Ordinals in googology

[gameoflifemaniac]

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.

So why do the Bachmann-Howard ordinal and ψ(Ω_ω) have separate pages?
And can it be explained using simpler functions?

[Moosey]

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.

So why do the Bachmann-Howard ordinal and ψ(Ω_ω) have separate pages?

Oh, whoops. The Bachmann-Howard ordinal is not in madore's psi but in weirmann's theta. Sorry. Yeah, psi(w_w) > psi(e_(W+1))

And can it be explained using simpler functions?

Depends on which one, but ψ(Ω_ω) is the PTO of P1,1-CA_0


Look, I'm sorry, but I don't understand many ordinals beyond gamma_0 (besides perhaps the ackermann ordinal, the SVO, or the LVO) very well. Which is the main reason why I only use ordinals up to gamma_0 when I am using ah-- I feel like I don't have the "rights" to do anything beyond that.

[Moosey]

Extending an extension!

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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)=U(n<ω)C_n,0(α)
ψ_0,0(α)=min{β∈Ω|β∉C,0(α)}
ψ_1,0(α)=min(β>Ω|β∉(C,0(α)))
ψ_2,0(α)=sup(C,0(α))

This definition is more ordinal-suited
C_0,m(α)={0,Ω}
C_n+1,m(α)={γ+δ,γδ,γ^δ,φ_γ(δ),ψ_0,m(η),ψ_2,o(γ),ψ_1,o(γ),ψ_0,o(γ),w_γ|γ,δ,η∈C_n,m+1(α);η<α,o<(m),m>1}
C,m(α)=U(n<ω)C_n,m(α)
ψ_0,m(α)=min{β∈Ω|β∉C,m(α)}
ψ_1,m(α)=min(β>Ω|β∉(C,m(α)))
ψ_2,m(α)=sup(C,m(α))

Now you can have ψ_0,w(α), which can access ψ_0,m(α) for any finite m

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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)=U(n<ω)C_n,0(α)
ψ_0,0(α)=min{β∈Ω|β∉C,0(α)}
ψ_1,0(α)=min(β>Ω|β∉(C,0(α)))
ψ_2,0(α)=sup(C,0(α))

C_0,m(α)={0,Ω}
C_n+1,m(α)={γ+δ,γδ,γ^δ,φ_γ(δ),ψ_0,m(η),ψ_2,o(γ),ψ_1,o(γ),ψ_0,o(γ),w_γ|γ,δ,η∈C_n,m+1(α);η<α,o<(m),m>1}
C,m(α)=U(n<ω)C_n,m(α)
ψ_0,m(α)=min{β∈Ω|β∉C,m(α)}
ψ_1,m(α)=min(β>Ω|β∉(C,m(α)))
ψ_2,m(α)=sup(C,m(α))

C_1,0,0(α)={0,Ω}
C_1,n+1,0(α)={γ+δ,γδ,γ^δ,φ_γ(δ),ψ_1,0,0(η),ψ_2,δ(γ),ψ_1,δ(γ),ψ_0,δ(γ),w_γ|γ,δ,η∈C_1,n,0(α);η<α}
C1,,0(α)=U(n<ω)C_n,m(α)
ψ_1,0,0(α)=min{β∈Ω|β∉C,m(α)}
ψ_1,1,0(α)=min(β>Ω|β∉(C,m(α)))
ψ_1,2,0(α)=sup(C,m(α))

ψ_1,0,0(α) can access ψ_0,γ(α), ψ_1,γ(α), and ψ_2,γ(α) as long as γ is constructible in ψ_1,0,0(β) for β<a

[gameoflifemaniac]

Oh, whoops. The Bachmann-Howard ordinal is not in madore's psi but in weirmann's theta. Sorry. Yeah, psi(w_w) > psi(e_(W+1))

The Wikipedia artivle about OCF's says that the Bachmann-Howard ordinal is the supremum of psi(W), psi(W^W), psi(W^W^W), psi(W^W^W^W)... and we can't go any further with the definition we were given. https://en.wikipedia.org/wiki/Ordinal_c ... te_ordinal
So what is ψ(Ω_ω)? The supremum of ψ?

