ConwayLife.com

Posted: **November 9th, 2019, 8:07 am**

gameoflifemaniac wrote: November 9th, 2019, 3:33 am Wow.

What is the Bachmann-Howard ordinal?

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))

Posted: **November 9th, 2019, 8:20 pm**

Posted: **November 11th, 2019, 10:13 pm**

Since my last attempt at a FPT was trash, I made another:

https://googology.wikia.org/wiki/User_b ... udop/FPT_2

Posted: **November 11th, 2019, 10:21 pm**

New OCF:

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))

ψ_0,m(a) can access ψ_0,(m-1) freely.
ψ_0,w(a) can access ψ_0,m for all m < w freely, etc.
Assuming this is well defined, it should be easy to extend.

Posted: **November 12th, 2019, 4:42 am**

What is an OCF?
I presume it’s something like Ordinal ____ Function.

Posted: **November 12th, 2019, 7:14 am**

PkmnQ wrote: November 12th, 2019, 4:42 am What is an OCF?
I presume it’s something like Ordinal ____ Function.

An ordinal collapsing function

Posted: **November 12th, 2019, 5:02 pm**

SVO is the fixed point of what?
I'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_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))

Posted: **November 12th, 2019, 5:05 pm**

gameoflifemaniac wrote: November 12th, 2019, 5:02 pm SVO is the fixed point of what?

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.

Posted: **November 13th, 2019, 6:00 am**

How small is reciprocal of transcendental integer?

Posted: **November 13th, 2019, 7:46 am**

PHPBB12345 wrote: November 13th, 2019, 6:00 am How small is reciprocal of transcendental integer?

There are many transcendental integers.
All of their reciprocals are incredibly small but nonzero, of course
(This doesn't seem to be directly related to ordinals though)

Posted: **November 13th, 2019, 7:51 am**

gameoflifemaniac wrote: November 12th, 2019, 5:02 pm I'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))

That guide is pretty good, but I want to say:
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

Posted: **November 13th, 2019, 11:00 am**

Moosey wrote: November 13th, 2019, 7:51 am

gameoflifemaniac wrote: November 12th, 2019, 5:02 pm I'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))

That guide is pretty good, but I want to say:
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

Thanks, and I will correct it.

Posted: **November 13th, 2019, 11:01 am**

PHPBB12345 wrote: November 13th, 2019, 6:00 am How small is reciprocal of transcendental integer?

What the

Also, if W is w_1, what is W_1?

Posted: **November 13th, 2019, 11:47 am**

gameoflifemaniac wrote: November 13th, 2019, 11:01 amAlso, if W is w_1, what is W_1?

W
generally W and w are treated as the same, if they have a subscript. It's just that w = w_0 and W = w_1, if there is no subscript

Posted: **November 13th, 2019, 12:36 pm**

Can 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?

Posted: **November 13th, 2019, 3:40 pm**

gameoflifemaniac wrote: November 13th, 2019, 12:36 pm Can 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?

It isn't

Posted: **November 13th, 2019, 4:36 pm**

I had 2 more questions in the previous post.
Does the function f_w_1CK(n) make any sense?

Posted: **November 13th, 2019, 8:41 pm**

gameoflifemaniac wrote: November 13th, 2019, 4:36 pm I had 2 more questions in the previous post.
Does the function f_w_1CK(n) make any sense?

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 something
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!

Posted: **November 14th, 2019, 7:51 pm**

What is w_1ck?

Posted: **November 14th, 2019, 10:37 pm**

PkmnQ wrote: November 14th, 2019, 7:51 pmWhat is w_1ck?

CK, fur Church-Kleene, the first non-recursive ordinal.

Posted: **November 15th, 2019, 10:43 am**

Moosey wrote: November 13th, 2019, 8:41 pm
gameoflifemaniac wrote: November 13th, 2019, 4:36 pm I had 2 more questions in the previous post.
Does the function f_w_1CK(n) make any sense?
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 something
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!

I meant f_w_1CK(n) in the fast-growing hierarchy.

Posted: **November 15th, 2019, 10:46 am**

Can the Bachmann-Howard ordinal be explained in terms of a simpler notation? What is ψ(Ω_ω)? I asked this before

Posted: **November 15th, 2019, 3:11 pm**

gameoflifemaniac wrote: November 15th, 2019, 10:43 amI meant f_w_1CK(n) in the fast-growing hierarchy.

Yes, I said it would make sense if you have a fundamental sequence for w_1ck and then explained a possibility

gameoflifemaniac wrote: November 15th, 2019, 10:46 am What is ψ(Ω_ω)? I asked this before

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

Posted: **November 15th, 2019, 3:32 pm**

Moosey wrote: November 15th, 2019, 3:11 pm
gameoflifemaniac wrote: November 15th, 2019, 10:43 amI meant f_w_1CK(n) in the fast-growing hierarchy.
Yes, I said it would make sense if you have a fundamental sequence for w_1ck and then explained a possibility

gameoflifemaniac wrote: November 15th, 2019, 10:46 am What is ψ(Ω_ω)? I asked this before
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

Yes, Madore's psi

But the Bachmann-Howard ordinal is ψ(Ω^Ω^Ω^Ω^...), and ψ(Ω_ω) is much bigger?

Posted: **November 15th, 2019, 4:08 pm**

I literally wrote that since the limit of Madore's psi is the Bachmann-Howard ordinal, psi(w_w) is no larger than the Bachmann-Howard ordinal.
With Madore's psi, a>b does not mean that psi(a) > psi(b). It just means psi(a) >= psi(b).

So in other words, psi(w_w) = the Bachmann-Howard ordinal.

ConwayLife.com

Ordinals in googology

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Re: Ordinals in googology

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Re: Ordinals in googology

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Re: Ordinals in googology

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Re: Ordinals in googology

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