Cardinality - Finite, Countable and Uncountable Sets

Finite, Countable and Uncountable Sets

If the axiom of choice holds, the law of trichotomy holds for cardinality. Thus we can make the following definitions:

  • Any set X with cardinality less than that of the natural numbers, or | X | < | N |, is said to be a finite set.
  • Any set X that has the same cardinality as the set of the natural numbers, or | X | = | N | = ℵ0, is said to be a countably infinite set.
  • Any set X with cardinality greater than that of the natural numbers, or | X | > | N |, for example | R | = c > | N |, is said to be uncountable.

Read more about this topic:  Cardinality

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