Submission date: 17 October 2017

Abstract:

Let L be a countable language. We characterize, in terms of definable closure, those countable theories Σ of L_{ω_1, ω}(L) for which there exists an S_∞-invariant probability measure on the collection of models of Σ with underlying set N. Restricting to L_{ω, ω}(L), this answers an open question of Gaifman from 1964, via a translation between S_∞-invariant measures and Gaifman's symmetric measure-models with strict equality. It also extends the known characterization in the case where Σ implies a Scott sentence. To establish our result, we introduce machinery for building invariant measures from a directed system of countable structures with measures.

Mathematics Subject Classification:

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Full text arXiv 1710.06128: pdf, ps.