it's phi(e_(W+1)) = psi(W^W^W^W^W^W^W^W^W...) in Madore's psi.
EDIT:
Thanks fluffy kitty
EDIT turns out it's actually defined in weirmann's theta:
theta(e_(W+1))
it's phi(e_(W+1)) = psi(W^W^W^W^W^W^W^W^W...) in Madore's psi.
Code: Select all
An old OCF:
C_0(α)={0,Ω}
C_n+1(α)={γ+δ,γδ,γ^δ,φ_γ(δ),ψ_0(η),w_γ|γ,δ,η∈C_n(α);η<α}
C(α)=⋃(n<ω)C_n(α)
ψ_0(α)=min{β∈Ω|β∉C(α)}
ψ_1(α)=sup(C(α))
Code: Select all
extension: (point out any errors)
C_0,0(α)={0,Ω}
C_n+1,0(α)={γ+δ,γδ,γ^δ,φ_γ(δ),ψ_0,0(η),w_γ|γ,δ,η∈C_n,0(α);η<α}
C,0(α)=⋃(n<ω)C_n,0(α)
ψ_0,0(α)=min{β∈Ω|β∉C,0(α)}
ψ_1,0(α)=sup(C,0(α))
C_0,m(α)={0,Ω}
C_n+1,m+1(α)={γ+δ,γδ,γ^δ,φ_γ(δ),ψ_0,m+1(η),ψ_1,o(γ),ψ_0,o(γ),w_γ|γ,δ,η∈C_n,m+1(α);η<α,o<(m+1)}
C,m(α)=⋃(n<ω)C_n,m(α)
ψ_0,m(α)=min{β∈Ω|β∉C,m(α)}
ψ_1,m(α)=sup(C,m(α))
Code: Select all
Function(‘a’+’lert(1)’)()
An ordinal collapsing function
Code: Select all
ωεζηφα Γ
0,1,2,3,4,5,6... ω
ω, ω+1, ω+2, ω+3, ω+4,... ω+ω = ω2
ω, ω2, ω3, ω4, ω5,... ω*ω = ω^2
ω^2, ω^3, ω^4, ω^5,... ω^ω
ω, ω^ω, ω^ω^ω, ω^ω^ω^ω,... ω^ω^ω^ω^ω^ω^ω^... = ε_0
ε_0+1, ω^(ε_0+1), ω^ω^(ε_0+1), ω^ω^ω^(ε_0+1),... ε_1
ε_1+1, ω^(ε_1+1), ω^ω^(ε_1+1), ω^ω^ω^(ε_1+1),... ε_2
ε_0, ε_1, ε_2, ε_3, ε_4, ε_5,... ε_ω, ε_ε_0
ε_ε_0, ε_ε_1, ε_ε_2, ε_ε_3,... ε_ε_ω
ε_0, ε_ε_0, ε_ε_ε_0, ε_ε_ε_ε_0, ε_ε_ε_ε_ε_0,... ε_ε_ε_ε_ε_ε_ε_... = ζ_0
(Cantor's ordinal)
ζ_0+1, ε_(ζ_0+1), ε_ε_(ζ_0+1), ε_ε_ε_(ζ_0+1),... ζ_1
ζ_0, ζ_1, ζ_2, ζ_3, ζ_4,... ζ_ω, ζ_ζ_0
ζ_ζ_0, ζ_ζ_1, ζ_ζ_2,... ζ_ζ_ω
ζ_0, ζ_ζ_0, ζ_ζ_ζ_0, ζ_ζ_ζ_ζ_0,... ζ_ζ_ζ_ζ_ζ_ζ_ζ_ζ_ζ_... = η_0
Table of φ(α,β)
0) 1) 2) 3) fixed point
φ(0, ω ω^ω ω^ω^ω ω^ω^ω^ω ω^ω^ω^ω^ω^... = ε_0 α = φ(0,α)
φ(1, ε_0 ε_1 ε_2 ε_3 ε_ε_ε_ε_... = ζ_0 α = φ(1,α) = ε_α
φ(2, ζ_0 ζ_1 ζ_2 ζ_3 ζ_ζ_ζ_ζ_... = η_0 α = φ(2,α) = ζ_α
φ(3, η_0 η_1 η_2 η_3 η_η_η_η_... = φ(4,0) α = η_α
...
