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 ≧ ℵL then it is shown that maximal sets do not exist.
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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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Lerman, M., Simpson, S.G. Maximal sets in α-recursion theory. Israel J. Math. 14, 236–247 (1973). https://doi.org/10.1007/BF02764882
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DOI: https://doi.org/10.1007/BF02764882