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8/24/2019: I defined a neat ordinal collapsing function:

S(A) ⇔ ∀ f : sup A ↦ sup A, ∃ α ∈ A, ∀ η ∈ α (f(η) ∈ α)

B(α, κ, 0) = κ ∪ {0, K}

B(α, κ, n+1) = {γ + δ | γ, δ ∈ B(α, κ, n)}
               ∪ {Ψ_η(μ) | μ ∈ B(α, κ, n) ∧ η ∈ α ∩ B(α, κ, n)}

B(α, κ) = ⋃ {B(α, κ, n) | n ∈ N}

Ξ(α) = {κ, K ∈ K′ | κ ∉ B(α, κ) ∧ α ∈ cl(B(α, κ)) ∧ S(⋂ {Ξ(η) ∩ κ | η ∈ B(α, κ) ∩ α})}

Ψ_α = enum(Ξ(α))

C(α, κ, 0) = κ ∪ {0, K}

C(α, κ, n+1) = {γ + δ | γ, δ ∈ C(α, κ, n)}
               ∪ {ψ^η_ξ(μ) | μ, ξ, η ∈ C(α, κ, n) ∧ η ∈ α}

C(α, κ) = ⋃ {C(α, κ, n) | n ∈ N}

ψ^α_π = enum{κ, K ∈ Ξ(π) | κ ∉ C(α, κ) ∧ α ∈ cl(C(α, κ))}

where K is a weakly compact cardinal and K' is the (K+1)th hyper-Mahlo or alternatively, the smallest ordinal larger than K closed under γ ↦ M(γ), where M(γ) is the first γ-Mahlo. On its own this doesn't make a notation for large countable ordinals, but it can be used with another ordinal collapsing function for such purpose.

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