Properly ergodic structures

Por • 1 nov, 2017 • Sección: Leyes

Nathanael Ackerman, Cameron Freer, Alex Kruckman, Rehana Patel

Abstract: We consider ergodic Sym(N)  -invariant probability measures on the space of L  -structures with domain N  (for L  a countable relational language), and call such a measure a properly ergodic structure when no isomorphism class of structures is assigned measure 1  . We characterize those theories in countable fragments of L ω 1 ,ω for which there is a properly ergodic structure concentrated on the models of the theory. We show that for a countable fragment F  of L ω 1 ,ω the almost-sure F  -theory of a properly ergodic structure has continuum-many models (an analogue of Vaught’s Conjecture in this context), but its full almost-sure L ω 1 ,ω -theory has no models. We also show that, for an F  -theory T  , if there is some properly ergodic structure that concentrates on the class of models of T  , then there are continuum-many such properly ergodic structures.

arXiv:1710.09336v1 [math.LO]

Logic (math.LO); Combinatorics (math.CO); Probability (math.PR)

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