10/22/2020: Recently I've taken a liking to bracketing methods for rootfinding and have even written my own code. It would seem most of the wellknown 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 hyperMahlo 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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My favorite topics include numericalmethods, sequencesandseries, calculus, bignumbers, divergentseries, summation, and specialfunctions on math.SE.
Some of my favorite posts:
Golf a number bigger than TREE(3)
Methods to compute $\sum_{k=1}^nk^p$ without Faulhaber's formula

SupporterMay 10, 2016

AutobiographerMay 10, 2016
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