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A simplicity theorem for amoebas over random reals

Author: Fred G. Abramson
Journal: Proc. Amer. Math. Soc. 78 (1980), 409-413
MSC: Primary 03E40; Secondary 28A20
MathSciNet review: 553385
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Abstract: Let M be a countable standard transitive model of ZFC, $ \mathcal{A}$ be an amoeba over M, and r be a random real over M.

Theorem. (a) There is no infinite set of reals X contained in the complement of $ \mathcal{A}$ with $ X \in M[r];(b)\;If\;\{ {x_n}\vert n < \omega \} \in M[r]$ is a sequence of distinct reals, then for all large enough n, $ \{ {x_i}\vert{2^n} \leqslant i < {2^{n + 1}}\} \cap \mathcal{A} \ne \emptyset $.

References [Enhancements On Off] (What's this?)

  • [D] A. Martin and R. M. Solovay [1970], Internal Cohen extensions, Ann. Math. Logic 2, 143-178. MR 0270904 (42:5787)
  • [R] M. Solovay [1970], A model of set theory in which every set of reals is Lebesgue measurable, Ann. of Math. (2) 92, 1-56. MR 0265151 (42:64)

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Keywords: Amoeba forcing, Solovay forcing, random reals
Article copyright: © Copyright 1980 American Mathematical Society

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