Submission date: 9 January 2014.

Abstract:

We characterize model theoretic properties of the Urysohn sphere in continuous logic. In particular, we show that the theory of the Urysohn sphere is SOP_n for all n\geq 3, but does not have the (finitary) strong order property. In the process, we give necessary and sufficient conditions for when a partially defined symmetric function on a finite set can be extended to a metric on that set. Our second main result is a geometric characterization of dividing independence in the theory of the Urysohn sphere. We further show that this characterization satisfies the extension axiom, and so forking and dividing are the same for complete types.

Mathematics Subject Classification:

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Full text arXiv 1401.2132: pdf, ps.