Logic Workshop

Fall 2017

CUNY Graduate Center
Room 6417
Fridays 2:00-3:30pm
Organized by:
Joel David Hamkins
Roman Kossak
Russell Miller
Philipp Rothmaler
Hans Schoutens

September 1
John Goodrick, Universidad de los Andes
New directions in the model theory of groups: tame topologies, orderings, and finite rank
We will discuss several recent results concerning the model theory of groups for which we assume some kind of tameness condition. First, we review some results (joint with A. Dolich) about ordered Abelian groups of finite burden, where “burden” or “inp-rank” is a generalization of weight which is useful in unstable theories; in this context, we can show that unary definable sets satisfy various desirable properties. Next, we present more recent results (joint with J. Dobrowolski) showing that inp-minimal groups with an ordering invariant under left translations are Abelian, and also showing that finite weight stable groups cannot be too far from being Abelian. Finally, time permitting, we will discuss definable uniform structures, introducing the new concept of “viscerality” which allows us to develop a tame topology for definable sets in arbitrary dimensions (also joint work with A. Dolich).

Joel David Hamkins, CUNY
Arithmetic potentialism and the universal algorithm
Consider the collection of all the models of arithmetic under the end-extension relation, which forms a potentialist system for arithmetic, a collection of possible arithmetic worlds or universe fragments, with a corresponding potentialist modal semantics. What are the modal validities? I shall prove that every model of arithmetic validates exactly S4 with respect to assertions in the language of arithmetic allowing parameters, but if one considers sentences only (no parameters), then some models can validate up to S5, thereby fulfilling the arithmetic maximality principle, which asserts for a model $M$ that whenever an arithmetic sentence is true in some end-extension of $M$ and all subsequent end-extensions, then it is already true in $M$. (We also consider other accessibility relations, such as arbitrary extensions or $\Sigma_n$-elementary extensions or end-extensions.)

The proof makes fundamental use of what I call the universal algorithm, a fascinating result due to W. Hugh Woodin, asserting that there is a computable algorithm that can in principle enumerate any desired finite sequence, if only it is undertaken in the right universe, and furthermore any given model of arithmetic can be end-extended so as to realize any desired additional behavior for that universal program. I shall give a simple proof of the universal algorithm theorem and explain how it can be used to determine the potentialist validities of a model of arithmetic. This is current joint work in progress with Victoria Gitman and Roman Kossak, and should be seen as an arithmetic analogue of my recent work on set-theoretic potentialism with Øystein Linnebo. The mathematical program is strongly motivated by philosophical ideas arising in the distinction between actual and potential infinity.

September 15
No seminar because of Simon Thomas: the first 60 years birthday conference.

September 22
No seminar because of CUNY holidays.

September 29
No seminar because of CUNY holidays.

October 6
Hans Schoutens, CUNY

October 13
Alf Dolich, CUNY

October 20
Daniel Turetsky, Notre Dame University

October 27
No seminar because of MAMLS Logic Friday.

November 3
Jennifer Chubb, University of San Francisco

November 17
Carolin Antos, University of Konstanz

November 24
No seminar because of CUNY holidays.