Depends on which one, but ψ(Ω_ω) is the PTO of P1,1-CA_0

Oh sorry I meant Backmann-Howard ordinal

[Moosey]

Oh, whoops. The Bachmann-Howard ordinal is not in madore's psi but in weirmann's theta. Sorry. Yeah, psi(w_w) > psi(e_(W+1))

The Wikipedia artivle about OCF's says that the Bachmann-Howard ordinal is the supremum of psi(W), psi(W^W), psi(W^W^W), psi(W^W^W^W)... and we can't go any further with the definition we were given. https://en.wikipedia.org/wiki/Ordinal_c ... te_ordinal
So what is ψ(Ω_ω)? The supremum of [ψ(a)]?

Either Wikipedia is wrong or the googology wiki is wrong; the googology wiki says ψ(w_w) = the limit of Madore's psi, and Wikipedia is claiming ψ(e_(W+1)) is. I don't know how to find the limit of an OCF. (Also, most of the people on the googology wiki agree with the latter, as if it weren't confusing enough.)

I'd say the most likely scenario is that ψ(w_w) is in a different OCF with a higher limit than Madore's psi and that someone on the googology wiki made a mistake in saying what OCF it was in, but I could be wrong

Depends on which one, but ψ(Ω_ω) is the PTO of P1,1-CA_0

Oh sorry I meant Bachmann-Howard ordinal

I don't know of any other simpler ways to construct the Bachmann-Howard ordinal than with OCFs

[gameoflifemaniac]

I'll post the guide once more for everybody who knows nothing about googology.

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ωεζηφαψΓ
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
0th fixed point of α = ω^α

ε_0+1, ω^(ε_0+1), ω^ω^(ε_0+1), ω^ω^ω^(ε_0+1),... ε_1
1st fixed point of α = ω^α

ε_1+1, ω^(ε_1+1), ω^ω^(ε_1+1), ω^ω^ω^(ε_1+1),... ε_2
2nd fixed point of α = ω^α

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) 0th fixed point of α = ε_α

ζ_0+1, ε_(ζ_0+1), ε_ε_(ζ_0+1), ε_ε_ε_(ζ_0+1),... ζ_1
1st fixed point of α = ε_α

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)+1),  Γ_Γ_(φ(1,1,0)+1),  Γ_Γ_Γ_(φ(1,1,0)+1),... 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([i]0[/i]), 0 0s (not counting the 1st and here only argument which is a 0 unfortunately)
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)), supremum of Veblen's hierarchy

Ordinal collapsing function (OCF) - a function used to write large ordinals using even larger uncountable ordinals
Ω = ω_1, the first uncountable ordinal (after any countable ordinal)

Definition of Madore's psi ψ(α)
C(α) = the set of all ordinals constructible using 0, 1, ω and Ω and finite applications of addition, multiplication, exponentiation and ψ(κ) (previously defined values), where κ is contained in C(β) and β < α.
ψ(α) is the smallest ordinal not in the set (NOT the ordinal coming after all of them)

ψ(0) = ε_0 (cannot be written as ω^ω^ω^ω^... in finite applications of exponentiation)
ψ(1) = ε_1 (ψ(0) = ε_0 is now in the set and still ε_1 cannot be constructed)
ψ(α) = ε_α (for α =< ζ_0)
ψ(ζ_0) = ζ_0 (because it's the fixed point of α = ε_α, now ε_α for any α is contained in C(α))
ψ(>ζ_0) = ζ_0 (ζ_0 cannot be in C(α) because it cannot be expressed as ε_ε_ε_ε_... in finite applications, so it's stuck at ζ_0 up to Ω)

ψ(Ω) = ζ_0
ψ(Ω+1) = ε_(ζ_0+1) (Ω, and for that matter, ψ(Ω) = ζ_0, are finally in C(α). This is the next number that cannot be constructed)
ψ(Ω+α) = ε_(ζ_0+α) (α =< ζ_1)
ψ(Ω+ζ_1) = ζ_1 (stuck at ζ_1 up to Ω2)
ψ(Ω+>ζ_1) = ζ_1 (The first ordinal not constructible from ε_α is ζ_1, ζ_α isn't contained in the set of functions yet)

ψ(Ω2) = ζ_1
ψ(Ω(1+α)) = ζ_α (α =< φ(3,0)) (now ζ_α is in the set of functions)
ψ(Ω*Ω) = ψ(Ω^2) = η_0 = φ(3,0)
ψ(Ω^α) = φ((α+1),0) (α =< Γ_0) (now φ(α,0) is in the set)
To be continued...

