Properly ergodic structures

2017/10/25 by Ackerman, Nathanael, Freer, Cameron, Kruckman, Alex +1
#Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO) #Probability (math.PR)

paper · doi:10.48550/arxiv.1710.09336

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_{ω1, 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_{ω1, 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_{ω1, 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.

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