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Consequences of arithmetic for set theory

Published online by Cambridge University Press:  12 March 2014

Lorenz Halbeisen  and
Saharon Shelah
Lorenz Halbeisen
Affiliation:
Department of Mathematics, Eldgen. Technische Hochschule, Zürich, Switzerland, E-mail:halbeisen@math.ethz.ch
Saharon Shelah
Affiliation:
Institute of Mathematics, Hebrew University Jerusalem, Jerusalem., Israel, E-mail:shelah@math.huji.ac.il
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Abstract

In this paper, we consider certain cardinals in ZF (set theory without AC, the axiom of choice). In ZFC (set theory with AC), given any cardinals and , either ≤ or ≤ . However, in ZF this is no longer so. For a given infinite set A consider seq1-1(A), the set of all sequences of A without repetition. We compare |seq1-1(A)|, the cardinality of this set, to ||, the cardinality of the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that ZF ⊢ ∀A(| seq1-1(A)| ≠ ||), and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then | fin(B)| < | (B*)| even though the existence for some infinite set B* of a function ƒ from fin(B*) onto (B*) is consistent with ZF.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 1 , March 1994 , pp. 30 - 40
Copyright
Copyright © Association for Symbolic Logic 1994

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References

REFERENCES

[Ba] Bachmann, H., Transfinite Zahlen, Springer-Verlag, Berlin, 1967.CrossRefGoogle Scholar
[Je1] Jech, Th., Set theory, Academic Press, New York, 1978.Google Scholar
[Je2] Jech, Th., The axiom of choice, North-Holland, Amsterdam, 1973.Google Scholar
[La] Läuchli, H., Auswahlaxiom in der Algebra, Commentarii Mathematici Hehetiei, vol. 37 (1962), pp. 1–18.Google Scholar
[Sl] Sloane, N. J. A., A handbook of integer sequence, Academic Press, New York, 1973.Google Scholar
[Sp1] Specker, E., Verallgemeinerte Kontinuumshypothese und Auswahlaxiom, Archiv der Mathematik, vol. 5 (1954), pp. 332–337.CrossRefGoogle Scholar
[Sp2] Specker, E., Zur Axiomatik der Mengenlehre, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 3 (1957), pp. 173–210.CrossRefGoogle Scholar