[Moosey]

Slight update to @@@
I'll put the whole definition of ah up to @@@ in this code:

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Part one:

define $_n as any entries (including no entries) in an array.
It’s my symbol for we-don’t-care entries.
In any one use of any one rule, if n is the same, $_n is the same

a#b = concatenation of a and b

n@m =
n, m = 1
{n}#(n@(m-1)), m > 1, m not a lim ord
n@(m[n]) if m is a lim ord

g(a,n,B) =
ah_g(a-1,n,B) {B}, a > 1;
n, a = 0

ah definition:
Rule 1.
ah^a_n {$_0} = g(a,n,$_0)
Rule 2.
ah_n ({$_1}#{z}) = ah_n {$_1}, z = 0
Rule 3.
ah_n{} = n+1
Rule 4.
ah_n{a+1,$_2} = ah^n_n{a,$_2}
Rule 5.
ah_n{a,$_3} = ah_n{a[n],$_3}, a a lim ord
Rule 6.
ah_n ((0@b)#{a+1,$_4}) = ah_(n+1) (((a+1)@b)#{a,$_4}), b > 0
Rule 7.
ah_n ((0@b)#{a,$_5}) = ah_n ((a@b)#{a[n],$_5}), a a lim ord & b > 0

The rest of the array rules:
if unspecified, n is the subscript in the first (closest) ah that is in front of the array.

Now we can redefine the rest of the array rules:

The rest of the array rules:
{$_0//(b+1),$_1} = ((($_0)(@^_n)ah_n{$_0//b})#{//b,$_1})
{$_0//0} = {$_0}
{$_0//b,$_1} = {$_0//(b[n]),$_1} for lim ord b
Else: apply ah's 7 rules, starting after legion bar. This includes nesting and incrementing the subscript (e.g. ah_n{$_0//,0,0,w+1} = ah_(n+1){$_0//,w+1,w+1,w})

($_1)(@^_n)a =
($_1)@(ah_n(($_1)(@^_n)(a-1))), a > 1,
ah_n{$_1}, a=1

{$_0@@0} = {$_0}
{$_0@@a,$_1} = {$_0@@a[n],$_1},a a lim ord
{$_0@@(a+1),$_1} = ({$_0//}#{$_0@@(a),$_1}
Apply basic ah rules otherwise to everything after the @@. See legion bar notes for more details

{$_0(@@b,$_2)0} = {$_0}
{$_0((@@b,$_2)a,$_1} = {$_0(@@b)a[n],$_1},a a lim ord
{$_0(@@0)(a+1),$_1} = ({$_0//}#{$_0(@@0)(a),$_1}
{$_0(@@b+1,$_2)(a+1),$_1} = ({$_0(@@b,$_2)}#{$_0@@(a),$_1}
{$_0(@@b,$_2)$_3} = {$_0(@@b[n],$_2)$_3}, if b is a lim ord
Apply basic ah rules otherwise to everything after the (@@$) or everything inside the (@@$) depending on what is necessary. See legion bar notes for more details.

(Now things such as ah_n{a(@@(a(@@a)a))a} are well defined.)

{$_0@@@(a+1),$_1} = {$_0(@@($_0@@@a,$_1))$_0,$_1}
{$_0@@@0} = {$_0}
{$_0@@@a,$_1} = {$_0@@@a[n],$_1}, a a lim ord.

