Randomness and Semigenericity

by Baldwin and Shelah. [BlSh:528]
Transactions American Math Soc, 1997
Let L contain only the equality symbol and let L^+ be an arbitrary finite symmetric relational language containing L . Suppose probabilities are defined on finite L^+ structures with ``edge probability'' n^{- alpha} . By T^alpha, the almost sure theory of random L^+-structures we mean the collection of L^+-sentences which have limit probability 1. T_alpha denotes the theory of the generic structures for K_alpha, (the collection of finite graphs G with delta_{alpha}(G)=|G|- alpha . | edges of G | hereditarily nonnegative.) THEOREM: T_alpha, the almost sure theory of random L^+-structures is the same as the theory T_alpha of the K_alpha-generic model. This theory is complete, stable, and nearly model complete. Moreover, it has the finite model property and has only infinite models so is not finitely axiomatizable.

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