Rosario Mennuni - Double-membership graphs of models of Anti-Foundation
|Dates:||25 September 2019|
|Times:||15:00 - 16:00|
|What is it:||Seminar|
|Organiser:||Department of Mathematics|
|Who is it for:||University staff, External researchers, Current University students|
Rosario Mennuni joins us for the Logic Seminar.
It is an old result that the "membership graph" of any countable
model of set theory, obtained by joining x and y if x is in y *or*
y is in x, is isomorphic to the random graph. This is true for
extremely weak set theories but, crucially, they have to satisfy
the Axiom of Foundation.
I will present recent work with Bea Adam-Day and John Howe in which
we study the class of "double-membership graphs", obtained by
joining x and y if x is in y *and* y is in x, in the case of set
theory with the Anti-Foundation Axiom. In contrast with the omega-
categorical class of "traditional" membership graphs, we show that
double-membership graphs are way less well-behaved: their theory is
incomplete and each of its completions has the maximum number of
countable models and is wild in the sense of neostability theory.
By using ideas from finite model theory, we characterise the
aforementioned completions, and show that the class of countable
double-edge graphs of Anti-Foundation is not even closed under
elementary equivalence among countable structures. This answers
some questions of Adam-Day and Cameron.
Organisation: University of Leeds
Travel and Contact Information
Frank Adams 1
Alan Turing Building