Maximal sets in α-recursion theory |
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Authors: | M. Lerman S. G. Simpson |
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Affiliation: | (1) Department of Mathematics, Yale University, New Haven, USA |
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Abstract: | Let α be an admissible ordinal, and leta * be the Σ1-projectum ofa. Call an α-r.e. setM maximal if α→M is unbounded and for every α→r.e. setA, eitherA∩(α-M) or (α-A)∩(α-M) is bounded. Call and α-r.e. setM amaximal subset of α* if α*−M is undounded and for any α-r.e. setA, eitherA∩(α*-M) or (⇌*-A)∩(α*-M) is unbounded in α*. Sufficient conditions are given both for the existence of maximal sets, and for the existence of maximal subset of α*. Necessary conditions for the existence of maximal sets are also given. In particular, if α ≧ ℵ L then it is shown that maximal sets do not exist. Research partially supported by NSF Grant GP-34088 X. Some of the results in this paper have been taken from the second author’s Ph. D. Thesis, written under the supervision of Gerald Sacks. |
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