Apply ah rules otherwise

Now I should start with multidimensional arrays

We will indicate a dimensional break as [[m]] for some m
{$_0[[1]]0} = {$_0}
{$_0[[1]]a+1,$_1} = {$_0@@@$_0[[1]]a,$_1}
{$_0[[1]]a,$_1} = {$_0,a[[1]]a[n],$_1}, a a lim ord
{$_0[[1]]0@m,a+1,$_1} = {$_0[[1]](ah_n{$_0[[1]](a+1)@m,a,$_1})@m,a,$_1}
{$_0[[1]]0@m,a,$_1} = {$_0[[1]]a@m,a[n],$_1}, a a lim ord

I'll figure out higher-dimensional arrays later

But here are the full rules:

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Part one:

define $_n as any entries (including no entries) in an array.
It’s my symbol for we-don’t-care entries.
In any one use of any one rule, if n is the same, $_n is the same

a#b = concatenation of a and b

n@m =
n, m = 1
{n}#(n@(m-1)), m > 1, m not a lim ord
n@(m[n]) if m is a lim ord

g(a,n,B) =
ah_g(a-1,n,B) {B}, a > 1;
n, a = 0

ah definition:
Rule 1.
ah^a_n {$_0} = g(a,n,$_0)
Rule 2.
ah_n ({$_1}#{z}) = ah_n {$_1}, z = 0
Rule 3.
ah_n{} = n+1
Rule 4.
ah_n{a+1,$_2} = ah^n_n{a,$_2}
Rule 5.
ah_n{a,$_3} = ah_n{a[n],$_3}, a a lim ord
Rule 6.
ah_n ((0@b)#{a+1,$_4}) = ah_(n+1) (((a+1)@b)#{a,$_4}), b > 0
Rule 7.
ah_n ((0@b)#{a,$_5}) = ah_n ((a@b)#{a[n],$_5}), a a lim ord & b > 0

The rest of the array rules:
if unspecified, n is the subscript in the first (closest) ah that is in front of the array.

Now we can redefine the rest of the array rules:

The rest of the array rules:
{$_0//(b+1),$_1} = ((($_0)(@^_n)ah_n{$_0//b})#{//b,$_1})
{$_0//0} = {$_0}
{$_0//b,$_1} = {$_0//(b[n]),$_1} for lim ord b
Else: apply ah's 7 rules, starting after legion bar. This includes nesting and incrementing the subscript (e.g. ah_n{$_0//,0,0,w+1} = ah_(n+1){$_0//,w+1,w+1,w})

($_1)(@^_n)a =
($_1)@(ah_n(($_1)(@^_n)(a-1))), a > 1,
ah_n{$_1}, a=1

{$_0@@0} = {$_0}
{$_0@@a,$_1} = {$_0@@a[n],$_1},a a lim ord
{$_0@@(a+1),$_1} = ({$_0//}#{$_0@@(a),$_1}
Apply basic ah rules otherwise to everything after the @@. See legion bar notes for more details

{$_0(@@b,$_2)0} = {$_0}
{$_0((@@b,$_2)a,$_1} = {$_0(@@b)a[n],$_1},a a lim ord
{$_0(@@0)(a+1),$_1} = ({$_0//}#{$_0(@@0)(a),$_1}
{$_0(@@b+1,$_2)(a+1),$_1} = ({$_0(@@b,$_2)}#{$_0@@(a),$_1}
{$_0(@@b,$_2)$_3} = {$_0(@@b[n],$_2)$_3}, if b is a lim ord
Apply basic ah rules otherwise to everything after the (@@$) or everything inside the (@@$) depending on what is necessary. See legion bar notes for more details.

(Now things such as ah_n{a(@@(a(@@a)a))a} are well defined.)

{$_0@@@(a+1),$_1} = {$_0(@@($_0@@@a,$_1))$_0,$_1}
{$_0@@@0} = {$_0}
{$_0@@@a,$_1} = {$_0@@@a[n],$_1}, a a lim ord.

