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- W2110165629 abstract "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 ." @default.
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- W2110165629 date "1976-09-01" @default.
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- W2110165629 title "Types of simple α-recursively enumerable sets" @default.
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- W2110165629 doi "https://doi.org/10.1017/s0022481200051240" @default.
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