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Interpretability and definability in therecursively enumerable degrees

Published online by Cambridge University Press:  01 September 1998

A Nies ,
RA Shore  and
TA Slaman
A Nies
Affiliation:
Department of Mathematics, 5734 S. University Avenue, University of Chicago, Chicago, IL 60637, USA. E-mail: nies@math.uchicago.edu
RA Shore
Affiliation:
Department of Mathematics, White Hall, Cornell University, Ithaca, NY 14853-7901, USA. E-mail: shore@math.cornell.edu
TA Slaman
Affiliation:
Department of Mathematics, 5734 S. University Avenue, University of Chicago, Chicago, IL 60637, USA. E-mail: nies@math.uchicago.edu Present address: Department of Mathematics, University of California at Berkeley, Berkeley CA 94720, USA. E-mail: slaman@math.berkeley.edu
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Abstract

We investigate definability in $\mathcal{R}$, the recursively enumerable Turing degrees, using codings of standard models of arithmetic (SMAs) as a tool. First we show that an SMA can be interpreted in $\mathcal{R}$ without parameters. Building on this, we prove that the recursively enumerable $T$-degrees satisfy a weak form of the bi-interpretability conjecture which implies that all jump classes $\mathrm{Low}_n$ and $\mathrm{High}_{n-1}$$(n\ge 2)$ are definable in $\mathcal{R}$ without parameters and, more generally, that all relations on $\mathcal{R}$ that are definable in arithmetic and invariant under the double jump are actually definable in $\mathcal{R}$. This partially answers Soare's Question 3.7 (R. Soare, {\emRecursively enumerable sets and degrees} (Springer, Berlin, 1987), Chapter XVI).

1991 Mathematics Subject Classification: primary 03D25, 03D35; secondary 03D30.

Keywords

recursively enumerable degreesr.e.\ degreesjump classeshomogeneity failuresdefinabilitybi-interpretabilitytrue arithmetic
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 77 , Issue 2 , September 1998 , pp. 241 - 291
Copyright
London Mathematical Society 1998

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