10/22/2020: Recently I've taken a liking to bracketing methods for root-finding and have even written my own code. 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(⋂ {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.

If you need me, you can find me here:

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My favorite topics include numerical-methods, sequences-and-series, calculus, big-numbers, divergent-series, summation, and special-functions on math.SE.

Some of my favorite posts:

Golf a number bigger than TREE(3)

Largest Number Printable

Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula

How to prove this $\pi$ formula?

Visually stunning math concepts which are easy to explain