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On forcing without the continuum hypothesis1

Published online by Cambridge University Press:  12 March 2014

Uri Abraham
Uri Abraham*
Affiliation:
Hebrew University Jerusalem, Israel University of California, Berkeley, California 94720
*
Ben Gurion University of the Negev, Beer Sheva, Israel
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Extract

ℵ2. One of the first examples of the forcing method is cardinal collapsing (A. Levy, see [5]), for example, collapsing ℵ2 to ℵ1: the poset P is the collection of all countable functions from a countable ordinal into ℵ2. As is well known, in VP, , and because P is closed under union of countable chains, remains a cardinal and, in fact, no new countable sets are added. But to prove that remains a cardinal we need to conclude ∣P∣ ≤ ℵ2 and hence ℵ3 is not collapsed. If it has been observed that is actually collapsed in VP. Hence the following theorem, which makes no assumptions on the continuum, is relevant.

1. Theorem. There is a poset R such that in VR ℵ2 becomes of cardinality ℵ1, but ℵ1 and the cardinals above ℵ2 are not collapsed.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 3 , September 1983 , pp. 658 - 661
Copyright
Copyright © Association for Symbolic Logic 1983

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Footnotes

1

This work is part of the author's Ph.D. Dissertation at The Hebrew University, Jerusalem. He is grateful to his supervisors, Professors A. Levy and S. Shelah.

References

REFERENCES

[1]Abraham, U., On the intersection of closed unbounded sets (in preparation).Google Scholar
[2]Abraham, U. and Shelah, S., Isomorphism types of Aronszajn trees, Israel Journal of Mathematics (to appear).Google Scholar
[3]Abraham, U. and Shelah, S., Forcing closed unbounded sets, this Journal, vol. 48 (1983), pp. 643–657.Google Scholar
[4]Baumgartner, J.E. and Taylor, A.D., Saturation properties of ideals in generic extensions. I. Transactions of the American Mathematical Society, vol. 270 (1982), pp. 557–574.CrossRefGoogle Scholar
[5]Jech, T.J., Set theory, Academic Press, New York, 1978.Google Scholar
[6]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin and New York, 1982.Google Scholar