The -theory of is undecidable

Authors:
Russell G. Miller, Andre O. Nies and Richard A. Shore

Journal:
Trans. Amer. Math. Soc. **356** (2004), 3025-3067

MSC (2000):
Primary 03D25, 03D35; Secondary 03D28

Published electronically:
October 8, 2003

MathSciNet review:
2052940

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Abstract: The three quantifier theory of , the recursively enumerable degrees under Turing reducibility, was proven undecidable by Lempp, Nies and Slaman (1998). The two quantifier theory includes the lattice embedding problem and its decidability is a long-standing open question. A negative solution to this problem seems out of reach of the standard methods of interpretation of theories because the language is relational. We prove the undecidability of a fragment of the theory of that lies between the two and three quantifier theories with but includes function symbols.

**Theorem.** *The two quantifier theory of , the r.e. degrees with Turing reducibility, supremum and infimum (taken to be any total function extending the infimum relation on ) is undecidable.*

The same result holds for various lattices of ideals of which are natural extensions of preserving join and infimum when it exits.

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Additional Information

**Russell G. Miller**

Affiliation:
Department of Mathematics, Queens College (CUNY), 65-30 Kissena Blvd., Flushing, New York 11367

Email:
Russell_Miller@qc.edu

**Andre O. Nies**

Affiliation:
Office 592, Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand

Email:
andre@cs.auckland.ac.nz

**Richard A. Shore**

Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853

Email:
shore@math.cornell.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-03-03406-8

Received by editor(s):
October 21, 2002

Published electronically:
October 8, 2003

Additional Notes:
The first author was partially supported by a VIGRE Postdoctoral Fellowship under NSF grant number 9983660 to Cornell University. The third author was partially supported by NSF Grant DMS-0100035. The authors offer their thanks to the referee for several helpful comments.

Article copyright:
© Copyright 2003
American Mathematical Society