User blog:ArtismScrub/Analysis of MOST

See here.

MOST, or Multiarrowed Ordinal Snowballing Train, is a simple ordinal notation created by the once sane Edwin Shade.

It is defined as follows:

α ↑γ β = α ↑γ-1 (α ↑γ (β - 1) + 1) if β is the successor of a limit ordinal (so as to not choke on ε0)

α ↑γ β = α ↑γ-1 (α ↑γ (β - 1)) if β is a successor ordinal but the previous rule does not apply

α ↑γ β = sup{α ↑γ γ|γ < β}

α ↑γ β = sup{α ↑δ β|δ < γ} if γ is a limit ordinal

Countable Index
ω↑↑ω = ε0

ω↑↑(ω+1) = ω↑(ε0+1)

ω↑↑(ω+n) = ω↑ω↑ω↑...↑ω↑ω↑ω↑(ε0+1) (n "ω"s)

ω↑↑(ω2) = ε1

ω↑↑(ωn) = εn-1

ω↑↑(ω2) = εω

ω↑↑(ω2+ω) = εω+1

ω↑↑(ω2+ωn) = εω+n

ω↑↑(ω22) = εω2

ω↑↑(ω2n) = εωn

ω↑↑(ω3) = εω2

ω↑↑(ωn) = εωn-1

ω↑↑(ωω) = εωω

ω↑↑↑3 = εε 0

ω↑↑↑n = εε ... ε 0   (n-1"ε"s)

ω↑↑↑ω = ζ0

ω↑↑↑↑ω = η0

ω{n}ω = φ(n-1,0)

ω{ω}ω = φ(ω,0)

ω{ω{ω}ω}ω = φ(φ(ω,0),0)

Limit of countable-indexed MOST is Γ0. Edwin then defines ω{Ω}ω as the fixed point of α ↦ ω{α}ω.

First Uncountable Cardinality
ω{Ω}ω = Γ0 = φ(1,0,0)

ω{Ω}(ω2) = Γ1 = φ(1,0,1)

ω{Ω+1}ω = φ(1,1,0)

ω{Ω+2}ω = φ(1,2,0)

ω{Ω+α}ω = φ(1,α,0)

ω{Ω2}ω = φ(2,0,0)

ω{Ωα}ω = φ(α,0,0)

ω{Ω2}ω = φ(1,0,0,0)

ω{Ω3}ω = φ(1,0,0,0,0)

ω{Ωω}ω = θ(Ωω)

ω{ΩΩ}ω = θ(ΩΩ)

ω{Ω↑↑ω}ω = θ(εΩ+1)

ω{Ω↑↑↑ω}ω = θ(ζΩ+1)

ω{Ω{ω}ω}ω = θ(φ(ω,Ω+1))

Up until this point, it's clear that ω{α}ω = θ(α).

ω{Ω{Ω}ω}ω = θ(Ω2)

ω{Ω{Ω{Ω}ω}ω}ω = θ(Ω3)

Limit: θ(Ωω). At this point, we can define Ωα as the fixed point of β ↦ Ωα-1{β}ω.

Beyond omega one
It's at this point that I'm not entirely certain how to work with the ordinal notations, so bear with me--this could be entirely wrong.

ω{Ω2}ω = θ(Ωω)

ω{Ω2}α = θ(Ωα)

ω{Ω2+1}ω = θ(ΩΩ)

ω{Ω2+1}(ω+1) = θ(Ωθ(Ω Ω) )

ω{Ω2+1}(ω2) = θ(ΩΩ 2 )

ω{Ω2+1}(ω2) = θ(ΩΩ ω )

ω{Ω2+1}(ω3) = θ(ΩΩ ω2 )

ω{Ω2+1}(ωω) = θ(ΩΩ ωω )

ω{Ω2+2}ω = θ(ΩΩ Ω )

ω{Ω2+3}ω = θ(ΩΩ Ω Ω )

ω{Ω2+ω}ω = ψ(ψ𝐈(0))

ω{Ω2+ω}(ω2) = ψ(ψ𝐈(1))

ω{Ω2+ω}(ω2) = ψ(ψ𝐈(ω))

ω{Ω2+ω}(ωω) = ψ(ψ𝐈(ωω))

ω{Ω2+ω}ω{Ω2+ω}ω = ψ(ψ𝐈(ψ(ψ𝐈(0))))

ω{Ω2+ω+1}ω = ψ(ψ𝐈(Ω))

ω{Ω2+ω+1}(ω2) = ψ(ψ𝐈(Ωω))

ω{Ω2+ω+2}ω = ψ(ψ𝐈(ΩΩ))

ω{Ω2+ω+3}ω = ψ(ψ𝐈(ΩΩ Ω ))

ω{Ω2+ω2}ω = ψ(ψ𝐈(ψ𝐈(0)))

ω{Ω2+ω3}ω = ψ(ψ𝐈(ψ𝐈(ψ𝐈(0))))

ω{Ω2+ω2}ω = ψ(𝐈)

ω{Ω2+ω2}(ω2) = ψ(𝐈×ω)

ω{Ω2+ω2+1}ω = ψ(𝐈×Ω)

ω{Ω2+ω2+2}ω = ψ(𝐈×ΩΩ)

ω{Ω2+ω2+ω}ω = ψ(𝐈×ψ𝐈(0))

ω{Ω2+ω2+ω2}ω = ψ(𝐈×ψ𝐈(ψ𝐈(0)))

ω{Ω2+ω22}ω = ψ(𝐈2)

ω{Ω2+ω3}ω = ψ(𝐈ω)

ω{Ω2+ωn}ω = ψ(𝐈ω n-2 )

And from this point on ω{Ω2+α}ω = ψ(𝐈α) for α ≥ ωω. So the limit of MOST is:

ω{ψ𝐈(0)}ω = ψ(𝐈ψ𝐈(0))