### 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.

Back to the list of publications