Model theory and probability
Shichang Song · Illinois Digital Environment for Access to Learning and Scholarship (University of Illinois at Urbana-Champaign) · 2012
This thesis presents a systematic study of the model theory of probability algebras, random variable structures, and adapted structures, with an emphasis on their atomless counterparts. In this thesis, the author uses a continuous version of first order logic that has been developed recently and that is better suited for applications to metric structures than classical first order logic. The set of truth values in continuous logic is the interval [0,1] instead of the truth values {True, False} in classical logic. The author studies axioms, type spaces, quantifier elimination, separable categoricity, saturated models, stability, and d-finiteness for the theories of atomless probability algebras and atomless random variable structures. Explicit formulas for the d*-metric between types in the theory of atomless random variable structures are given. For the theory of atomless adapted structures, the author studies axioms, type spaces, quantifier elimination, separably categoricity, and d-finiteness.