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Definability and initial segments of c-degrees

  • Robert S. Lubarsky (a1)

Abstract

We combine two techniques of set theory relating to mininal degrees of constructibility. Jensen constructed a minimal real which is additionally a singleton. Groszek built an initial segment of order type 1 + α*, for any ordinal α. This paper shows how to force a singleton such that the c-degrees beneath it, all represented by reals, are of type 1 + α*, for many ordinals α. We also examine the definability α needs to be so represented by a real.

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[A]Adamowicz, Zofia, Constructible semi-lattices of degrees of constructibility, Set theory and hierarchy theory. V (Lachlan, A.et al., editors), Lecture Notes in Mathematics, vol. 619, Springer-Verlag, Berlin, 1977, pp. 143.
[G]Groszek, Marcia, in preparation.
[J]Jensen, Ronald, Definable sets of minimal degree, Mathematical logic and foundations of set theory (Bar-Hillel, Y., editor), North-Holland, Amsterdam, 1970, pp. 122128.
[Le]Lerman, Manuel, Degrees of unsolvability. Springer-Verlag, Berlin, 1983.
[Lu]Lubarsky, Robert, Lattices of c-degrees, Annals of Pure and Applied Logic, vol. 36 (1987), pp. 115118.
[Sa]Sacks, Gerald E., Countable admissible ordinals and hyperdegrees, Advances in Mathematics, vol. 20 (1976), pp. 213262.
[Sa1]Sacks, Gerald E., Forcing with perfect closed sets, Axiomatic set theory (Scott, D. S., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 331355.

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