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# Fine spectra and limit laws II First-order 0--1 laws.

Published:1997-08-01
Printed: Aug 1997
• Stanley Burris
• Kevin Compton
• Andrew Odlyzko
• Bruce Richmond
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## Abstract

Using Feferman-Vaught techniques a condition on the fine spectrum of an admissible class of structures is found which leads to a first-order 0--1 law. The condition presented is best possible in the sense that if it is violated then one can find an admissible class with the same fine spectrum which does not have a first-order 0--1 law. If the condition is satisfied (and hence we have a first-order %% 0--1 law)
 MSC Classifications: 03N45 - unknown classification 03N4511N45 - Asymptotic results on counting functions for algebraic and topological structures 11N80 - Generalized primes and integers 05A15 - Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05A16 - Asymptotic enumeration 11M41 - Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 11P81 - Elementary theory of partitions [See also 05A17]