Limit Ordinals

Set Theory · Axiom Academy

Ordinals with no immediate predecessor: approached from below by infinitely many smaller ordinals 1. Definition of Limit Ordinal A limit ordinal is a non-zero ordinal that is not a successor of any ordinal. Key property: A limit ordinal has no largest element. For any β < λ, there exists γ such that β < γ < λ. In other words, you can always find ordinals between β and λ. The ordinal ω (omega) is the first infinite ordinal and the first limit ordinal. No finite ordinal n satisfies ω = n⁺, because n⁺ = n+1 is still finite ω = sup 0, 1, 2, 3, ... —it's the least upper bound of all natural numbers For any n < ω, we have n < n+1 < ω, so ω has no largest element Beyond ω, there are infinitely many larger limit ordinals. ω = 0, 1, 2, 3, ... - First infinite ordinal ω·2 = 0, 1, 2, ..., ω, ω+1, ω+2, ... - "Twice omega" ω·3 = Three copies of ω concatenated ω^ω - Omega to the omega power ε₀ = sup ω, ω^ω, ω^ω^ω, ... - First epsilon number Each of these ordinals is "approached from below" by infinitely many smaller ordinals but is not itself a successor. Every limit ordinal is the supremum (least upper bound) of the ordinals below it. ω = sup 0, 1, 2, 3, ... = ⋃ 0, 1, 2, 3, ... ω·2 = sup 0, 1, ..., ω, ω+1, ω+2, ... This is fundamentally different from successor ordinals: α⁺ is not the supremum of anything infinite—it's just α with one more element added. 5. Approaching but Never Reaching

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