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Ordinals in googology

Posted: September 26th, 2019, 7:12 am
by testitemqlstudop
...This topic is taking over BOTH the LTCFC and TFNCAQ so now it deserves its own thread.

Primarily I'm concerned about large ordinals here.

For example, the first fixed point of ∂ |-> w∂ is w^^2. Hence we can make a function:
f(i) = the i-th fixed point of ∂ |-> w∂
What would f(2) be? f(w)? Would the last fixed point of ∂ |-> w∂ be well-defined?

At this point there is a function that diagonalizes from epsilon 0:
f(i) =the 1st fixed point of ∂ |-> {w, ∂, i, 1}
...if I got BEAF correct.

Then we effectively get a new fast growing function t(i), where assuming FGH is the fgh function,
t(i) = FGH_f(i)(2)

Re: Ordinals

Posted: September 26th, 2019, 7:20 am
by Moosey
testitemqlstudop wrote:...This topic is taking over BOTH the LTCFC and TFNCAQ so now it deserves its own thread.

Primarily I'm concerned about large ordinals here.

For example, the first fixed point of ∂ |-> w∂ is w^^2. Hence we can make a function:
f(i) = the i-th fixed point of ∂ |-> w∂
What would f(2) be? f(w)? Would the last fixed point of ∂ |-> w∂ be well-defined?

At this point there is a function that diagonalizes from epsilon 0:
f(i) =the 1st fixed point of ∂ |-> {w, ∂, i, 1}
...if I got BEAF correct.

Then we effectively get a new fast growing function t(i), where assuming FGH is the fgh function,
t(i) = FGH_f(i)(2)
I believe that f(2) would be w^w^w judging by the fact that e_1 = e_0^^w

EDIT:
it's probably a good idea to mention my ord function up here.

Formal definition:
oddpart(n)=

Code: Select all

oddpart(n/2), n/2 ∈ natural numbers
n, n/2 ∈ not-natural numbers
twoeypart(n)=

Code: Select all

n/oddpart(n)
ord(n)=

Code: Select all

(ord(log_2(twoeypart(n))),ord((oddpart(n)-1)/2)), n>0
0, n=0

Re: Ordinals

Posted: September 26th, 2019, 7:24 am
by testitemqlstudop
Which f

Re: Ordinals

Posted: September 26th, 2019, 3:51 pm
by Moosey
testitemqlstudop wrote:Which f
the first one

EDIT:
I think it's worth mentioning the (informal) definitions of fundamental sequences, the fgh, and a crappy hierarchy I came up with.

The fgh, sgh, and HH are all defined in the general format

f_n(x) =

Code: Select all

x, n = 0
f_n[x](x), n a lim ord
[something involving f_n-1(x)] otherwise
for instance, the fgh is:
f_n(x) =

Code: Select all

x+1, n = 0
f_n[x](x), n a lim ord
f^x_n-1(x) otherwise
and the HH is:
h_n(x) =

Code: Select all

x, n = 0
h_n[x](x), n a lim ord
h_n-1(x+1) otherwise
Which is surprisingly totally different and much more faster than the sgh:
g_n(x)=

Code: Select all

x, n = 0
g_n[x](x), n a lim ord
g_n-1(x)+1 otherwise
About your second f:
testitempimplesquidIDyoulot wrote:f(i) =the 1st fixed point of ∂ |-> {w, ∂, i, 1}
...if I got BEAF correct.
f(1) > gamma_0 (which is {w,w,1,1}. The reader could refer to the bottom of this section for ordinal beaf.) since it is a |-> {w, a, 1, 1}

Here's something which grows faster than t(i)
tmod(i) = fgh_f(i)(i)
Which is just t but it's of I rather than 2, making it larger for large values

Re: Ordinals

Posted: September 27th, 2019, 1:07 am
by Hdjensofjfnen
I've posted the fifth post in this thread.

Re: Ordinals

Posted: September 27th, 2019, 3:08 am
by testitemqlstudop
Ok screw it that's not what I mean

Re: Ordinals

Posted: September 27th, 2019, 6:34 am
by Moosey
Hdjensofjfnen wrote:I've posted the fifth post in this thread.
Well done
Hopefully we can get to w posts soon...
(For those who are itching to make a lot of posts, please don't. Go to random posts or somewhere else)

Re: Ordinals in googology

Posted: September 27th, 2019, 6:45 am
by Moosey
Here's what one could call the
hardy, fast and slow-growing hierarchy:
hsf_n(x)=