ω = φ(0,0), ε_0 = φ(1,0), ζ_0 = φ(2,0), η_0 = φ(3,0), φ(4,0), φ(5,0),... φ(ω,0)
φ(ω,0), φ(φ(ω,0),0), φ(φ(φ(ω,0),0),0),... φ(φ(φ(φ(... ...),0),0),0),0) = Γ_0 = φ(1,0,0)
(Feferman–Schütte ordinal) 0th fixed point of α = φ(α,0)
Γ_0+1, φ(Γ_0+1,0), φ(φ(Γ_0+1,0),0), φ(φ(φ(Γ_0+1,0),0),0)... Γ_1 = φ(1,0,1)
1st fixed point of α = φ(α,0)
Γ_0 = φ(1,0,0), Γ_1 = φ(1,0,1), Γ_2 = φ(1,0,2), Γ_3, Γ_4, Γ_5,... Γ_ω
Γ_ω, Γ_Γ_ω, Γ_Γ_Γ_ω,... Γ_Γ_Γ_Γ_Γ_Γ_... = φ(1,1,0)
0th fixed point of α = φ(1,0,α)
φ(1,1,0)+1, Γ_φ(1,1,0), Γ_Γ_φ(1,1,0), Γ_Γ_Γ_φ(1,1,0),... φ(1,1,1)
1st fixed point of α = φ(1,0,α)
φ(1,1,0), φ(1,1,1), φ(1,1,2), φ(1,1,3),... φ(1,1,φ(1,1,φ(1,1,φ(1,1,...)))) = φ(1,2,0)
0th fixed point of α = φ(1,1,α)
φ(1,0,0), φ(1,1,0), φ(1,2,0), φ(1,3,0),... φ(1,φ(1,φ(1,φ(1,...,0),0),0),0) = φ(2,0,0)
0th fixed point of α = φ(1,α,0)
φ(1,0,0), φ(2,0,0), φ(3,0,0),... φ(φ(φ(φ(...,0,0),0,0),0,0),0,0), = φ(1,0,0,0)
(Ackermann's ordinal) 0th fixed point of α = φ(α,0,0)
φ(1,0), φ(1,0,0), φ(1,0,0,0), φ(1,0,0,0,0),... φ(1,0,0,0,0,0,0,0,0,...) = SVO
(Small Veblen ordinal)
ω = φ(0)
φ(1,0,0,0,0,0,0,0,...), ω 0s = SVO
φ(1,0,0,0,0,0,0,0,...), SVO 0s
φ(1,0,0,0,0,0,0,0,...), w/ that many 0s
φ(1,0,0,0,0,0,0,0,...), w/ that many 0s
...
φ(1,0,0,0,0,0,0,0,...) (infinite layers) = LVO (Large Veblen ordinal)
0th fixed point of α = φ(1,0,0,0,0,0,0,0,...(w/ α 0s))
Code: Select all
x = 21, y = 5, rule = B3/S23
b2o2bo2b2o2b2o2bobobo$o3bobobobobobobobobo$bo2bobob2o2b2o3bo2bo$2bobob
obobobobo2bo$2o3bo2bobobobo2bo2bo!
As far as I know it's not really a fixed point so much as the limit of the finitary (finitely-many-entries) veblen function. Obviously I guess you could find some fixed point and point out that it's the SVO.
There are many transcendental integers.
That guide is pretty good, but I want to say:gameoflifemaniac wrote: ↑November 12th, 2019, 5:02 pmI've got a complete guide to ordinal numbers up to LVO. For those who know nothing about them.Code: Select all
ωεζηφα Γ 0,1,2,3,4,5,6... ω ω, ω+1, ω+2, ω+3, ω+4,... ω+ω = ω2 ω, ω2, ω3, ω4, ω5,... ω*ω = ω^2 ω^2, ω^3, ω^4, ω^5,... ω^ω ω, ω^ω, ω^ω^ω, ω^ω^ω^ω,... ω^ω^ω^ω^ω^ω^ω^... = ε_0 ε_0+1, ω^(ε_0+1), ω^ω^(ε_0+1), ω^ω^ω^(ε_0+1),... ε_1 ε_1+1, ω^(ε_1+1), ω^ω^(ε_1+1), ω^ω^ω^(ε_1+1),... ε_2 ε_0, ε_1, ε_2, ε_3, ε_4, ε_5,... ε_ε_0 ε_0, ε_ε_0, ε_ε_ε_0, ε_ε_ε_ε_0, ε_ε_ε_ε_ε_0,... ε_ε_ε_ε_ε_ε_ε_... = ζ_0 (Cantor's ordinal) ζ_0+1, ε_(ζ_0+1), ε_ε_(ζ_0+1), ε_ε_ε_(ζ_0+1),... ζ_1 ζ_0, ζ_1, ζ_2, ζ_3, ζ_4,... ζ_ζ_0, ζ_ζ_1, ζ_ζ_2,... ζ_ζ_ζ_0, ζ_ζ_ζ_ζ_0, ζ_ζ_ζ_ζ_ζ_0,... ζ_ζ_ζ_ζ_ζ_ζ_ζ_ζ_ζ_... = η_0 Table of φ(α,β) 0) 1) 2) 3) fixed point φ(0, ω ω^ω ω^ω^ω ω^ω^ω^ω ω^ω^ω^ω^ω^...