A sufficient condition for super-simplicity of almost-sure theories
نویسنده
چکیده
In Hill [11], it was shown that an almost-sure theory for a Fraïssé class K relative to an asymptotic probability measure with “independent sampling” is necessarily super-simple of finite rank with D(x=x) = 1. In this note, we show that this independent sampling hypothesis is much stronger than necessary. By examining the Aldous-Hoover-Kallenberg (AHK) presentation of an asymptotic probability measure, we find that a certain “ample-fibers” condition is sufficient for super-simplicity. This ample-fibers condition expresses, as a statement about the “hidden” randomness of the AHK-presentation, the assurance that every extension axiom of the generic theory of K is satisfied.
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