Scala

Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if and only if there is a bijection between them. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.

(1 ≤ ν and κ ≤ μ) → (ν^κ ≤ ν[sup]μ) and

In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. counting).

There are many technical advantages to this restriction, and little generality is lost, because essentially all mathematical concepts can be modeled by pure sets. Sets in the von Neumann universe are organized into a cumulative hierarchy, based on how deeply their members, members of members, etc. are nested.

Outside set theory, the word "class" is sometimes used synonymously with "set". This usage dates from a historical period where classes and sets were not distinguished as they are in modern set-theoretic terminology.

First we might try to proceed as if X were finite. If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X. Next we might try specifying the least element from each set.