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BASIS THEOREMS FOR ${\rm{\Sigma }}_2^1$ -SETS

  • CHI TAT CHONG (a1), LIUZHEN WU (a2) and LIANG YU (a3)

Abstract

We prove the following two basis theorems for ${\rm{\Sigma }}_2^1$ -sets of reals:

  1. (1)Every nonthin ${\rm{\Sigma }}_2^1$ -set has a perfect ${\rm{\Delta }}_2^1$ -subset if and only if it has a nonthin ${\rm{\Delta }}_2^1$ -subset, and this is equivalent to the statement that there is a nonconstructible real.
  2. (2)Every uncountable ${\rm{\Sigma }}_2^1$ -set has an uncountable ${\rm{\Delta }}_2^1$ -subset if and only if either every real is constructible or $\omega _1^L$ is countable.

We also apply the method that proves (2) to show that if there is a nonconstructible real, then there is a perfect ${\rm{\Pi }}_2^1$ -set with no nonempty ${\rm{\Pi }}_2^1$ -thin subset, strengthening a result of Harrington [4].

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[2]Chong, C. T. and Yu, L., Recursion Theory, Computational Aspects of Definability, De Gruyter Series in Logic and its Applications, vol. 8, De Gruyter, Berlin, 2015.10.1515/9783110275643
[3]Groszek, M. J. and Slaman, T. A., A basis theorem for perfect sets. Bulletin of Symbolic Logic , vol. 4 (1998), no. 2, pp. 204209.10.2307/421023
[4]Harrington, L., ${\rm{\Pi }}_2^1$ sets and ${\rm{\Pi }}_2^1$ singletons. Proceedings of the American Mathematical Society, vol. 52 (1975), pp. 356360.
[5]Jech, T., Set Theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
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[7]Mansfield, R., Perfect subsets of definable sets of real numbers. Pacific Journal of Mathematics, vol. 35 (1970), pp. 451457.10.2140/pjm.1970.35.451
[8]Martin, D. A., The axiom of determinateness and reduction principles in the analytical hierarchy. Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687689.10.1090/S0002-9904-1968-11995-0
[9]Martin, D. A., Proof of a conjecture of Friedman. Proceedings of the American Mathematical Society, vol. 55 (1976), no. 1, p. 129.10.1090/S0002-9939-1976-0406785-9
[10]Sacks, G. E., Higher Recursion Theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990.10.1007/978-3-662-12013-2
[11]Simpson, S. G., Minimal covers and hyperdegrees. Transactions of the American Mathematical Society, vol. 209 (1975), pp. 4564.10.1090/S0002-9947-1975-0392534-3
[12]Solovay, R. M., On the cardinality of ${\rm{\Delta }}_2^1$ sets of reals, Foundations of Mathematics (Symposium Commemorating Kurt Gödel, Columbus, Ohio, 1966) (Bulloff, J. J., Holyoke, T. C., and Hahn, S. W., editors), Springer, New York, 1969, pp. 5873.

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