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Sequences of n-diagrams

Published online by Cambridge University Press:  12 March 2014

Valentina S. Harizanov ,
Julia F. Knight  and
Andrei S. Morozov
Valentina S. Harizanov
Affiliation:
Department of Mathematics, The George Washington University, Washington, D.C. 20052, USA, E-mail: [email protected]
Julia F. Knight
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA, E-mail: [email protected]
Andrei S. Morozov
Affiliation:
Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia, E-mail: [email protected]
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Extract

We consider only computable languages, and countable structures, with universe a subset of ω, which we think of as a set of constants. We identify sentences with their Gödel numbers. Thus, for a structure , the complete (elementary) diagram, Dc(), and the atomic diagram, D(), are subsets of ω. We classify formulas as usual. A formula is both Σ0 and Π0 if it is open. For n > 0, a formula, in prenex normal form, is Σn, or Πn, if it has n blocks of like quantifiers, beginning with ∃, or ∀. For a formula θ, in prenex normal form, we let neg(θ) denote the dual formula that is logically equivalent to ¬θ—if θ is Σn, then neg(θ) is Πn, and vice versa.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 3 , September 2002 , pp. 1227 - 1247
Copyright
Copyright © Association for Symbolic Logic 2002

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References

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