# This Week in Logic at CUNY

Set theory seminar
Friday, September 27, 2013 10:00 am 6417
Speaker: Gunter Fuchs The City University of New York
Title: A self-specializing Souslin tree

I will present a construction, assuming Jensen’s combinatorial principle diamond, of a Souslin tree T which, after forcing with T, will be Ahronszajn off the generic branch. More precisely, forcing with T will add a cofinal branch b through T, yet in the generic extension by b, whenever p is a node of T which does not belong to b, then the subtree of T which lies above p will be Q-embeddable, meaning that there is an order preserving function from that subtree to the rationals. This shows that the rigidity property of being Souslin off the generic branch is strictly stronger than the unique branch property, two notions of rigidty previously studied in joint work with Joel Hamkins, where it was conjectured that it would be possible to construct such a self specializing Souslin tree.

Model theory seminar
Friday, September 27, 2013 12:30 pm GC 6417
Speaker: Diana Ojeda Aristizabal Cornell University
Title: Finite forms of Gowers’ Theorem on the oscillation stability of c_0

We give a constructive proof of the finite version of Gowers’ FIN_k Theorem and analyze the corresponding upper bounds. The FIN_k Theorem is closely related to the oscillation stability of c_0. The stabilization of Lipschitz functions on arbitrary finite dimensional Banach spaces was proved well before by V. Milman. We compare the finite FIN_k Theorem with the Finite Stabilization Principle found by Milman in the case of spaces of the form ell_{infty}^n, ninN, and establish a much slower growing upper bound for the finite stabilization principle in this particular case.

CUNY Logic Workshop
Friday, September 27, 2013 2:00 pm GC 6417
Speaker: Joel David Hamkins The City University of New York
Title: Satisfaction is not absolute
I will discuss a number of theorems showing that the satisfaction relation of first-order logic is less absolute than might have been supposed. Two models of set theory $M_1$ and $M_2$, for example, can agree on their natural numbers $langlemathbb{N},{+},{cdot},0,1,{lt}rangle^{M_1}=langlemathbb{N},{+},{cdot},0,1,{lt}rangle^{M_2}$, yet disagree on arithmetic truth: they have a sentence $sigma$ in the language of arithmetic that $M_1$ thinks is true in the natural numbers, yet $M_2$ thinks $negsigma$ there. Two models of set theory can agree on the natural numbers $mathbb{N}$ and on the reals $mathbb{R}$, yet disagree on projective truth. Two models of set theory can have the same natural numbers and have a computable linear order in common, yet disagree about whether this order is well-ordered. Two models of set theory can have a transitive rank initial segment $V_delta$ in common, yet disagree about whether this $V_delta$ is a model of ZFC. The theorems are proved with elementary classical methods.