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    Eslami, Esfandiar and Golshani, Mohammad 2012. Shelah's strong covering property and CH in V[r]. Mathematical Logic Quarterly, Vol. 58, Issue. 3, p. 153.

    Velickovic, Boban and Woodin, W.Hugh 1998. Complexity of reals in inner models of set theory. Annals of Pure and Applied Logic, Vol. 92, Issue. 3, p. 283.

    Brendle, Jörg 1995. Combinatorial properties of classical forcing notions. Annals of Pure and Applied Logic, Vol. 73, Issue. 2, p. 143.

    Gitik, Moti 1995. Some results on the nonstationary ideal. Israel Journal of Mathematics, Vol. 92, Issue. 1-3, p. 61.

    Bukovský, Lev and Copláková-Hartová, Eva 1990. Minimal collapsing extensions of models of zfc. Annals of Pure and Applied Logic, Vol. 46, Issue. 3, p. 265.

    Gitik, Moti 1989. The negation of the singular cardinal hypothesis from o(K)=K++. Annals of Pure and Applied Logic, Vol. 43, Issue. 3, p. 209.

    Gitik, Moti 1988. On the Mitchell and Rudin-Kiesler orderings of ultrafilters. Annals of Pure and Applied Logic, Vol. 39, Issue. 2, p. 175.

    Shelah, Saharon 1987. Classification Theory.


Forcing the failure of CH by adding a real

  • Saharon Shelah (a1) and Hugh Woodin (a2)
  • DOI:
  • Published online: 01 March 2014

We prove several independence results relevant to an old question in the folklore of set theory. These results complement those in [Sh, Chapter XIII, §4]. The question is the following. Suppose V ⊨ “ZFC + CH” and r is a real not in V. Must V[r] ⊨ CH? To avoid trivialities assume = .

We answer this question negatively. Specifically we find pairs of models (W, V) such that W ⊨ ZFC + CH, V = W[r], r a real, = and V ⊨ ¬CH. Actually we find a spectrum of such pairs using ZFC up to “ZFC + there exist measurable cardinals”. Basically the nicer the pair is as a solution, the more we need to assume in order to construct it.

The relevant results in [Sh, Chapter XIII] state that if a pair (of inner models) (W, V) satisfies (1) and (2) then there is an inaccessible cardinal in L; if in addition V ⊨ 20 > ℵ2 then 0# exists; and finally if (W, V) satisfies (1), (2) and (3) with V ⊨ 20 > ℵω, then there is an inner model with a measurable cardinal.

Definition 1. For a pair (W, V) we shall consider the following conditions:

(1) V = W[r], r a real, = , W ⊨ ZFC + CH but CH fails in V.

(2) W ⊨ GCH.

(3) W and V have the same cardinals.

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[BJW]A. Beller , R. B. Jensen and P. Welch , Coding the universe, London Mathematical Society Lecture Note Series, no. 47, Cambridge University Press, Cambridge, 1982.

[Sh]S. Shelah , Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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