7 research outputs found
On double-membership graphs of models of Anti-Foundation
We answer some questions about graphs which are reducts of countable models
of Anti-Foundation, obtained by considering the binary relation of
double-membership . We show that there are continuum-many such
graphs, and study their connected components. We describe their complete
theories and prove that each has continuum-many countable models, some of which
are not reducts of models of Anti-Foundation.Comment: 15 pages, 2 figure
Undirecting membership in models of Anti-Foundation
It is known that, if we take a countable model of ZermeloâFraenkel set theory ZFC and âundirectâ the membership relation (that is, make a graph by joining x to y if either xây or yâx), we obtain the ErdĆsâRĂ©nyi random graph. The crucial axiom in the proof of this is the Axiom of Foundation; so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczelâs Anti-Foundation Axiom). The resulting graph may fail to be simple; it may have loops (if xâx for some x) or multiple edges (if xây and yâx for some distinct x, y). We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the ârandom loopy graphâ (which is â”0-categorical and homogeneous), but if we keep multiple edges, the resulting graph is not â”0-categorical, but has infinitely many 1-types. Moreover, if we keep only loops and double edges and discard single edges, the resulting graph contains countably many connected components isomorphic to any given finite connected graph with loops.Publisher PDFPeer reviewe