The Definability of Cardinal Numbers

  • Azriel Lévy


One says that the sets x and y are equinumerous (in symbols, x ≈ y) if there is a 1 – 1 function mapping x on y. The notion of the cardinal |x| of x is obtained from equinumerosity by abstraction. The use of |x| does usually not require any special apparatus. E.g., when we say |x| — ℵ0 we mean to say that there is a 1 – 1 function mapping x on the set of all natural numbers; when we say |x| < |y| we mean that there are 1 – 1 functions mapping x into y, but none of them is onto y; etc. As is common in mathematics, one tends to pass from the abstract notion of cardinal numbers to real cardinal numbers, i.e., one wants to regard the cardinal numbers as objects of the mathematical system. This is where one encounters the problem of how to define the cardinal | x | of x as an object of set theory.


Free Variable Cardinal Number Axiom Schema Operation Symbol Membership Relation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asser, G.: Theorie der logischen Auswahlfunktionen. Z. math. Logik Grundlagen Math. 3, 30–68 (1957).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Fraenkel, A. A., and H. Bar-Hillel: Foundation of set theory. Amsterdam: North Holland Publishing Co. 1958.Google Scholar
  3. 3.
    Levy, A.: The interdependence of some consequences of the axiom of choice. Fundamenta Mathematicae 54, 135–157 (1964).MathSciNetMATHGoogle Scholar
  4. 4.
    Mendelson, E.: The idenpendence of a weak axiom of choice. J. Symbolic Logic 21, 350–366 (1956).MathSciNetCrossRefGoogle Scholar
  5. 5.
    Mendelson, E.— A semantic proof of the eliminability of descriptions. Z. math. Logik Grundlagen Math. 6, 199–200 (1960).MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Mostowski, A.: Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip. Fundamenta Mathematicae 32, 201–252 (1939).Google Scholar
  7. 7.
    Scott, D.: Definitions by abstraction in axiomatic set theory (abstract). Bull. Amer. Math. Soc. 61, 442 (1955).CrossRefGoogle Scholar
  8. 8.
    Specker, E.: Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom). Z. math. Logik Grundlagen Math. 3, 173–210 (1957).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Sttppes, P.: Introduction to logic. Princeton: D. Van Nostrand Co. 1957.Google Scholar
  10. 10.
    Tarski, A.: General principles of induction and recursion in axiomatic set theory (abstract). Bull. Amer. Math. Soc. 61, 442–443 (1955).Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1969

Authors and Affiliations

  • Azriel Lévy

There are no affiliations available

Personalised recommendations