Ordinal Number - Downward Closed Sets of Ordinals

Downward Closed Sets of 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.

Examples:

  • The set of ordinals less than 3 is 3 = { 0, 1, 2 }, the smallest ordinal not less than 3.
  • The set of finite ordinals is infinite, the smallest infinite ordinal: ω.
  • The set of countable ordinals is uncountable, the smallest uncountable ordinal: ω1.

Read more about this topic:  Ordinal Number

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