Randomness and semigenericity
Abstract
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.
Cite
@article{arxiv.math/9607226,
title = {Randomness and semigenericity},
author = {John T. Baldwin and Saharon Shelah},
journal= {arXiv preprint arXiv:math/9607226},
year = {2016}
}