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Models of set theory with more real numbers than ordinals

Published online by Cambridge University Press:  12 March 2014

Paul E. Cohen
Paul E. Cohen*
Affiliation:
Institute for Advanced Study, Princeton, New Jersey 08540
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Extract

Suppose M is a countable standard transitive model of set theory. P. J. Cohen [2] showed that if κ is an infinite cardinal of M then there is a one-to-one function Fκ from κ into the set of real numbers such that M[Fκ] is a model of set theory with the same cardinals as M.

If Tκ is the range of Fκ then Cohen also showed [2] that M[Tκ] fails to satisfy the axiom of choice. We will give an easy proof of this fact.

If κ, λ are infinite we will also show that M[Tκ] is elementarily equivalent to M[Tλ] and that (] in M[Fλ]) is elementarily equivalent to (] in M[FK]).

Finally we show that there may be an N ∈ M[GK] which is a standard model of set theory (without the axiom of choice) and which has, from the viewpoint of M[GK], more real numbers than ordinals.

We write ZFC and ZF for Zermelo-Fraenkel set theory, respectively with and without the axiom of choice (AC). GBC is Gödel-Bernays' set theory with AC. DC and ACℵo are respectively the axioms of dependent choice and of countable choice defined in [6].

Lower case Greek characters (other than ω) are used as variables over ordinals. When α is an ordinal, R(α) is the set of all sets with rank less than α.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 39 , Issue 3 , September 1974 , pp. 579 - 583
Copyright
Copyright © Association for Symbolic Logic 1974

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References

BIBLIOGRAPHY

[1]Cohen, P. E., Some applications of forcing in set theory, Doctoral Dissertation, University of Illinois, Champaign-Urbana, 1972.Google Scholar
[2]Cohen, P. J., Set theory and the continuum hypothesis, Benjamin, New York, 1966.Google Scholar
[3]Scott, D., Measurable cardinals and constructive sets, Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 7 (1961), pp. 145–149.Google Scholar
[4]Shoenfield, J., Unramified forcing, Proceedings of Symposia in Pure Mathematics, Vol. 13, Part 1 (University of California, Los Angeles, 1967), American Mathematical Society, Providence, R.I., 1971, pp. 357–381.Google Scholar
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[6]Takeuti, G., Hypotheses on power set, Proceedings of Symposia in Pure Mathematics, Vol. 13, Part 1 (University of California, Los Angeles, 1967), American Mathematical Society, Providence, R.I. 1971, pp. 439–446.Google Scholar
[7]Takeuti, G. and Zaring, W., Axiomatic set theory, Springer-Verlag, New York, 1973.CrossRefGoogle Scholar