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Types of simple α-recursively enumerable sets

Published online by Cambridge University Press:  12 March 2014

Anne Leggett  and
Richard A. Shore
Anne Leggett
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Richard A. Shore
Affiliation:
Cornell University, Ithaca, New York 14853
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Extract

One general program of α-recursion theory is to determine as much as possible of the lattice structure of (α), the lattice of α-r.e. sets under inclusion. It is hoped that structure results will shed some light on whether or not the theory of (α) is decidable with respect to a suitable language for lattice theory. Fix such a language ℒ.

Many of the basic results about the lattice structure involve various sorts of simple α-r.e. sets (we use definitions which are definable in ℒ over (α)). It is easy to see that simple sets exist for all admissible α. Chong and Lerman [1] have found some necessary and some sufficient conditions for the existence of hhsimple α-r.e. sets, although a complete determination of these conditions has not yet been made. Lerman and Simpson [9] have obtained some partial results concerning r-maximal α-r.e. sets. Lerman [6] has shown that maximal α-r.e. sets exist iff a is a certain sort of constructibly countable ordinal. Lerman [5] has also investigated the congruence relations, filters, and ideals of (α). Here various sorts of simple sets have also proved to be vital tools. The importance of simple α-r.e. sets to the study of the lattice structure of (α) is hence obvious.

Lerman [6, Q22] has posed the following problem: Find an admissible α for which all simple α-r.e. sets have the same 1-type with respect to the language ℒ. The structure of (α) for such an α would be much less complicated than that of (ω). Lerman [7] showed that such an α could not be a regular cardinal of L. We show that there is no such admissible α.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 3 , September 1976 , pp. 681 - 694
Copyright
Copyright © Association for Symbolic Logic 1976

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Footnotes

1

Research partially supported by NSF grants GP–43706X and MPS 74–06378 respectively.

References

REFERENCES

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