10/22/2020: Recently I`ve taken a liking to bracketing methods for root-finding and have even written . It would seem most of the well-known bracketing methods suffer from myriad problems, including very suboptimal orders of convergence and insufficiently intelligent conditions for using bisection. 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(â‹‚ {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_Ï€ = enum{κ, K ∈ Ξ(Ï€) 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. If you need me, you can find me here: or on . My favorite topics include , , , , , , and on . Some of my favorite posts: ©