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Degrees coded in jumps of orderings

Published online by Cambridge University Press:  12 March 2014

Julia F. Knight
Julia F. Knight*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
*
Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
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Extract

All structures to be considered here have universe ω, and all languages come equipped with Gödel numberings. If is a structure, then D(), the open diagram of , can be thought of as a subset of ω, and it makes sense to talk about the Turing degree deg(D()). This depends on the presentation as well as the isomorphism type of .

For example, consider the ordering = (ω, <). For any B ⊆ ω, it is possible to code B in a copy of as follows: Let π be the permutation of ω such that for each n ∈ ω, π leaves 2n and 2n + 1 fixed if n ∈ B and switches 2n with 2n + 1 if n ∉ B. Let be the copy of such that ≃π. Then n ∈ B iff the sentence 2n < 2n + 1 is in D(). In §4, this idea will be used to show that for any structure that is not completely trivial, {deg(D()): ≃ } is closed upwards.

It would be satisfying to have a way of assigning Turing degrees to structures such that the degree assigned to a given structure measured the recursion-theoretic complexity of the isomorphism type and was independent of the presentation. Jockusch suggested the following.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 4 , December 1986 , pp. 1034 - 1042
Copyright
Copyright © Association for Symbolic Logic 1986

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References

REFERENCES

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