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Update infinite-infinities.md

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Co-authored-by: Copilot <175728472+Copilot@users.noreply.github.com>
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Alexa Amundson
2026-02-27 11:55:10 -06:00
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@@ -80,11 +80,10 @@ The sequence ℵ₀, ℵ₁, ℵ₂, ... is strictly increasing. **□**
**Proof:** **Proof:**
By the Axiom of Foundation (Regularity), every non-empty set A contains an element By the Axiom of Foundation (Regularity), every non-empty set A contains an element
m ∈ A such that m ∩ A = ∅. This prohibits infinite descending ∈-chains. m ∈ A such that m ∩ A = ∅. This prohibits infinite descending ∈-chains and makes ∈
well-founded: every non-empty collection of sets has an ∈-minimal element.
Without infinite ∈-chains, every set is built from simpler sets in finitely many steps. This well-foundedness lets us assign to each set a rank via transfinite (well-founded)
By transfinite induction: recursion. By transfinite induction on rank:
- ∅ ∈ V₁ (rank 0). **Base case.** - ∅ ∈ V₁ (rank 0). **Base case.**
- If every element of x has a rank, then x ∈ V_{α+1} where α = sup_{y∈x} ρ(y). **Inductive step.** - If every element of x has a rank, then x ∈ V_{α+1} where α = sup_{y∈x} ρ(y). **Inductive step.**