Mathematical Logic III The Joy of Sets § 3 . 0 - § 3 . 5

Also, for any two ordinals X,Y either X ∈ Y or Y ∈ X. As we saw before with ordinals ∈ is simply < or ⊂. Every well-ordered set is isomorphic to an ordinal. The first ordinal is 0, the second 1 = {0}, . . . the n’th ordinal is {0, 1, 2, . . . ,n − 1}. The first infintite ordinal is ω = {0, 1, 2, . . . , n, n + 1, . . .}. The second infinite ordinal is ω + 1 = {0, 1, . . . , n, . . . , ω}. In general, any ordinal γ = α+ 1 (i.e., γ = α ∪ {α}) is called a successor ordinal. Any ordainal that is not a successor ordinal is called a limit ordinal. That is, a limit ordinal is an ordinal with no immediate predicessor (examples: 0, ω, ω, . . .). The following set-theorhetic characterization of ordinals is very usefull. A set X is called transitive iff