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On a Π0 1 Set of Positive Measure

  • Hisao Tanaka (a1)

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Some basis results for arithmetic, hyperarithmetic (HA) or sets which have positive measure (or which are not meager, i.e., of the second Baire category) have been obtained by several authors. For example, every non-meager set must have a recursive element (Shoenfield-Hinman, Hinman [2]) but there exists a non-meager set (as well as of measure 1) that contains no recursive element (Shoenfield [7]), and every set (i.e., arithmetic set) of positive measure contains an arithmetic element (Sacks [5], and Tanaka [12]).

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References

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[1] Addison, J.W., Some consequences of the axiom of constructibility, Fund. Math., 46 (1959), 337357.
[2] Hinman, P.G., Some applications of forcing to hierarchy problems in arithmetic, to appear.
[3] Kreisel, G., The axiom of choice and the class of hyperarithmetic functions, Indag. Math., 24 (1962), 308319.
[4] Mathias, A.R.D., A survey of recent results in set theory, Mimeographed note, Stanford University, (1968).
[5] Sacks, G.E., Measure-theoretic uniformity in recursion theory and set theory, to appear. Summary of results in Bull. Amer. Math. Soc., 73 (1967), 169174.
[6] Sampei, Y., A proof of Mansfield’s Theorem by forcing method, to appear.
[7] Shoenfield, J.R., The class of recursive functions, Proc. Amer. Math. Soc., 9 (1958), 690692.
[8] Shoenfield, J.R., Mathematical Logic, Addison-Wesley Company (1967).
[9] Spector, C., Recursive well-orderings, J. Symbolic Logic, 20 (1955), 151163.
[10] Spector, C., Hyperarithmetical quantifiers, Fund. Math., 48 (1960), 313320.
[11] Solovay, R., On the cardinality of sets of reals, to appear.
[12] Tanaka, H., Some results in the effective descriptive set theory, Publ. RIMS, Kyoto Univ., Ser. A, 3 (1967), 1152.
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On a Π0 1 Set of Positive Measure

  • Hisao Tanaka (a1)

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