Talk by Sy Friedman (LTCS)

Short Title: 
Talk by Sy Friedman
Event Date(s): 
Wednesday, 9. May 2018 - 10:15 to 11:30

Room B078, ExWi, Sidlerstrasse 5, 3012 Bern

Prof. Dr. Sy Friedman (University of Vienna)

SUPERsystems of Second Order Arithmetic

There has been an extensive study of SUBsystems of second order arithmetic due to their strong connections to mathematical practice. In this talk I'll however focus on systems which extend Z_2, the axioms for second-order arithematic with full second-order comprehension, focusing on three topics. Let AC be the scheme saying that if for each natural number n there is a set X such that phi(n,X) then there is a single Y such that phi(n,Y_n) for all n. Let DC be the scheme saying that if for all sets X there is a set Y such that phi(X,Y) then for all X there is a Z such that Z_0 = X and for all n, phi(Z_n,Z_n+1).

1. Feferman-Levy showed that Z_2 does not prove AC, but their proof used the existence of aleph_omega, a rather substantial piece of set theory. I'll show how one can just start with a model of Z_2.

2. Simpson conjectured that Z_2 + AC does not prove DC. Vika Gitman and I verified this using a tree of Jensen-forcing iterations, extending work of Kanovei-Lyubetsky. Our proof takes place in third-order arithematic; I don't know how to do it starting with just a model of Z_2.

3. A beta-model of Z_2 is *minimal* if for some set X it is the smallest beta-model of Z_2 containing X. A beta-model M' of Z_2 is a *fair* extension of a beta-model M of Z_2 if M' contains M and every wellorder in M' is isomorphic (in M') to a wellorder in M. Carolin Antos and I showed that any countable beta-model of Z_2 + DC has a minimal fair extension. I don't know if this holds for Z_2 + AC. Our proof works by replacing M by its ''companion model'' of ZFC - Powerset and applying coding techniques to this model.