User:C7X/Buchholz increasing

Basic equalities
"Nonstandard: (expression)" means the expression given is equal to the ordinal in that row, but not in normal form.

"Incorrect: (expression)" means the expression given is not equal to the ordinal in that row.

Justification for comparisons
Lemma (Buchholz): ψ₀(α)&lt;ψ₀(β) iff α∈C₀(α).

Lemma (Buchholz): If 0∈α∈C₀(α), then ψ₀(α+1) is the next additive principal after ψ₀(α).

Lemma (Buchholz): Each ψ function is continuous.

Claim: ψ₀(Ω)=ε₀.


 * Proof: For α≤ε₀ we have ψ₀(α)=ωα, and for any ε₀&lt;α<&Omega; we have α∉C₀(α) and therefore ψ₀(ε₀)=ψ₀(α). We have ψ₀(Ω) = ψ₀(sup{α | α&lt;Ω}) = sup{ψ₀(α) | α&lt;Ω} = ε₀ by continuousness of ψ₀.