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Least upper bounds for minimal pairs of α-R.E. α-degrees

  • Manuel Lerman (a1)

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The application of priority arguments to study the structure of the upper semilattice of α-r.e. α-degrees for all admissible ordinals α was first done successfully by Sacks and Simpson [5] who proved that there exist incomparable α-r.e. α-degrees. Lerman and Sacks [3] studied the existence of minimal pairs of α-r.e. α-degrees, and proved their existence for all admissible ordinals α which are not refractory. We continue the study of the α-r.e. α-degrees, and prove that no minimal pair of α-r.e. α-degrees can have as least upper bound the complete α-r.e. α-degree.

The above-mentioned theorem was first proven for α = ω by Lachlan [1]. Our proof for α = ω differs from Lachlan's in that we eliminate the use of the recursion theorem. The proofs are similar, however, and a knowledge of Lachlan's proof will be of considerable aid in reading this paper.

We assume that the reader is familiar with the basic notions or α-recursion theory, which can be found in [2] or [5].

Throughout the paper a will be an arbitrary admissible ordinal. We identify a set A ⊆ α with its characteristic function, A(x) = 1 if xA, and A(x) = 0 if xA.

If A ⊆ α and B ⊆ α, then AB will denote the set defined by

AB(x) = A(y) if x = λ + 2z, λ is a limit ordinal, z < ω and y = λ + z,

= B(y) if x = λ + 2z + 1, λ is a limit ordinal, z < ω, and y = λ + z.

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[1]Lachlan, A. H., Lower bounds for pairs of r.e. degrees, Proceedings of the London Mathematical Society, vol. 16 (1966), pp. 537569.
[2]Lerman, M., On suborderings of the α-recursively enumerable α-degrees, Annals of Mathematical Logic, vol. 4 (1972), pp. 369392.
[3]Lerman, M. and Sacks, G. E., Some minimal pairs of α-recursively enumerable degrees, Annals of Mathematical Logic, vol. 4 (1972), pp. 415442.
[4]Sacks, G. E., Post's problem, admissible ordinals, and regularity, Transactions of the American Mathematical Society, vol. 124 (1966), pp. 123.
[5]Sacks, G. E. and Simpson, S., The α-finite injury priority method, Annals of Mathematical Logic, vol. 4 (1972), pp. 343367.

Least upper bounds for minimal pairs of α-R.E. α-degrees

  • Manuel Lerman (a1)

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