[FOM] cardinality beyond regularity and Choice

Irving ianellis at iupui.edu
Mon Dec 14 13:21:30 EST 2009

Zuhair Abdul Ghafoor Al-Johar writes that:

"(1) Frege-Russell's Cardinals:

Cardinality(A) is the class of all sets equinumerous to A.

      Those are incompatible with Z, since
      they entail the existence of the set of all sets in Z, or in
      NBG\MK they would be proper classes. However in
      NF and related systems they are as general as the primitive
      concept of Cardinality, but the problem with these theories is
      that they are very complex, and difficult to understand, using
      concepts of stratification of formulas which is not desirable,
      even the finite axiomatization of NFU , though its axioms
      do not use stratification, but yet most of its theorems
      relies on it.

      Those Cardinals were the first defined cardinals
      in history of human kind."

Speaking strictly historical from the standpoint of the history of set 
theory, it was of course Georg Cantor who first introduced cardinal 
numbers and defined the concept of cardinality (as the "Machtigkeit" of 
a set). Cantor's original treatment of the concept of cardinality and 
his definition of cardinal numbers is to be found in his "Ein Beitrag 
zur Mannigfaltigkeitslehre", Journal für die reine und angewandte 
Mathematik 84 (1878), 242-258. A much more mature version is in his 
"Beiträge zur Begründung der transfiniten Mengenlehre", Mathematische 
Annalen XLVI (1895), 481-512.

Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
URL: http://www.irvinganellis.info

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