Order Type of Well-orderings
Every well-ordered set is order-equivalent to exactly one ordinal number. The ordinal numbers are taken to be the canonical representatives of their classes, and so the order type of a well-ordered set is usually identified with the corresponding ordinal. For example, the order type of the natural numbers is ω.
The order type of a well-ordered set V is sometimes expressed as ord(V).
For example, consider the set of even ordinals less than ω·2+7, which is:
- V = {0, 2, 4, 6, ...; ω, ω+2, ω+4, ...; ω·2, ω·2+2, ω·2+4, ω·2+6}.
Its order type is:
- ord(V) = ω·2+4 = {0, 1, 2, 3, ...; ω, ω+1, ω+2, ...; ω·2, ω·2+1, ω·2+2, ω·2+3}.
Because there are 2 separate lists of counting and 4 in sequence at the end.
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