alternative definition of cardinality
The concept of cardinality comes from the notion of equinumerosity of sets. To define the cardinality of a set , one desirable property is that is equinumerous to precisely when . The first attempt, due to Frege and Russel, is to define a relation on the class of sets so that iff there is a bijection from to . This relation is an equivalence relation on . Then we can define as the equivalence class containing the set . However, is not a set, so we can’t do much with in ZF.
The second attempt, due to Von Neumann, defines to be the smallest ordinal equinumerous to . Now, exists if is well-orderable. But in general, we do not know if is well-orderable unless the well-ordering principle is applied, which is just another form of the axiom of choice. Thus, this definition depends on AC, and, in everyday mathematical usage (which assumes ZFC), suffices.
The third way, due to Scott, of looking at , without AC, is to modify the first attempt somewhat, so that is a set. Recall that the rank of a set is the least ordinal such that in the cumulative hierarchy. A set having a rank is said to be grounded. By the axiom of foundation, every set is grounded. For any set , let . Then , as a class of ordinals, has a least element . So . Next, we define (borrowing the terminology used in the first reference below)
and set . Since every element in is a subset of , , so that is a set. This method is known as Scott’s trick. It can also be used in defining other isomorphism types on sets. It is easy to see that iff . However, with this definition, in general, where is a natural number.
Nevertheless, it is known that every finite set is well-orderable, and so we come to the fourth definition of the cardinality of a set: given a set :
The one big advantage of this definition is clear: it does not require AC, and with AC, it is identical to the second definition above. At the same time, it also resolves the conflict with our intuitive notion about cardinality: the cardinality of a finite set is the number of elements in the set. However, the one big disadvantage in this definition is that we do not have in general (of course, is infinite). There is no way, without AC, to find a definition of , such that iff , and at the same time.
- 1 H. Enderton, Elements of Set Theory, Academic Press, Orlando, FL (1977).
- 2 T. J. Jech, Set Theory, 3rd Ed., Springer, New York, (2002).
- 3 A. Levy, Basic Set Theory, Dover Publications Inc., (2002).
|Title||alternative definition of cardinality|
|Date of creation||2013-03-22 18:50:11|
|Last modified on||2013-03-22 18:50:11|
|Last modified by||CWoo (3771)|