Code: Select all

x, n = 0
hsf_n[x](x), n a lim ord
hsf^x_n-1(x+1)+1 otherwise
More fgh variants:

f(x) =

Code: Select all

x+1, n = 0
f_n[f_n[x](x)](x), n a lim ord
f^x_n-1(x) otherwise
(Grows significantly faster than the normal fgh)
f(x) =

Code: Select all

x+1, n = 0
f_g(x,n,x)(x), n a lim ord
f^x_n-1(x) otherwise

define g(m,n,x) =
n[f_g(m-1,n,x)(x)], m>0,
x, m less than or = to 0 
(An extension of the previous)

Or something similar:
f(x) =

Code: Select all

x+1, n = 0
f_g(x,n,x)(x), n a lim ord
f^x_n-1(x) otherwise

define g(m,n,x) =
n[g(m-1,n,x)], m>0,
x, m less than or = to 0 

(g is just n[n[n[n[n[n[n[....x]]]]], with m ns)
Which is probably less powerful but sill pretty simple and much more powerful than the fgh (for inputs >= w2, since w[n] = n)

Re: Ordinals in googology

Posted: September 27th, 2019, 7:47 am
by testitemqlstudop
Congratulations, we now need an fgh for the fgh.

Re: Ordinals in googology

Posted: September 27th, 2019, 8:39 am
by Moosey
testitemqlstudop wrote:Congratulations, we now need an fgh for the fgh.
Nah, we can approximate values in what I'll call the weak moose-growing hierarchy
wm_n(x)=

Code: Select all

x+1, n = 0
wm_g(x,n,x)(x), n a lim ord
wm^x_n-1(x) otherwise

define g(m,n,x) =
n[g(m-1,n,x)], m>0,
x, m less than or = to 0 

(g is just n[n[n[n[n[n[n[....x]]]]], with m ns)
<late edit>oh crumbs, wm isn't defined for a lot of ordinals when you get to stuff like n[w^w]</late edit>

And strong moose-growing hierarchy:
sm_n(x)=

Code: Select all

x+1, n = 0
sm_g(x,n,x)(x), n a lim ord
sm^x_n-1(x) otherwise

define g(m,n,x) =
n[sm_g(m-1,n,x)(x)], m>0,
x, m less than or = to 0
In terms of the fgh

What is sm_w(n) in the fgh?

(Note to self: remember to add a link to this (viewtopic.php?f=12&t=4131&p=83326#p83326) with a name like "sm and wm" to the fluffykitty conwaylife googology googology wiki page; I'm at school and they block wikia so I can't do it right now.)

Re: Ordinals in googology

Posted: September 27th, 2019, 9:41 am
by BlinkerSpawn
testitemqlstudop wrote:...This topic is taking over BOTH the LTCFC and TFNCAQ so now it deserves its own thread.

Primarily I'm concerned about large ordinals here.

For example, the first fixed point of ∂ |-> w∂ is w^w. Hence we can make a function:
f(i) = the i-th fixed point of ∂ |-> w∂
What would f(2) be? f(w)? Would the last fixed point of ∂ |-> w∂ be well-defined?
∂ |-> w∂ is the same as w^∂ |-> w^(1+∂), so f(i) is w^{i'th limit of successors} = w^(wi). The "last fixed point" does not exist because there is no greatest ordinal.

Re: Ordinals

Posted: September 27th, 2019, 10:20 am
by PkmnQ
Moosey wrote: The fgh, sgh, and HH are all defined in the general format
f_n(x) =

Code: Select all

x, n = 0
f_n[x](x), n a lim ord
[something involving f_n-1(x)] otherwise
What if I...

Ę_m_n(x) =

Code: Select all

x, m=0 and n=0
Ę_m_n[x](x), n a limit ord
Ę_m_n-1(x+1) n not a limit ord
Ę_m[x]_m(x), n=0 and m a limit ord
Ę_m-1_m(x+1), n=0 and m not a limit ord

Code: Select all

Ę_w_w(1)
Ę_w_1(1)
Ę_w_0(2)
Ę_2_w(2)
Ę_2_2(2)
Ę_2_1(3)
Ę_2_0(4)
Ę_1_2(5)
Ę_1_1(6)
Ę_1_0(7)
Ę_0_1(8)
Ę_0_0(9)
9

Code: Select all

Ę_w_w^2+1(2)
Ę_w_w^2(3)
Ę_w_w3(3)
Ę_w_9(3)
Ę_w_0(12)
Ę_12_w(12)
Ę_12_12(12)
Ę_12_0(24)
Ę_11_12(25)
Ę_11_0(37)
Ę_10_11(38)
Ę_10_0(49)
Ę_9_10(50)
Ę_9_0(60)
Ę_8_9(61)
Ę_8_0(70)
Ę_7_8(71)
Ę_7_0(79)
Ę_6_7(80)
Ę_6_0(87)
Ę_5_6(88)
Ę_5_0(94)
Ę_4_5(95)
Ę_4_0(100)
Ę_3_4(101)
Ę_3_0(105)
Ę_2_3(106)
Ę_2_0(109)
Ę_1_2(110)
Ę_1_0(112)
Ę_0_1(113)
Ę_0_0(114)
114

Re: Ordinals in googology

Posted: September 27th, 2019, 8:10 pm
by testitemqlstudop
Any limit ordinal ∂ is the supremum of all elements in its fundamental sequence?

