Global properties of the lattice of classes

Authors:
Douglas Cenzer and André Nies

Journal:
Proc. Amer. Math. Soc. **132** (2004), 239-249

MSC (2000):
Primary 03D25

DOI:
https://doi.org/10.1090/S0002-9939-03-06984-3

Published electronically:
May 7, 2003

MathSciNet review:
2021268

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the lattice of classes of reals. We show there are exactly two possible isomorphism types of end intervals, . Moreover, finiteness is first order definable in .

**1.**D. Cenzer, R. Downey, C. J. Jockusch and R. Shore.

Countable Thin Classes.*Ann. Pure Appl. Logic*, 59:79-139, 1993. MR**93m:03075****2.**D. Cenzer and C. J. Jockusch.

Classes-Structure and Applications.*Computability Theory and Its Applications*, eds. P. Cholak, S. Lempp, M. Lerman and R. Shore, Contemporary Mathematics 257:39-59, 2000. MR**2001h:03074****3.**D. Cenzer and A. Nies.

Initial segments of the lattice of classes.*Journal of Symbolic Logic*, 66: 1749-1765, 2001. MR**2002k:03067****4.**D. Cenzer and J. B. Remmel. Index sets for classes.*Ann. Pure Appl. Logic*, 93:3-61, 1998. MR**99m:03080****5.**D. Cenzer and J. B. Remmel. Classes in Mathematics.*Handbook of Recursive Mathematics, vol. 2*, Stud. Logic Found. Math. 139:623-821, Elsevier, 1998. MR**2001d:03108****6.**P. Cholak, R. Coles, R. Downey and E. Hermann. Automorphisms of the Lattice of Classes.*Transactions Amer. Math. Soc.*, 353:4899-4924, 2001. MR**2002f:03080****7.**C. C. Chang and H. J. Keisler.*Model Theory*.

North-Holland Publishing Co., Amsterdam, 1973. MR**53:12927****8.**R. Downey and J. B. Remmel.

Questions in Computable Algebra and Combinatorics.*Computability Theory and Its Applications*, eds. P. Cholak, S. Lempp, M. Lerman and R. Shore, Contemporary Mathematics 257:95-125, 2000. MR**2001i:03094****9.**L. A. Harrington and A. Nies.

Coding in the partial order of enumerable sets.*Adv. Math.*, 133:133-162, 1998. MR**99c:03063****10.**A. Nies.

Relativizations of structures arising from recursion theory.

In*Computability, Enumerability and Unsolvability (Proc. Leeds Logic Year*, eds. S.B. Cooper, T.A. Slaman and S.S. Wainer, Cambridge University Press, London Math. Society Lecture Notes 224:219-232, 1996. MR**98g:03104****11.**A. Nies.

Intervals of the lattice of computably enumerable sets and effective Boolean algebras.*Bull. Lond. Math. Soc.*, 29:683-92, 1997. MR**98j:03057****12.**A. Nies.

Effectively dense Boolean algebras and their applications.*Trans. Amer. Math. Soc.*, 352:4989-5012, 2000. MR**2001i:03066**

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Additional Information

**Douglas Cenzer**

Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611

Email:
cenzer@math.ufl.edu

**André Nies**

Affiliation:
Department of Mathematics, The University of Chicago, 5734 S. University Ave., Chicago, Illinois 60637

Address at time of publication:
Department of Computer Science, University of Auckland, Private Bag 92019, Auckland 1020, New Zealand

Email:
nies@math.uchicago.edu, andre@cs.auckland.ac.nz

DOI:
https://doi.org/10.1090/S0002-9939-03-06984-3

Keywords:
$\Pi^0_1$ classes,
definability,
end segments

Received by editor(s):
June 4, 2002

Received by editor(s) in revised form:
August 19, 2002

Published electronically:
May 7, 2003

Additional Notes:
The second author was partially supported by NSF grant DMS–9803482

Communicated by:
Carl G. Jockusch, Jr.

Article copyright:
© Copyright 2003
American Mathematical Society