Rosario Mennuni: Model theory of double-membership
Thursday, 21.01.2021 10:30 im Raum via Zoom
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.
In joint work with Bea Adam-Day and John Howe 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,
supersimple 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.
Angelegt am Monday, 11.01.2021 09:57 von Martina Pfeifer
Geändert am Monday, 11.01.2021 09:57 von Martina Pfeifer
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