= ε_0 α = φ(0,α) φ(1, ε_0 ε_1 ε_2 ε_3 ε_ε_ε_ε_... = ζ_0 α = φ(1,α) = ε_α φ(2, ζ_0 ζ_1 ζ_2 ζ_3 ζ_ζ_ζ_ζ_... = η_0 α = φ(2,α) = ζ_α φ(3 η_0 η_1 η_2 η_3 η_η_η_η_... = φ(4,0) α = η_α ... ω = φ(0,0), ε_0 = φ(1,0), ζ_0 = φ(2,0), η_0 = φ(3,0), φ(4,0), φ(5,0),... φ(ω,0) φ(ω,0), φ(φ(ω,0),0), φ(φ(φ(ω,0),0),0),... φ(φ(φ(φ(... ...),0),0),0),0) = Γ_0 = φ(1,0,0) (Feferman–Schütte ordinal) 0th fixed point of α = φ(α,0) Γ_0+1, φ(Γ_0+1,0), φ(φ(Γ_0+1,0),0), φ(φ(φ(Γ_0+1,0),0),0)... Γ_1 = φ(1,0,1) 1st fixed point of α = φ(α,0) Γ_0 = φ(1,0,0), Γ_1 = φ(1,0,1), Γ_2 = φ(1,0,2), Γ_3, Γ_4, Γ_5,... Γ_ω, Γ_Γ_ω, Γ_Γ_Γ_ω,... Γ_Γ_Γ_Γ_Γ_Γ_... = φ(1,1,0) 0th fixed point of α = φ(1,0,α) φ(1,1,0)+1, Γ_φ(1,1,0), Γ_Γ_φ(1,1,0), Γ_Γ_Γ_φ(1,1,0),... φ(1,1,1) 1st fixed point of α = φ(1,0,α) φ(1,1,0), φ(1,1,1), φ(1,1,2), φ(1,1,3),... φ(1,1,φ(1,1,φ(1,1,φ(1,1,...)))) = φ(1,2,0) 0th fixed point of α = φ(1,1,α) φ(1,0,0), φ(1,1,0), φ(1,2,0), φ(1,3,0),... φ(1,φ(1,φ(1,φ(1,...,0),0),0),0) = φ(2,0,0) 0th fixed point of α = φ(1,α,0) φ(1,0,0), φ(2,0,0), φ(3,0,0),... φ(φ(φ(φ(...,0,0),0,0),0,0),0,0), = φ(1,0,0,0) (Ackermann's ordinal) 0th fixed point of α = φ(α,0,0) φ(1,0), φ(1,0,0), φ(1,0,0,0), φ(1,0,0,0,0),... φ(1,0,0,0,0,0,0,0,0,...) = SVO (Small Veblen ordinal) ω = φ(0) φ(1,0,0,0,0,0,0,0,...), ω 0s = SVO φ(1,0,0,0,0,0,0,0,...), SVO 0s φ(1,0,0,0,0,0,0,0,...), w/ that many 0s φ(1,0,0,0,0,0,0,0,...), w/ that many 0s ... φ(1,0,0,0,0,0,0,0,...) (infinite layers) = LVO (Large Veblen ordinal) 0th fixed point of α = φ(1,0,0,0,0,0,0,0,...(w/ α 0s))
Code: Select all
ε_0, ε_1, ε_2, ε_3, ε_4, ε_5,... ε_w
Thanks, and I will correct it.Moosey wrote: ↑November 13th, 2019, 7:51 amThat guide is pretty good, but I want to say:gameoflifemaniac wrote: ↑November 12th, 2019, 5:02 pmI've got a complete guide to ordinal numbers up to LVO. For those who know nothing about them.Code: Select all