Re: Ordinals in googology

Posted: September 28th, 2019, 2:18 am
by testitemqlstudop
BlinkerSpawn wrote:
∂ |-> w∂ is the same as w^∂ |-> w^(1+∂), so f(i) is w^{i'th limit of successors} = w^(wi). The "last fixed point" does not exist because there is no greatest ordinal.
There has to be something wrong with that logic, because if ∂ = w2 = w+w, then it is implied that w+w = w(w+w), or w+w = w^2+w^2.

Re: Ordinals in googology

Posted: September 28th, 2019, 7:05 am
by Moosey
testitemqlstudop wrote:
BlinkerSpawn wrote:
∂ |-> w∂ is the same as w^∂ |-> w^(1+∂), so f(i) is w^{i'th limit of successors} = w^(wi). The "last fixed point" does not exist because there is no greatest ordinal.
There has to be something wrong with that logic, because if ∂ = w2 = w+w, then it is implied that w+w = w(w+w), or w+w = w^2+w^2.
You can't just assign arbitrary values unless you follow the such that.
Of course it won't work for ∂ = w2
testitemqlstudop wrote:Any limit ordinal ∂ is the supremum of all elements in its fundamental sequence?
By definition, yes.

Re: Ordinals in googology

Posted: September 30th, 2019, 4:43 pm
by BlinkerSpawn
testitemqlstudop wrote:
BlinkerSpawn wrote:
∂ |-> w∂ is the same as w^∂ |-> w^(1+∂), so f(i) is w^{i'th limit of successors} = w^(wi). The "last fixed point" does not exist because there is no greatest ordinal.
There has to be something wrong with that logic, because if ∂ = w2 = w+w, then it is implied that w+w = w(w+w), or w+w = w^2+w^2.
Of course w2 doesn't work, it's not of form w^(wa).

Re: Ordinals in googology

Posted: September 30th, 2019, 4:51 pm
by Moosey
BlinkerSpawn wrote:
testitemqlstudop wrote:
BlinkerSpawn wrote:
∂ |-> w∂ is the same as w^∂ |-> w^(1+∂), so f(i) is w^{i'th limit of successors} = w^(wi). The "last fixed point" does not exist because there is no greatest ordinal.
There has to be something wrong with that logic, because if ∂ = w2 = w+w, then it is implied that w+w = w(w+w), or w+w = w^2+w^2.
Of course w2 doesn't work, it's not of form w^(wa).
Moosey wrote:
testitemqlstudop wrote:
BlinkerSpawn wrote:
∂ |-> w∂ is the same as w^∂ |-> w^(1+∂), so f(i) is w^{i'th limit of successors} = w^(wi). The "last fixed point" does not exist because there is no greatest ordinal.
There has to be something wrong with that logic, because if ∂ = w2 = w+w, then it is implied that w+w = w(w+w), or w+w = w^2+w^2.
You can't just assign arbitrary values unless you follow the such that.
Of course it won't work for ∂ = w2

Re: Ordinals in googology

Posted: October 3rd, 2019, 4:55 pm
by Moosey
Does this notation I created do what I want?
It's sm but the g analogue preserves iteration

Code: Select all

ssm_n(x,y)=

x+y, n = 0

ssm_sg(x,n,x,y)(x,y), n a lim ord

ssm_n-1(x,xy) otherwise

define sg(m,n,x,y) =

n[ssm_sg(m-1,n,x)(x,y)], m>0,

x, m less than or = to 0


Remember:

Code: Select all

sm_n(x)=

x+1, n = 0
sm_g(x,n,x)(x), n a lim ord
sm^x_n-1(x) otherwise

define g(m,n,x) =
n[sm_g(m-1,n,x)(x)], m>0,
x, m less than or = to 0
Speaking of which, I should demonstrate the awesome power of sm:
sm_w(3)=

Code: Select all

sm_g(3,w,3)(3)=
sm_w[sm_g(2,w,3)(3)](3)=
sm_w[sm_w[sm_g(1,w,3)(3)](3)](3)=
sm_w[sm_w[sm_w[sm_g(0,w,3)(3)](3)](3)=
sm_w[sm_w[sm_w[sm_3(3)](3)](3)=
f_(f_(f_(f_3(3))(3))(3))(3))

f_3(3)= f_2(f_2(f_2(3)))

f_2(3)=(2^3)*3 = 24

f_3(3) = f_2(f_2(24)) =
f_2((2^24)*24) =
f_2(402653184) =
(2^402653184)*402653184

f_(f_(f_(f_3(3))(3))(3))(3)) =
f_(f_(f_(2^402653184)*402653184(3))(3))(3)) ...