Apply ah rules otherwise

Dimensional arrays: 2D
{$_0[[1]]0} = {$_0}
{$_0[[1]]a+1,$_1} = {$_0@@@$_0[[1]]a,$_1}
{$_0[[1]]a,$_1} = {$_0,a[[1]]a[n],$_1}, a a lim ord
{$_0[[1]]0@m,a+1,$_1} = {$_0[[1]](ah_n{$_0[[1]](a+1)@m,a,$_1})@m,a,$_1}
{$_0[[1]]0@m,a,$_1} = {$_0[[1]]a@m,a[n],$_1}, a a lim ord

[gameoflifemaniac]

I searched one more source http://quibb.blogspot.com/2012/03/infin ... tions.html
and it also says the BHO is ψ(Ω^Ω^Ω^Ω^...) = ψ(ε_Ω+1).
But it says that ψ1(0) = ε_Ω+1 (which is uncountable), not ψ(ε_Ω+1). Is that a mistake?

[Moosey]

I searched one more source http://quibb.blogspot.com/2012/03/infin ... tions.html
and it also says the BHO is ψ(Ω^Ω^Ω^Ω^...) = ψ(ε_Ω+1).

That wasn't the question-- the question is whether that's the limit of Madore's psi

But it says that ψ1(0) = ε_Ω+1 (which is uncountable), not ψ(ε_Ω+1). Is that a mistake?

No

[gameoflifemaniac]

That wasn't the question-- the question is whether that's the limit of Madore's psi

Then BHO is the supremum of Madore's psi, and what the heck is ψ(Ω_ω)

But it says that ψ1(0) = ε_Ω+1 (which is uncountable), not ψ(ε_Ω+1). Is that a mistake?

No

Then I don't understand.

[Moosey]

That wasn't the question-- the question is whether that's the limit of Madore's psi

Then BHO is the supremum of Madore's psi, and what the heck is ψ(Ω_ω)

But it says that ψ1(0) = ε_Ω+1 (which is uncountable), not ψ(ε_Ω+1). Is that a mistake?

No

Then I don't understand.

The BHO = ψ(ψ_1(0))

[gameoflifemaniac]

That wasn't the question-- the question is whether that's the limit of Madore's psi

Then BHO is the supremum of Madore's psi, and what the heck is ψ(Ω_ω)

No

Then I don't understand.

The BHO = ψ(ψ_1(0))

Now everything has become clear for me
Or is it?
Madore's psi C(a) set includes ordinals in the form of psi(a). So what makes e_e_(z_0+1) invalid in the set? Is it because if it's used it can be the only function and used just once?

[Moosey]

I'm gonna try to figure out how much more powerful ah_n{a[[1]]a} is than f_a(n)

ah_n {a@n} > f_an(n) which is around f_aw(n)
ah_n {a@a@n} > (assuming reasonable ah rule extensions) ah_n {a@an} > f_((a^2)*n)(n)
ah_n {a//w} = ah_n {a//n} probably > f_a^w(n) (probably even without the reasonable extensions, since I skipped a good layer of recursion)
ah_n {a//a//n} probably > f_a^a^w(n) (maybe?)
ah_n {a(@@n)a} probably ~ f_a{n}a(n) (assuming reasonable extensions of hyper operators to ordinals) or, very roughly, ~ f_phi_n(0)(n) assuming a< phi_n(0)
ah_n {a@@@n} probably ~ f_gamma_0(n) assuming a is considerably less than gamma_0

That was better than I expected, if I was right in my analysis

[Moosey]

BUMP

Is my analysis of ah correct?
If not, how powerful is @@@ actually?

[Moosey]

After learning about mahlos on the googology discord:

Code: Select all

B_0,0(a,b) = {0,M} U b
B_m,n+1(a,b) = B_m,n(a,b) U {y+d,yd,y^d,chi_m(h),chi_o(y);y,d,h in B_m,n(a,b),o<m,h<b}
B_m(a,b) = U(n<w) B_m,n(a,b)
chi_m(a) = min b|b= intersection(M,B_m(a,b)), b is regular, b is uncountable

C_0,0(α)={0,Ω,M}
C_n+1,0(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,0(η)),w_γ,chi_γ(δ)|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)=min{β>Ω|β∉C,0(α)}
ψ_2,0(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_2,o(γ)),(ψ_1,o(γ)),(ψ_0,o(γ)),w_γ,chi_γ(δ)| γ,δ,η∈(C_n,m+1(α));η<α,o<(m+1)}
C,m(α)=⋃(n<ω)(C_n,m(α))

How crappy is it?*

*admittedly it's a big leap from my best OCF previously, but...*
*also, sorry about the sudden change in style between psi and chi--I copied and modified modified a psi I defined on gwiki*
*WHAT ARE YOU DOING READING FINE PRINT?!? PLEASE TELL ME YOU ZOOMED IN TO READ THIS!!!