ωεζηφα Γ 0,1,2,3,4,5,6... ω ω, ω+1, ω+2, ω+3, ω+4,... ω+ω = ω2 ω, ω2, ω3, ω4, ω5,... ω*ω = ω^2 ω^2, ω^3, ω^4, ω^5,... ω^ω ω, ω^ω, ω^ω^ω, ω^ω^ω^ω,... ω^ω^ω^ω^ω^ω^ω^... = ε_0 ε_0+1, ω^(ε_0+1), ω^ω^(ε_0+1), ω^ω^ω^(ε_0+1),... ε_1 ε_1+1, ω^(ε_1+1), ω^ω^(ε_1+1), ω^ω^ω^(ε_1+1),... ε_2 ε_0, ε_1, ε_2, ε_3, ε_4, ε_5,... ε_ε_0 ε_0, ε_ε_0, ε_ε_ε_0, ε_ε_ε_ε_0, ε_ε_ε_ε_ε_0,... ε_ε_ε_ε_ε_ε_ε_... = ζ_0 (Cantor's ordinal) ζ_0+1, ε_(ζ_0+1), ε_ε_(ζ_0+1), ε_ε_ε_(ζ_0+1),... ζ_1 ζ_0, ζ_1, ζ_2, ζ_3, ζ_4,... ζ_ζ_0, ζ_ζ_1, ζ_ζ_2,... ζ_ζ_ζ_0, ζ_ζ_ζ_ζ_0, ζ_ζ_ζ_ζ_ζ_0,... ζ_ζ_ζ_ζ_ζ_ζ_ζ_ζ_ζ_... = η_0 Table of φ(α,β) 0) 1) 2) 3) fixed point φ(0, ω ω^ω ω^ω^ω ω^ω^ω^ω ω^ω^ω^ω^ω^...= ε_0 α = φ(0,α) φ(1, ε_0 ε_1 ε_2 ε_3 ε_ε_ε_ε_... = ζ_0 α = φ(1,α) = ε_α φ(2, ζ_0 ζ_1 ζ_2 ζ_3 ζ_ζ_ζ_ζ_... = η_0 α = φ(2,α) = ζ_α φ(3 η_0 η_1 η_2 η_3 η_η_η_η_... = φ(4,0) α = η_α ... ω = φ(0,0), ε_0 = φ(1,0), ζ_0 = φ(2,0), η_0 = φ(3,0), φ(4,0), φ(5,0),... φ(ω,0) φ(ω,0), φ(φ(ω,0),0), φ(φ(φ(ω,0),0),0),... φ(φ(φ(φ(... ...),0),0),0),0) = Γ_0 = φ(1,0,0) (Feferman–Schütte ordinal) 0th fixed point of α = φ(α,0) Γ_0+1, φ(Γ_0+1,0), φ(φ(Γ_0+1,0),0), φ(φ(φ(Γ_0+1,0),0),0)... Γ_1 = φ(1,0,1) 1st fixed point of α = φ(α,0) Γ_0 = φ(1,0,0), Γ_1 = φ(1,0,1), Γ_2 = φ(1,0,2), Γ_3, Γ_4, Γ_5,... Γ_ω, Γ_Γ_ω, Γ_Γ_Γ_ω,... Γ_Γ_Γ_Γ_Γ_Γ_... = φ(1,1,0) 0th fixed point of α = φ(1,0,α) φ(1,1,0)+1, Γ_φ(1,1,0), Γ_Γ_φ(1,1,0), Γ_Γ_Γ_φ(1,1,0),... φ(1,1,1) 1st fixed point of α = φ(1,0,α) φ(1,1,0), φ(1,1,1), φ(1,1,2), φ(1,1,3),... φ(1,1,φ(1,1,φ(1,1,φ(1,1,...)))) = φ(1,2,0) 0th fixed point of α = φ(1,1,α) φ(1,0,0), φ(1,1,0), φ(1,2,0), φ(1,3,0),... φ(1,φ(1,φ(1,φ(1,...,0),0),0),0) = φ(2,0,0) 0th fixed point of α = φ(1,α,0) φ(1,0,0), φ(2,0,0), φ(3,0,0),... φ(φ(φ(φ(...,0,0),0,0),0,0),0,0), = φ(1,0,0,0) (Ackermann's ordinal) 0th fixed point of α = φ(α,0,0) φ(1,0), φ(1,0,0), φ(1,0,0,0), φ(1,0,0,0,0),... φ(1,0,0,0,0,0,0,0,0,...) = SVO (Small Veblen ordinal) ω = φ(0) φ(1,0,0,0,0,0,0,0,...), ω 0s = SVO φ(1,0,0,0,0,0,0,0,...), SVO 0s φ(1,0,0,0,0,0,0,0,...), w/ that many 0s φ(1,0,0,0,0,0,0,0,...), w/ that many 0s ... φ(1,0,0,0,0,0,0,0,...) (infinite layers) = LVO (Large Veblen ordinal) 0th fixed point of α = φ(1,0,0,0,0,0,0,0,...(w/ α 0s))
Shouldn't it be this on line 11 or so?Code: Select all
ε_0, ε_1, ε_2, ε_3, ε_4, ε_5,... ε_w
Code: Select all
x = 21, y = 5, rule = B3/S23
b2o2bo2b2o2b2o2bobobo$o3bobobobobobobobobo$bo2bobob2o2b2o3bo2bo$2bobob
obobobobo2bo$2o3bo2bobobobo2bo2bo!
