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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Randomness and semigenericity
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by John T. Baldwin and Saharon Shelah PDF
Trans. Amer. Math. Soc. 349 (1997), 1359-1376 Request permission


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 $\mathbb {K}_{\alpha }$ (the collection of finite graphs $G$ with $\delta _\alpha (G) = |G| - \alpha \cdot |\text {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 $\mathbb {K}_{\alpha }$-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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Additional Information
  • Saharon Shelah
  • MR Author ID: 160185
  • ORCID: 0000-0003-0462-3152
  • Received by editor(s): September 7, 1994
  • Additional Notes: Partially supported by NSF grant 9308768 and a visit to Simon Fraser University.
    This is paper 528. Both authors thank Rutgers University and the Binational Science Foundation for partial support of this research.
  • © Copyright 1997 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 349 (1997), 1359-1376
  • MSC (1991): Primary 03C10, 05C80
  • DOI:
  • MathSciNet review: 1407480