[gameoflifemaniac]

Why are the values of higher psi functions considered countable when they're don't seem to be?

[Moosey]

they aren't
why?

Also, here's an extension I posted on discord:

Code: Select all

B_p,n,m(x,y) = {0,J}Uy, where J is the smallest p-Mahlo
B_p,n,m+1(x,y) = {a+b,theta_p,0(h),theta_q,o(a)|a,b,h in C_p,n,m(a,b),q<p,o<n,h<y}
B_p,n(x,y) = U(m<w) C_p,n,m(x,y)
Theta_p+1,n(x) = min b|b=intersection(J,B_p,n(x,b)),b is p-Mahlo, J is the smallest p+1-mahlo
Theta_0,n(x) = min b|b=intersection(M,B_p,n(x,b)),b is Regular and uncountable

C_0,m(α)={0,Ω,M}
C_n+1,0(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,0(η)),w_γ,theta_{γ,0}(δ),minimum γ-Mahlo|γ,δ,η∈(C_n,0(α));η<α}
C,0(α)=⋃(n<ω)C_n,0(α)
ψ_0,m(α)=min{β∈Ω|β∉C,m(α)}
ψ_1,m(α)=min{β>Ω|β∉C,m(α)}
ψ_2,m(α)=sup(C,m(α))
C_n+1,m(α)={γ+δ,γδ,γ^δ,φ_γ(δ),(ψ_0,m(η)),(ψ_2,o(γ)),(ψ_1,o(γ)),(ψ_0,o(γ)),w_γ,theta_{γ,o}(δ),minimum γ-Mahlo|γ,δ,η∈(C_n,m(α));η<α,o<(m)}
C,m(α)=⋃(n<ω)(C_n,m(α))

probably rather strong

It can collapse higher-order mahlos down to lower ones, and they form a hierarchy where each function can access lower-level ones

[gameoflifemaniac]

So how are countable ordinals larger than the Bachmann-Howard ordinal defined?

[testitemqlstudop]

To start, you can add any other countable ordinal to it.

How is PTO rigorously defined? I thought it was defined as "biggest ordinal provably well-ordered", but w_1 is obviously also well-ordered.

[Moosey]

To start, you can add any other countable ordinal to it.

How is PTO rigorously defined? I thought it was defined as "biggest ordinal provably well-ordered", but w_1 is obviously also well-ordered.

Smallest ordinal not probably well-ordered, I think

[gameoflifemaniac]

Can anyone explain me ordinals significantly bigger than the BHO?

[Moosey]

Can anyone explain me ordinals significantly bigger than the BHO?

W is the first uncountable ordinal and the supremum of all countable ordinals.

[BlinkerSpawn]

Can anyone explain me ordinals significantly bigger than the BHO?

In many commonly used ordinal notations, W_1 acts as a fixed point of psi_0.
We can then define psi_1 which creates expressions with W_1, and define W_2 as a fixed point of psi_1.
Many strong notations, such as Pi and Pair Sequence System, make use of this "omega-upgrading" up to its first limit, psi_0(W_w).

[gameoflifemaniac]

Can anyone explain me ordinals significantly bigger than the BHO?

In many commonly used ordinal notations, W_1 acts as a fixed point of psi_0.
We can then define psi_1 which creates expressions with W_1, and define W_2 as a fixed point of psi_1.
Many strong notations, such as Pi and Pair Sequence System, make use of this "omega-upgrading" up to its first limit, psi_0(W_w).

Wouldn't that give you uncountable ordinals?

[gameoflifemaniac]

they aren't
why?

Because someone tried to diagonalize f_psi_1(1) (3) in a video. How can you diagonalize an uncountable number?