>f_(f_(2{2^402653184*402653184-1}3)(3))(3))... (using bowers' notation)

> 2{2{2{2^402653184*402653184-1}3-1}3-1}3
So yeah...

Code: Select all

#C the 3-1's aren't twos -- you have to follow order of operations.
#C presumably more knuth arrows = higher priority in order of operations, but the point is that you're subtracting from the 2{huge thing}3 mess
x = 167, y = 24, rule = B/S012345678
63bobob3ob3ob3ob3ob3obob3obobo$63bobobobo3bobo3bo5bobobobobobo$63b3obo
bob3ob3ob3ob3obob3ob3o$65bobobobo3bobo3bo3bobobobo3bo$65bob3ob3ob3ob3o
b3obob3o3bo$59b3o39bobob3ob3ob3ob3ob3obob3obobo5bo$61bo39bobobobo3bobo
3bo5bobobobobobo5bo$59b3o35bobob3obobob3ob3ob3ob3obob3ob3ob3obo$59bo
38bo4bobobobo3bobo3bo3bobobobo3bo5bo$51b3o2bo2b3o35bobo3bob3ob3ob3ob3o
b3obob3o3bo5bo2b3o5bo$53bob3o86bo5bo$51b3o2bo85b3ob3obo$51bo4bo87bo5bo
$43b3o2bo2b3o2bo85b3o5bo2b3o5bo$45bob3o105bo5bo$43b3o2bo104b3ob3obo$
25b2o16bo4bo106bo5bo$14bob3obo6b2o6b3o2bo2b3o2bo104b3o5bo2b3o$b2o2b2o
6bo4bo2bo7bo7bob3o124bo$bo2b2o7bo2b3o2bo5b2o6b3o2bo123b3o$2o2bo3bobo2b
o4bo2bo3b2o8bo4bo125bo$7bo3bo2bob3obo14b3o2bo123b3o$7bobobo13b5o$8b3o!



while f_w(3) = (2^402653184)*402653184. When you have limit ordinals, sm becomes much much much more powerful than the fgh, even though it behaves the same for finite numbers

PkmnQ wrote:
September 27th, 2019, 10:20 am
Moosey wrote: The fgh, sgh, and HH are all defined in the general format
f_n(x) =

Code: Select all

x, n = 0
f_n[x](x), n a lim ord
[something involving f_n-1(x)] otherwise
What if I...

Ę_m_n(x) =

Code: Select all

x, m=0 and n=0
Ę_m_n[x](x), n a limit ord
Ę_m_n-1(x+1) n not a limit ord
Ę_m[x]_m(x), n=0 and m a limit ord
Ę_m-1_m(x+1), n=0 and m not a limit ord
Well done
Here's a stronger sm analogue:

smĘ_m_n(x)=

Code: Select all

x+1, m=0 and n=0
smĘ_m_ge(x,n,x)(x), n a lim ord
smĘ^x_m_n-1(x), n not a lim ord
smĘ_ge(x,m,x)_m(x), n=0 and m a lim ord
smĘ_m-1_m(x+1), n=0 and m not a lim ord

define ge(m,n,x) =
n[smĘ_ge(m-1,n,x)(x)], m>0,
x, m less than or = to 0
You could call it the "hardy strong-moose Ę hierarchy"

Re: Ordinals in googology

Posted: October 4th, 2019, 3:03 pm
by Moosey
Anyone care to evaluate
smĘ_w_w(1)?

Re: Ordinals in googology

Posted: October 4th, 2019, 6:31 pm
by Moosey
Here's a new thing

The decord (decreasing ordinal) sequence:

start with an ordinal c as term 1.
Rules:
Iff term n is 0: stop
Else:
Iff term n (call it a) is a limit ord, whose minimum "term" (eg. in w^w + w^4 + w2 = w^w + w^4 + w + w, the minimum term is w) is b, and for whom the rest of the terms of a are called r:
term n+1 = r+b[n] + b[n-1] + b[n-2] ... b[1],
Else:
Term n+1 = a-1

Demo:

Code: Select all

w+1
w
3
2
1
0

Code: Select all

w^w+1
w^w
w^2+w
w^2+6
w^2+5
w^2+4
w^2+3
w^2+2
w^2+1
w^2 (10th term)
w55
w54+66
...
w54 (78th term)
w53+3081
...
w53 (3160th term)
...

