Borel OD sets of reals are OD-Borel in some simple models

Kanovei, Vladimir; Lyubetsky, Vassily

doi:10.1090/proc/14286

Borel OD sets of reals are OD-Borel in some simple models
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by Vladimir Kanovei and Vassily Lyubetsky PDF

Proc. Amer. Math. Soc. 147 (2019), 1277-1282 Request permission

Abstract:

It is true in the Cohen, Solovay-random, and Sacks generic extensions that every ordinal-definable Borel set of reals has a Borel code in the ground model, and hence if non-empty, then has an element in the ground model.

References

Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
Ronald Jensen, Definable sets of minimal degree, Mathematical logic and foundations of set theory (Proc. Internat. Colloq., Jerusalem, 1968) North-Holland, Amsterdam, 1970, pp. 122–128. MR 0306002
Vladimir Kanovei and Vassily Lyubetsky, A definable $\mathsf {E}_0$ class containing no definable elements, Arch. Math. Logic 54 (2015), no. 5-6, 711–723. MR 3372617, DOI 10.1007/s00153-015-0436-9
V. G. Kanoveĭ and V. A. Lyubetskiĭ, A definable countable set containing no definable elements, Mat. Zametki 102 (2017), no. 3, 369–382 (Russian, with Russian summary); English transl., Math. Notes 102 (2017), no. 3-4, 338–349. MR 3691702, DOI 10.4213/mzm10842
Vladimir Kanovei and Vassily Lyubetsky, Countable OD sets of reals belong to the ground model, Arch. Math. Logic 57 (2018), no. 3-4, 285–298. MR 3778960, DOI 10.1007/s00153-017-0569-0
Vladimir Kanovei, Borel equivalence relations, University Lecture Series, vol. 44, American Mathematical Society, Providence, RI, 2008. Structure and classification. MR 2441635, DOI 10.1090/ulect/044
Alexander S. Kechris, Measure and category in effective descriptive set theory, Ann. Math. Logic 5 (1972/73), 337–384. MR 369072, DOI 10.1016/0003-4843(73)90012-0
Kenneth Kunen, Set theory, Studies in Logic (London), vol. 34, College Publications, London, 2011. MR 2905394
Alain Louveau, A separation theorem for $\Sigma ^{1}_{1}$ sets, Trans. Amer. Math. Soc. 260 (1980), no. 2, 363–378. MR 574785, DOI 10.1090/S0002-9947-1980-0574785-X
Yiannis N. Moschovakis, Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam-New York, 1980. MR 561709
Jack H. Silver, Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Ann. Math. Logic 18 (1980), no. 1, 1–28. MR 568914, DOI 10.1016/0003-4843(80)90002-9
Jacques Stern, On Lusin’s restricted continuum problem, Ann. of Math. (2) 120 (1984), no. 1, 7–37. MR 750715, DOI 10.2307/2007070