Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-22T08:43:30.635Z Has data issue: false hasContentIssue false
  • English
  • Français

Relations Between Some Cardinals in the Absence of the Axiom of Choice

Published online by Cambridge University Press:  15 January 2014

Lorenz Halbeisen  and
Saharon Shelah
Lorenz Halbeisen
Affiliation:
Department of Pure Mathematics, Queen's University Belfast, Belfast BT7 1NN, Northern Ireland Department of Mathematics, University of Californiaat Berkeley, Berkeley, California 94720, USAE-mail:[email protected]
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University Jerusalem, Jerusalem 91904, IsraelE-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

If we assume the axiom of choice, then every two cardinal numbers are comparable. In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using the axiom of choice.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 7 , Issue 2 , June 2001 , pp. 237 - 261
Copyright
Copyright © Association for Symbolic Logic 2001

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.)

References

REFERENCES

[1] Bachmann, Heinz, Transfinite Zahlen, Springer-Verlag, Berlin, Heidelberg, New York, 1967.Google Scholar
[2] Cantor, Georg, Über eine elementare Frage der Mannigfaltigkeitslehre, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 1 (1891), pp. 75–78.Google Scholar
[3] Cohen, Paul J., Set theory and the continuum hypothesis, Benjamin, New York, 1966.Google Scholar
[4] Feferman, Solomon and Lévy, Azriel, Independence results in set theory by Cohen's method II, Notices of the American Mathematical Society, vol. 10 (1963), p. 593.Google Scholar
[5] Gödel, Kurt, The consistency of the axiom of choice and of the generalized continuum-hypothesis, Proceedings of the National Academy of Sciences (U.S.A.), vol. 24 (1938), pp. 556–557.Google Scholar
[6] Goldstern, Martin, Strongly amorphous sets and dual Dedekind infinity, Mathematical Logic Quarterly, vol. 43 (1997), pp. 39–44.Google Scholar
[7] Halbeisen, Lorenz and Shelah, Saharon, Consequences of arithmetic for set theory, The Journal of Symbolic Logic, vol. 59 (1994), pp. 30–40.CrossRefGoogle Scholar
[8] Hartogs, Friedrich, Über das Problem der Wohlordnung, Mathematische Annalen, vol. 76(1915), pp. 438–443.CrossRefGoogle Scholar
[9] Hausdorff, Felix, Bemerkung über den Inhalt von Punktmengen, Math. Ann., vol. 75 (1914), pp. 428–433.CrossRefGoogle Scholar
[10] Hausdorff, Felix, Grundzüge der Mengenlehre, de Gruyter, Leipzig, 1914 (reprint: Chelsea, New York, 1965).Google Scholar
[11] Hodges, Wilfried, Model theory, Cambridge University Press, Cambridge, 1993.Google Scholar
[12] Howard, Paul and Rubin, Jean E., Consequences of the axiom of choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, 1998.CrossRefGoogle Scholar
[13] Jech, Thomas, The axiom of choice, North-Holland, Amsterdam, 1973.Google Scholar
[14] Jech, Thomas, Set theory, Pure and Applied Mathematics, Academic Press, London, 1978.Google Scholar
[15] Jech, Thomas and Sochor, Antonín, Applications of the θ-model, Bulletin de l'Académie Polonaise des Sciences, Série des Sciences Mathematiques, Astronomiques et Physiques, vol. 14 (1966), pp. 351–355.Google Scholar
[16] Kunen, Kenneth, Set theory, an introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, North Holland, Amsterdam, 1983.Google Scholar
[17] Läuchli, Hans, Ein Beitrag zur Kardinalzahlarithmetik ohne Auswahlaxiom, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 7 (1961), pp. 141–145.CrossRefGoogle Scholar
[18] Läuchli, Hans, Auswahlaxiom in der Algebra, Commentarii Mathematici Helvetia, vol. 37 (1962), pp. 1–18.Google Scholar
[19] Lebesgue, Henri, Sur les fonctions représentables analytiquement, Journal de Mathématiques Pures et Appliquées (6ème série), vol. 1 (1905), pp. 139–216.Google Scholar
[20] Lindenbaum, Adolf and Tarski, Alfred, Communication sur les recherches de la théorie des ensembles, Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 19 (1926), pp. 299–330.Google Scholar
[21] Moore, Gregory H., Zermelo's axiom of choice, Springer-Verlag, Berlin - New York, 1982.CrossRefGoogle Scholar
[22] Ramsey, Frank P., On a problem of formal logic, Proceedings of the London Mathematical Society, Ser. II, vol. 30 (1929), pp. 264–286.Google Scholar
[23] Sierpiński, Wacław, Cardinal and ordinal numbers, Państwowe Wydawnictwo Naukowe, Warszawa, 1958.Google Scholar
[24] Specker, Ernst, Verallgemeinerte Kontinuumshypothese und Auswahlaxiom, Archiv der Mathematik, vol. 5 (1954), pp. 332–337.CrossRefGoogle Scholar
[25] Specker, Ernst, Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom), Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 3 (1957), pp. 173–210.Google Scholar
[26] Spišiak, Ladislav and Vojtáš, Peter, Dependences between definitions of finiteness, Czechoslovak Mathematical Journal, vol. 38 (113) (1988), pp. 389–397.Google Scholar
[27] Tarski, Alfred, Sur quelques théorèmes qui équivalent à l'axiome du choix, Fundamenta Mathematicae, vol. 5 (1924), pp. 147–154.Google Scholar