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Cardinal Relations in ZF Only

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Combinatorial Set Theory

Part of the book series: Springer Monographs in Mathematics ((SMM))

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Abstract

In this chapter we shall compare different cardinal numbers in Zermelo–Fraenkel Set Theory, which is Set Theory without the Axiom of Choice. For example, it will be shown that for any infinite set A, the cardinality of the set of finite subsets of A is always strictly smaller than the cardinality of the power set of A.

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References

  1. Heinz Bachmann: Transfinite Zahlen. Springer, Berlin (1967)

    Book  MATH  Google Scholar 

  2. Felix Bernstein: Untersuchungen aus der Mengelehre. Dissertation, University of Göttingen (Germany) (1901)

    Google Scholar 

  3. Felix Bernstein: Untersuchungen aus der Mengelehre. Math. Ann. 61, 117–155 (1905)

    Article  MathSciNet  MATH  Google Scholar 

  4. Georg Cantor: Beiträge zur Begründung der transfiniten Mengenlehre. I./II. Math. Ann. 46/49, 481–512 (1895/1897), 207–246 (see [5] for a translation into English)

    Google Scholar 

  5. Georg Cantor: Contributions to the Founding of the Theory of Transfinite Numbers (translation into English of [4]) [translated, and provided with an introduction and notes, by Philip E.B. Jourdain]. Open Court Publishing Company, Chicago (1915) [reprint: Dover, New York (1952)]

    Google Scholar 

  6. Georg Cantor: Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, Mit Erläuternden Anmerkungen sowie mit Ergänzungen aus dem Briefwechsel Cantor–Dedekind, edited by E. Zermelo. Julius Springer, Berlin (1932)

    Google Scholar 

  7. Alonzo Church: Alternatives to Zermelo’s assumption. Trans. Am. Math. Soc. 29, 178–208 (1927)

    MathSciNet  MATH  Google Scholar 

  8. Richard Dedekind: Was sind und was sollen die Zahlen. Vieweg, Braunschweig (1888) (see also [9, pp. 335–390])

    Google Scholar 

  9. Richard Dedekind: Gesammelte mathematische Werke III, edited by R. Fricke, E. Noether, Ö. Ore. Vieweg, Braunschweig (1932)

    Google Scholar 

  10. Georg Faber: Über die Abzählbarkeit der rationalen Zahlen. Math. Ann. 60, 196–203 (1905)

    Article  MathSciNet  MATH  Google Scholar 

  11. Thomas E. Forster: Finite-to-one maps. J. Symb. Log. 68, 1251–1253 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Abraham A. Fraenkel: Abstract Set Theory. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1961)

    MATH  Google Scholar 

  13. Reuben L. Goodstein: On the restricted ordinal theorem. J. Symb. Log. 9, 33–41 (1944)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lorenz Halbeisen: Vergleiche zwischen unendlichen Kardinalzahlen in einer mengenlehre ohne Auswahlaxiom. Diplomarbeit, University of Zürich (Switzerland) (1990)

    Google Scholar 

  15. Lorenz Halbeisen: A number-theoretic conjecture and its implication for set theory. Acta Math. Univ. Comen. 74, 243–254 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Lorenz Halbeisen, Norbert Hungerbühler: Number theoretic aspects of a combinatorial function. Notes Number Theory Discrete Math. 5, 138–150 (1999)

    MathSciNet  Google Scholar 

  17. Lorenz Halbeisen, Saharon Shelah: Consequences of arithmetic for set theory. J. Symb. Log. 59, 30–40 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lorenz Halbeisen, Saharon Shelah: Relations between some cardinals in the absence of the axiom of choice. Bull. Symb. Log. 7, 237–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lauri Kirby, Jeff B. Paris: Accessible independence results for Peano arithmetic. Bull. Lond. Math. Soc. 14, 285–293 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  20. Djuro (George) Kurepa: On a characteristic property of finite sets. Pac. J. Math. 2, 323–326 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hans Läuchli: Ein Beitrag zur Kardinalzahlarithmetik ohne Auswahlaxiom. Z. Math. Log. Grundl. Math. 7, 141–145 (1961)

