ordinal number

Noun

ordinal number (plural ordinal numbers)

1. A word that expresses the relative position of an item in a sequence.
   • First, second and third are the ordinal numbers corresponding to one, two and three.
2. A natural number used to denote position in a sequence.
   • In the expression a₃, the "3" is an ordinal number.

   Synonyms:

   ordinal, ordinal numeral

3. Such a number generalised to correspond to any cardinal number (the size of some set); formally, the order type of some well-ordered set of some cardinality a, which represents an equivalence class of well-ordered sets (exactly those of cardinality a) under the equivalence relation "existence of an order-preserving bijection".
   • 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...1950, Frederick Bagemihl (translator), Erich Kamke, Theory of Sets, Dover (Dover Phoenix), 2006, page 137
   • 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...