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- W2766167949 abstract "We consider ergodic $mathrm{Sym}(mathbb{N})$-invariant probability measures on the space of $L$-structures with domain $mathbb{N}$ (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 $mathcal{L}_{omega_1, omega}$ for which there is a properly ergodic structure concentrated on the models of the theory. We show that for a countable fragment $F$ of $mathcal{L}_{omega_1, omega}$ 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 $mathcal{L}_{omega_1, omega}$-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." @default.
- W2766167949 created "2017-11-10" @default.
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- W2766167949 date "2017-10-25" @default.
- W2766167949 modified "2023-09-27" @default.
- W2766167949 title "Properly ergodic structures" @default.
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