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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 11:54:51 -06:00
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@@ -59,11 +59,15 @@ No function A → P(A) is surjective. Therefore |P(A)| > |A|. **□**
**Proof:** **Proof:**
ℵ₀ = |ℕ|. By Lemma 1, |P(ℕ)| > ℵ₀. Define ℵ₁ = the least cardinal greater than ℵ₀ ℵ₀ = |ℕ|. By Lemma 1, |P(ℕ)| > ℵ₀. Working in ZF, there is a least uncountable
(its existence follows from the Well-Ordering Theorem / Axiom of Choice). ordinal, usually denoted ω₁; its cardinality is the first uncountable cardinal,
which we call ℵ₁.
By induction: given ℵ_α, define ℵ_{α+1} = the least cardinal greater than ℵ_α. More generally, define the alephs as initial ordinals: given ℵ_α, let ℵ_{α+1} be
At limit ordinals λ: ℵ_λ = sup_{α<λ} ℵ_α. the least ordinal whose cardinality is strictly greater than |ℵ_α|; at limit
ordinals λ, set ℵ_λ = sup_{α<λ} ℵ_α. (If we assume the Axiom of Choice, every
set can be well-ordered, and every infinite cardinal is the cardinality of a
unique initial ordinal, so the sequence (ℵ_α)_α enumerates all infinite cardinals.)
The sequence ℵ₀, ℵ₁, ℵ₂, ... is strictly increasing. **□** The sequence ℵ₀, ℵ₁, ℵ₂, ... is strictly increasing. **□**