell's proposal was taken up by Weyl and Feferman in their development of the foundational program predicativist mathematics. The system $$\Pi_1^1-CA_0$$ (resp. arithmetical comprehension $$ACA_{0}$$) is a textbook example of impredicative (resp. predicative) mathematics. In this paper, we show that $$\Pi_1^1-CA_0$$ can be viewed as an instance of nonstandard arithmetical comprehension. We also show that bar recursion, another inherently impredicative operation, may be viewed as the highly predicative notion of primitive recursion, but with nonstandard numbers. In other words, predicativism seems to be contingent on whether the framework at hand accommodates Nonstandard Analysis, arguably an undesirable feature for a foundational philosophy.

The Midlands Logic Seminar was founded in 2011 and aims to cover all areas of mathematical logic, as well as related areas of theoretical computer science and philosophy of mathematics.

Logistics

All meetings for Term 2, 2014-15 will be Tuesday from 16:00-18:00 in room LT1, 52 Pritchatts Road on the campus of the University of Birmingham.
• campus map
To get to campus, take the train from Birmingham New Street to the University stop (7 minutes).

Organizers

Dr Richard Kaye
School of Mathematics
University of Birmingham

http://web.mat.bham.ac.uk/R.W.Kaye/

Dr Walter Dean
Department of Philosophy
University of Warwick

http://go.warwick.ac.uk/whdean