Finite ordinal number

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Ordinal numbers are numbers which designate a place in a sequence: first, second, third. They can be distinguished from the cardinal numbers which are the number of items in a set: one, two, three.

When dealing with finite numbers the two concepts agree, for every ordinal number there is a corresponding cardinal number, their use will depend on the context. These finite numbers are the natural numbers.

These two concepts have been extended to cover infinite sets, sometimes called transfinite sets. Here the correspondence between the two concepts breaks down. Two sets will have the same cardinality if there is a one two one correspondence between the sets. It can be shown that the natural numbers and rational numbers have the same cardinality called , aleph-null, the smallest infinite cardinal. The real numbers have a different cardinality. The use of the two terms in the set theoretic case are covered in ordinal number and cardinal number.