    Article  MATH  Google Scholar 

  22. Henri Lebesgue: Sur les fonctions représentables analytiquement. J. Math. Pures Appl. (6ème sér.) 1, 139–216 (1905)

    Google Scholar 

  23. Azriel Lévy: The independence of various definitions of finiteness. Fundam. Math. 46, 1–13 (1958)

    MATH  Google Scholar 

  24. Adolf Lindenbaum, Alfred Tarski: Communication sur les recherches de la théorie des ensembles. C. R. Séances Soc. Sci. et des Lettres de Varsovie, Classe III 19, 299–330 (1926)

    MATH  Google Scholar 

  25. John von Neumann: Die Axiomatisierung der Mengenlehre. Math. Z. 27, 669–752 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jeff B. Paris: Combinatorial statements independent of arithmetic. In: Mathematics of Ramsey Theory, J. Nešetřil, V. Rödl (eds.), pp. 232–245. Springer, Berlin (1990)

    Chapter  Google Scholar 

  27. Ernst Schröder: Über zwei Definitionen der Endlichkeit und G. Cantor’sche Sätze. Nova Acta, Abh. Kais. Leopoldinisch - Carolinisch Deutsch. Akad. Naturforscher 71, 301–362 (1898)

    Google Scholar 

  28. Wacław Sierpiński: Sur l’égalité \(2\mathfrak {m}= 2\mathfrak {n}\) pour les nombres cardinaux. Fundam. Math. 3, 1–6 (1922)

    MATH  Google Scholar 

  29. Wacław Sierpiński: Sur une décomposition effective d’ensembles. Fundam. Math. 29, 1–4 (1937)

    Google Scholar 

  30. Wacław Sierpiński: Sur l’implication \((2\mathfrak {m}\le 2\mathfrak {n}) \rightarrow (\mathfrak {m}\le \mathfrak {n})\) pour les nombres cardinaux. Fundam. Math. 34, 148–154 (1946)

    Google Scholar 

  31. Wacław Sierpiński: Sur la division des types ordinaux. Fundam. Math. 35, 1–12 (1948)

    MATH  Google Scholar 

  32. Wacław Sierpiński: Sur les types d’ordre des ensembles linéaires. Fundam. Math. 37, 253–264 (1950)

    MATH  Google Scholar 

  33. Wacław Sierpiński: Sur un type ordinal dénombrable qui a une infinite indénombrable de divisenrs gauches. Fundam. Math. 37, 206–208 (1950)

    MATH  Google Scholar 

  34. Wacław Sierpiński: Cardinal and Ordinal Numbers. Państwowe Wydawnictwo Naukowe, Warszawa (1958)

    MATH  Google Scholar 

  35. Ernst Specker: Verallgemeinerte Kontinuumshypothese und Auswahlaxiom. Arch. Math. 5, 332–337 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ernst Specker: Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom). Z. Math. Log. Grundl. Math. 3, 173–210 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ladislav Spišiak, Peter Vojtáš: Dependences between definitions of finiteness. Czechoslov. Math. J. 38(113), 389–397 (1988)

    Google Scholar 

  38. Alfred Tarski: Sur les ensembles finis. Fundam. Math. 6, 45–95 (1924)

    Google Scholar 

  39. Alfred Tarski: Cancellation laws in the arithmetic of cardinals. Fundam. Math. 36, 77–92 (1949)

    MathSciNet  MATH  Google Scholar 

  40. John K. Truss: Dualisation of a result of Specker’s. J. Lond. Math. Soc. (2) 6, 286–288 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  41. John K. Truss: Classes of Dedekind finite cardinals. Fundam. Math. 84, 187–208 (1974)

    MathSciNet  MATH  Google Scholar 

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  1. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Lorenz J. Halbeisen

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  1. Lorenz J. Halbeisen
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Correspondence to Lorenz J. Halbeisen .

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Halbeisen, L.J. (2012). Cardinal Relations in ZF Only. In: Combinatorial Set Theory. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-1-4471-2173-2_4

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