Bilangan ordinal: Perbedaan antara revisi

When <math>\langle \alpha_{\iota} | \iota < \gamma \rangle</math> is an ordinal-indexed sequence, indexed by a limit γ and the sequence is ''increasing'', i.e. <math>\alpha_{\iota} < \alpha_{\rho}\!</math> whenever <math>\iota < \rho,\!</math> we define its ''limit'' to be the least upper bound of the set <math>\{ \alpha_{\iota} | \iota < \gamma \},\!</math> that is, the smallest ordinal (it always exists) greater than any term of the sequence. In this sense, a limit ordinal is the limit of all smaller ordinals (indexed by itself). Put more directly, it is the supremum of the set of smaller ordinals.

A set is [[downward closed]] if anything less than an element of the set is also in the set. If a set of ordinals is downward closed, then that set is an ordinal—the least ordinal not in the set. <!-- Suppose a property is such that if is true for an ordinal it is true for all smaller ones. This means that it holds for a [[downward closed]] set of ordinals, which is itself an ordinal. Thus the smallest ordinal that does not have the property is the set of ordinals that has the property. -->