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Transactions of the American Mathematical Society

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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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