Successor Ordinals

Set Theory · Axiom Academy

Building the next ordinal: the successor operation α⁺ = α ∪ α The successor of an ordinal α is the smallest ordinal greater than α. This means we take all the elements of α (which are all ordinals less than α) and add α itself as a new element. Why does this work? Since α = β : β < α , adding α to this set gives us β : β < α ∪ α , which is exactly the set of all ordinals less than or equal to α. This becomes the new ordinal α⁺. 2. Every Ordinal Has a Successor A fundamental property of ordinals: there is no "largest" ordinal. We can always go one step further. There is no ordinal β such that α < β < α⁺ α⁺ is the smallest ordinal greater than α The successor operation is injective: if α⁺ = β⁺, then α = β Let's compute successors for both finite and infinite ordinals. n⁺ = 0, 1, ..., n-1 ∪ n = 0, 1, ..., n = n+1 ω⁺ = ω ∪ ω = 0, 1, 2, ..., ω = ω + 1 4. Successor vs Limit Ordinals Ordinals come in two flavors: successors and limits. Limit ordinals have no immediate predecessor—they are "approached from below" by infinitely many smaller ordinals. Let's visualize how the successor operation builds up ordinals. This process works identically for infinite ordinals: Start with ω = 0, 1, 2, 3, ... Result: 0, 1, 2, 3, ..., ω = ω+1

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