The -singleton conjecture
Abstract: The real is a natural example of a nonconstructible definable real. Moreover has a definition that is absolute: for some formula is the unique real such that . Solovay conjectured that there is a real such that and also has such an absolute definition. We prove his conjecture by constructing a -singleton , . A variant of our construction produces a countable nonempty set of reals not containing a -singleton. The latter result answers a question of Kechris.
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- R. David, A very absolute -singleton, Ann. Pure and Appl. Logic 23 (1982), 101-120. MR 701122 (84m:03057)
- S. Friedman, An Immune partition of the ordinals, Recursion Theory Week, Lecture Notes in Math., vol. 1144, Springer-Verlag, New York, 1986, pp. 141-147. MR 820778 (87h:03072)
- -, Minimal coding, Ann. Pure and Appl. Logic 41 (1989), 233-297. MR 984629 (90i:03056)
- R. Jensen and R. Solovay, Some applications of almost disjoint sets, Mathematical Logic and the Foundations of Set Theory, North-Holland, 1968, pp. 84-104. MR 0289291 (44:6482)
- A. Kechris and W. H. Woodin, On thin sets, handwritten note, 1983.
- M. Stanley, An absolute -singleton (to appear).
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