What the
Code: Select all
x = 21, y = 5, rule = B3/S23
b2o2bo2b2o2b2o2bobobo$o3bobobobobobobobobo$bo2bobob2o2b2o3bo2bo$2bobob
obobobobo2bo$2o3bo2bobobobo2bo2bo!
W
Code: Select all
x = 21, y = 5, rule = B3/S23
b2o2bo2b2o2b2o2bobobo$o3bobobobobobobobobo$bo2bobob2o2b2o3bo2bo$2bobob
obobobobo2bo$2o3bo2bobobobo2bo2bo!
It isn'tgameoflifemaniac wrote: ↑November 13th, 2019, 12:36 pmCan the Bachmann-Howard ordinal be explained in terms of a simpler notation? What is ψ(Ω_ω)? BHO is the supremum of Madore's psi, so how can this be any bigger?
Code: Select all
x = 21, y = 5, rule = B3/S23
b2o2bo2b2o2b2o2bobobo$o3bobobobobobobobobo$bo2bobob2o2b2o3bo2bo$2bobob
obobobobo2bo$2o3bo2bobobobo2bo2bo!
Yes, theoretically. But only if you have a fundamental sequence for w_1CK--which is still possible. E.g. w_1ck[n] = min a > all ordinals definable in n symbols in some language like set theory, or somethinggameoflifemaniac wrote: ↑November 13th, 2019, 4:36 pmI had 2 more questions in the previous post.
Does the function f_w_1CK(n) make any sense?
CK, fur Church-Kleene, the first non-recursive ordinal.
I meant f_w_1CK(n) in the fast-growing hierarchy.Moosey wrote: ↑November 13th, 2019, 8:41 pmYes, theoretically. But only if you have a fundamental sequence for w_1CK--which is still possible. E.g. w_1ck[n] = min a > all ordinals definable in n symbols in some language like set theory, or somethinggameoflifemaniac wrote: ↑November 13th, 2019, 4:36 pmI had 2 more questions in the previous post.
Does the function f_w_1CK(n) make any sense?
e.g. the symbols for
for all
there exists
not
equals
less than & greater than
in
supremum
successor
0
{
}
variables
Union
Implies
And
Allow you to define w thusly: min a|{forall n, n <a implies S(n)<a}
w^2 = min a|(forall b<a,(forall n, n <b implies S(n)<b) implies (there exists a c|(forall n, n <c implies S(n)<c) and b<c<a))
Imagine what you could do with actual FOST or something!
Code: Select all
x = 21, y = 5, rule = B3/S23
b2o2bo2b2o2b2o2bobobo$o3bobobobobobobobobo$bo2bobob2o2b2o3bo2bo$2bobob
obobobobo2bo$2o3bo2bobobobo2bo2bo!
Code: Select all
x = 21, y = 5, rule = B3/S23
b2o2bo2b2o2b2o2bobobo$o3bobobobobobobobobo$bo2bobob2o2b2o3bo2bo$2bobob
obobobobo2bo$2o3bo2bobobobo2bo2bo!
Yes, I said it would make sense if you have a fundamental sequence for w_1ck and then explained a possibilitygameoflifemaniac wrote: ↑November 15th, 2019, 10:43 amI meant f_w_1CK(n) in the fast-growing hierarchy.
Depends on which of a gazillon ordinal collapsing functions you're using-- if it's madore's psi, it's the limit of madore's psi-- the Bachmann-Howard Ordinal.
Yes, Madore's psiMoosey wrote: ↑November 15th, 2019, 3:11 pmYes, I said it would make sense if you have a fundamental sequence for w_1ck and then explained a possibilitygameoflifemaniac wrote: ↑November 15th, 2019, 10:43 amI meant f_w_1CK(n) in the fast-growing hierarchy.
Depends on which of a gazillon ordinal collapsing functions you're using-- if it's madore's psi, it's the limit of madore's psi-- the Bachmann-Howard Ordinal.
You see, all OCFs eventually freeze up at some point, so if you choose a point well beyond that point you still will only be at the limit of the OCF
Code: Select all
x = 21, y = 5, rule = B3/S23
b2o2bo2b2o2b2o2bobobo$o3bobobobobobobobobo$bo2bobob2o2b2o3bo2bo$2bobob
obobobobo2bo$2o3bo2bobobobo2bo2bo!