Sam Sanders

Sam Sanders, Ghent and Munich, will give a lecture with the tittle

On the contingency of predicativism.



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 devel-
opment of the foundational program predicativist mathematics.  The system $\Pi_1^1-CA_0$ (resp. arithmetical comprehen-
sion $ACA_{0}$) is a textbook example of impredicative (resp. predicative) mathematics.  In this talk, 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.   


Matematisk Institutt v/ Dag Normann
Tags: Matematisk Logikk
Published Mar. 4, 2015 10:51 AM - Last modified Mar. 4, 2015 10:51 AM