Obviously this leads to a new function:
the term in the decord sequence for the ordinal a at which it becomes zero = dco(a) (decord)
dco(w+1) = 6, and dco(w^w+1) is ~tri^53(3160), give or take an iteration or two, where tri(n) = the nth triangular number. This suggests a reasonable bound of dco(w^w+1) << 3160^106 (though not in a googolplex-vs-g_64 kind of <<)

Since a[n] < a for all finite n, all decord sequences strictly decrease over time. Thus, dco(a) is finite for all a

Re: Ordinals in googology

Posted: October 4th, 2019, 8:51 pm
by PkmnQ
Moosey wrote:
October 4th, 2019, 3:03 pm
Anyone care to evaluate
smĘ_w_w(1)?
I will, after I see what I can do with
Moosey wrote:
October 4th, 2019, 6:31 pm
Here's a new thing

The decord (decreasing ordinal) sequence:

start with an ordinal c as term 1.
Rules:
Iff term n is 0: stop
Else:
Iff term n (call it a) is a limit ord, whose minimum "term" (eg. in w^w + w^4 + w2 = w^w + w^4 + w + w, the minimum term is w) is b, and for whom the rest of the terms of a are called r:
term n+1 = r+b[n] + b[n-1] + b[n-2] ... b[1],
Else:
Term n+1 = a-1

Demo:

Code: Select all

w+1
w
3
2
1
0

Code: Select all

w^w+1
w^w
w^2+w
w^2+6
w^2+5
w^2+4
w^2+3
w^2+2
w^2+1
w^2 (10th term)
w55
w54+66
...
w54 (78th term)
w53+3081
...
w53 (3160th term)
...

Obviously this leads to a new function:
the term in the decord sequence for the ordinal a at which it becomes zero = dco(a) (decord)
dco(w+1) = 6, and dco(w^w+1) is ~tri^53(3160), give or take an iteration or two, where tri(n) = the nth triangular number. This suggests a reasonable bound of dco(w^w+1) << 3160^106 (though not in a googolplex-vs-g_64 kind of <<)

Since a[n] < a for all finite n, all decord sequences strictly decrease over time. Thus, dco(a) is finite for all a

Re: Ordinals in googology

Posted: October 4th, 2019, 9:10 pm
by PkmnQ
Here’s what I see about dco(n) right now.

₩(w) = 0
₩(w+1) = 1
₩(w^2+14233221) = 14233221
n < w dco(n) = n+1
w < n < w2 dco(n) = 2₩(n+1)+1

Re: Ordinals in googology

Posted: October 4th, 2019, 9:15 pm
by PkmnQ
Moosey wrote:
October 3rd, 2019, 4:55 pm

smĘ_m_n(x)= (m and n)

Code: Select all

x+1, m=0 and n=0
smĘ_m_ge(x,n,x)(x), n a lim ord
smĘ^x_m_n-1(x), n not a lim ord
smĘ_ge(x,m,x)_m(x), n=0 and m a lim ord
smĘ_m-1_m(x+1), n=0 and m not a lim ord

define ge(m,n,x) =
n[smĘ_ge(m-1,n,x)(x)], m>0, (you only provided m)
x, m less than or = to 0
You could call it the "hardy strong-moose Ę hierarchy"

Re: Ordinals in googology

Posted: October 5th, 2019, 9:11 am
by Moosey
PkmnQ wrote:
October 4th, 2019, 9:10 pm
Here’s what I see about dco(n) right now.

₩(w) = 0
₩(w+1) = 1
₩(w^2+14233221) = 14233221
n < w dco(n) = n+1
w < n < w2 dco(n) = 2₩(n+1)+1
what is the definition of ₩?

Re: Ordinals in googology

Posted: October 5th, 2019, 10:17 pm
by PkmnQ
Moosey wrote:
October 5th, 2019, 9:11 am
PkmnQ wrote:
October 4th, 2019, 9:10 pm
Here’s what I see about dco(n) right now.

n < w dco(n) = n+1
w < n < w2 dco(n) = 2₩(n+1)+1
what is the definition of ₩?
I don’t reslly know how to give a definition, so that’s why I put
₩(w) = 0
₩(w+1) = 1
₩(w^2+14233221) = 14233221