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METHOD:PUBLISH
BEGIN:VEVENT
DTSTAMP:20191007T130502Z
DTSTART:20190925T140000Z
DTEND:20190925T150000Z
SUMMARY:Rosario Mennuni - Double-membership graphs of models of Anti-Foun
dation
UID:{http://www.columbasystems.com/customers/uom/gpp/eventid/}on2-k0wl7sa
9-o2i64g
DESCRIPTION:Rosario Mennuni joins us for the Logic Seminar.\n\nIt is an o
ld result that the "membership graph" of any countable\nmodel of set the
ory\, obtained by joining x and y if x is in y *or*\ny is in x\, is isom
orphic to the random graph. This is true for\nextremely weak set theorie
s but\, crucially\, they have to satisfy\nthe Axiom of Foundation. \nI w
ill present recent work with Bea Adam-Day and John Howe in which\nwe stu
dy the class of "double-membership graphs"\, obtained by\njoining x and
y if x is in y *and* y is in x\, in the case of set\ntheory with the Ant
i-Foundation Axiom. In contrast with the omega-\ncategorical class of "t
raditional" membership graphs\, we show that\ndouble-membership graphs a
re way less well-behaved: their theory is\nincomplete and each of its co
mpletions has the maximum number of\ncountable models and is wild in the
sense of neostability theory. \nBy using ideas from finite model theory
\, we characterise the\naforementioned completions\, and show that the c
lass of countable\ndouble-edge graphs of Anti-Foundation is not even clo
sed under\nelementary equivalence among countable structures. This answe
rs\nsome questions of Adam-Day and Cameron.\n\n
STATUS:TENTATIVE
TRANSP:TRANSPARENT
CLASS:PUBLIC
LOCATION:Frank Adams 1\, Alan Turing Building\, Manchester
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