### 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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