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Forcing the failure of CH by adding a real

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Affiliation:
The Hebrew University, Jerusalem, Israel
Hugh Woodin
Affiliation:
California Institute of Technology, Pasadena, California 91125

Extract

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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1984

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References

[BJW]Beller, A., Jensen, R. B. and Welch, P., Coding the universe, London Mathematical Society Lecture Note Series, no. 47, Cambridge University Press, Cambridge, 1982.CrossRefGoogle Scholar
[JS]Jensen, R. B. and Solovay, R., Some applications of almost disjoint sets, Mathematical logic and the foundations of set theory (proceedings of an international colloquium, Jerusalem, 1968; Bar-Hillel, Y., editor), North-Holland, Amsterdam, 1970, pp. 84104.Google Scholar
[Sh]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin, 1982.CrossRefGoogle Scholar
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