# Talk by Sy Friedman (LTCS)

**Venue:**

Room B078, ExWi, Sidlerstrasse 5, 3012 Bern

**Speaker:**

Prof. Dr. Sy Friedman (University of Vienna)

**Title:**

SUPERsystems of Second Order Arithmetic

**Abstract:**

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.

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