Skip to main content
×
×
Home

Initial segments of the lattice of Π10 classes

  • Douglas Cenzer (a1) and Andre Nies (a2)
Abstract.

We show that in the lattice of classes there are initial segments [∅, P] = (P) which are not Boolean algebras, but which have a decidable theory. In fact, we will construct for any finite distributive lattice L which satisfies the dual of the usual reduction property a class P such that L is isomorphic to the lattice (P)*, which is (P). modulo finite differences. For the 2-element lattice, we obtain a minimal class, first constructed by Cenzer, Downey, Jockusch and Shore in 1993. For the simplest new class P constructed, P has a single, non-computable limit point and (P)* has three elements, corresponding to ∅, P and a minimal class P0P, The element corresponding to P0 has no complement in the lattice. On the other hand, the theory of (P) is shown to be decidable.

A class P is said to be decidable if it is the set of paths through a computable tree with no dead ends. We show that if P is decidable and has only finitely many limit points, then (P)* is always a Boolean algebra. We show that if P is a decidable class and (P) is not a Boolean algebra, then the theory of (P) interprets the theory of arithmetic and is therefore undecidable.

Copyright
References
Hide All
[1]Burris, S. and Sankappanavar, H. P., Lattice-theoretic decision problems in universal algebras, Algebra Universalis, vol. 5 (1975), pp. 163177.
[2]Cenzer, D., classes in computability theory, Handbook of computability (Griffor, E., editor), Studies in Logic, no. 140, North-Holland, 1999, pp. 3785.
[3]Cenzer, D., Downey, R., Jockusch, C., and Shore, R., Countable thin classes, Annals of Pure and Applied Logic, vol. 59 (1993), pp. 79139.
[4]Cenzer, D. and Jockusch, C., classes – structure and application, Proceedings of the Boulder American Mamthematical Society conference (Lerman, M., editor), Contemporary Mathematics, American Mathematical Society, to appear.
[5]Cenzer, D. and Remmel, J., classes in mathematics, Handbook of recursive mathematics (Ershov, Y., Goncharov, S., Remmel, J., and Nerode, A., editors), Studies in Logic, no. 139, North-Holland, 1998, pp. 623821.
[6]Downey, R., Maximal theories, Annals of Pure and Applied Logic, vol. 33 (1987), pp. 245282.
[7]Downey, R., Jockusch, C. G., and Stob, M., Array nonrecursive sets and multiple permitting arguments, Recursion theory week (Oberwolfach, March 1989) (Ambos-Spies, K., Muller, G., and Sacks, G. E., editors), Lecture Notes in Mathematics, no. 142, Springer, 1990, pp. 141173.
[8]Hermann, E.. Unpublished manuscript.
[9]Kripke, S. and Pour-El, M., Deduction-preserving recursive isomorphisms between theories, Fundamenta Mathematicae, vol. 61 (1967), pp. 141163.
[10]Lachlan, A., On the lattice of recursively enumerable sets, Transactions of the American Mathematical Society, vol. 130 (1968), pp. 137.
[11]Martin, D. and Pour-El, M., Axiomatizable theories with few axiomatizable extensions, this Journal, vol. 35 (1970), pp. 205209.
[12]Nies, A., Intervals of the lattice of computably enumerable sets, Bulletin of the London Mathematical Society, vol. 29 (1997), pp. 683692.
[13]Nies, A., Effectively dense boolean algebras and their applications, Transactions of the American Mathematical Society, vol. 352 (2000), pp. 49895012.
[14]Soare, R., Recursively enumerable sets and degrees, Springer-Verlag, 1987.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 5 *
Loading metrics...

Abstract views

Total abstract views: 70 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd May 2018. This data will be updated every 24 hours.