Hostname: page-component-797576ffbb-vjhkx Total loading time: 0 Render date: 2023-12-05T12:37:53.975Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "useRatesEcommerce": true } hasContentIssue false

Consequences of arithmetic for set theory

Published online by Cambridge University Press:  12 March 2014

Lorenz Halbeisen
Department of Mathematics, Eldgen. Technische Hochschule, Zürich, Switzerland,
Saharon Shelah
Institute of Mathematics, Hebrew University Jerusalem, Jerusalem., Israel,


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.

Research Article
Copyright © Association for Symbolic Logic 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



[Ba] Bachmann, H., Transfinite Zahlen, Springer-Verlag, Berlin, 1967.Google 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.Google Scholar
[Sp2] Specker, E., Zur Axiomatik der Mengenlehre, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 3 (1957), pp. 173–210.Google Scholar