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Turing computable embeddings

Published online by Cambridge University Press:  12 March 2014

Julia F. Knight ,
Sara Miller  and
M. Vanden Boom
Julia F. Knight
Affiliation:
University of Notre Dame, Department of Mathematics, 255 Hurley Building, Notre Dame, IN 46556-4618, USA. E-mail: Julia.F.Knight.l@nd.edu
Sara Miller
Affiliation:
University of Notre Dame, Department of Mathematics, 255 Hurley Building, Notre Dame, IN 46556-4618, USA. E-mail: smiller9@nd.edu
M. Vanden Boom
Affiliation:
University of Notre Dame, and, 14904 Lynbrook Drive, Baton Rouge, LA 70816, USA. E-mail: mvandenb@nd.edu
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Abstract

In [3]. two different effective versions of Borel embedding are defined. The first, called computable embedding, is based on uniform enumeration reducibility. while the second, called Turing computable embedding, is based on uniform Turing reducibility. While [3] focused mainly on computable embeddings, the present paper considers Turing computable embeddings. Although the two notions are not equivalent, we can show that they behave alike on the mathematically interesting classes chosen for investigation in [3]. We give a “Pull-back Theorem”, saying that if Ф is a Turing computable embedding of K into K′, then for any computable infinitary sentence φ in the language of K′, we can find a computable infinitary sentence φ* in the language of K such that for all A ∈ K A ⊨ φ* iff Φ (A) ⊨ φ and φ* has the same “complexity” as φ (i.e., if φ is computable Σα or computable Πα, for α ≥ 1, then so is φ*). The Pull-back Theorem is useful in proving non-embeddability, and it has other applications as well.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 3 , September 2007 , pp. 901 - 918
Copyright
Copyright © Association for Symbolic Logic 2007

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References

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