Ordinal number

First, second and third are the ordinal numbers corresponding to one, two and three.

In the expression a₃, the "3" is an ordinal number.

1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, Dover (Dover Phoenix), 2006, page 137, For not only do the antinomies a) to e) disappear when we admit as elements of sets only such sets, ordinal numbers, and cardinal numbers as are bounded above by a fixed cardinal number, but we see also that paradoxes always arise if we collect into a set any sets, cardinal numbers, or ordinal numbers which are not bounded above by a fixed cardinal number.

1960 [D. Van Nostrand], Paul R. Halmos, Naive Set Theory, 2017, Dover, Republication, page 80, Is there a set that consists exactly of all the ordinal numbers? It is easy to see that the answer must be no. If there were such a set, then we could form the supremum of all ordinal numbers. That supremum would be an ordinal number greater than or equal to every ordinal number. Since, however, for each ordinal number there exists a strictly greater one (for example, its successor), this is impossible; it makes no sense to speak of the "set" of all ordinals.

If #92;alpha is an ordinal number, then by definition any two well-ordered sets of type #92;alpha are similar, i.e., there exists a one-to-one mapping from one set to the other. Consequently these sets have the same cardinality. Consequently to any ordinal number #92;alpha we may assign a cardinal number, the common cardinality of all well-ordered sets of type #92;alpha.