## Ordinals in googology

- testitemqlstudop
**Posts:**1241**Joined:**July 21st, 2016, 11:45 am**Location:**in catagolue-
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### Ordinals in googology

...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)

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)

Last edited by testitemqlstudop on September 27th, 2019, 3:08 am, edited 1 time in total.

- Moosey
**Posts:**2789**Joined:**January 27th, 2019, 5:54 pm**Location:**A house, or perhaps the OCA board. Or [click to not expand]-
**Contact:**

### Re: Ordinals

I believe that f(2) would be w^w^w judging by the fact that e_1 = e_0^^wtestitemqlstudop 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)

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
```

Code: Select all

`n/oddpart(n)`

Code: Select all

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

Last edited by Moosey on October 5th, 2019, 9:19 am, edited 2 times in total.

I am a prolific creator of many rather pathetic googological functions

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

- testitemqlstudop
**Posts:**1241**Joined:**July 21st, 2016, 11:45 am**Location:**in catagolue-
**Contact:**

### Re: Ordinals

Which f

- Moosey
**Posts:**2789**Joined:**January 27th, 2019, 5:54 pm**Location:**A house, or perhaps the OCA board. Or [click to not expand]-
**Contact:**

### Re: Ordinals

the first onetestitemqlstudop wrote:Which f

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
```

f_n(x) =

Code: Select all

```
x+1, n = 0
f_n[x](x), n a lim ord
f^x_n-1(x) otherwise
```

h_n(x) =

Code: Select all

```
x, n = 0
h_n[x](x), n a lim ord
h_n-1(x+1) otherwise
```

g_n(x)=

Code: Select all

```
x, n = 0
g_n[x](x), n a lim ord
g_n-1(x)+1 otherwise
```

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}testitempimplesquidIDyoulot wrote:f(i) =the 1st fixed point of ∂ |-> {w, ∂, i, 1}

...if I got BEAF correct.

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

I am a prolific creator of many rather pathetic googological functions

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

- Hdjensofjfnen
**Posts:**1370**Joined:**March 15th, 2016, 6:41 pm**Location:**r cis θ

### Re: Ordinals

I've posted the

**fifth**post in this thread."A man said to the universe:

'Sir, I exist!'

'However,' replied the universe,

'The fact has not created in me

A sense of obligation.'" -Stephen Crane

'Sir, I exist!'

'However,' replied the universe,

'The fact has not created in me

A sense of obligation.'" -Stephen Crane

Code: Select all

```
x = 7, y = 5, rule = B3/S2-i3-y4i
4b3o$6bo$o3b3o$2o$bo!
```

- testitemqlstudop
**Posts:**1241**Joined:**July 21st, 2016, 11:45 am**Location:**in catagolue-
**Contact:**

### Re: Ordinals

Ok screw it that's not what I mean

- Moosey
**Posts:**2789**Joined:**January 27th, 2019, 5:54 pm**Location:**A house, or perhaps the OCA board. Or [click to not expand]-
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### Re: Ordinals

Well doneHdjensofjfnen wrote:I've posted thefifthpost in this thread.

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)

I am a prolific creator of many rather pathetic googological functions

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

- Moosey
**Posts:**2789**Joined:**January 27th, 2019, 5:54 pm**Location:**A house, or perhaps the OCA board. Or [click to not expand]-
**Contact:**

### Re: Ordinals in googology

Here's what one could call the

hardy, fast and slow-growing hierarchy:

hsf_n(x)=
More fgh variants:

f(x) =
(Grows significantly faster than the normal fgh)

f(x) =
(An extension of the previous)

Or something similar:

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

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
```

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
```

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
```

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)
```

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

- testitemqlstudop
**Posts:**1241**Joined:**July 21st, 2016, 11:45 am**Location:**in catagolue-
**Contact:**

### Re: Ordinals in googology

Congratulations, we now need an fgh for the fgh.

- Moosey
**Posts:**2789**Joined:**January 27th, 2019, 5:54 pm**Location:**A house, or perhaps the OCA board. Or [click to not expand]-
**Contact:**

### Re: Ordinals in googology

Nah, we can approximate values in what I'll call the weak moose-growing hierarchytestitemqlstudop wrote:Congratulations, we now need an fgh for the fgh.

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)
```

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
```

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.)

Last edited by Moosey on September 27th, 2019, 4:57 pm, edited 1 time in total.

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

- BlinkerSpawn
**Posts:**1929**Joined:**November 8th, 2014, 8:48 pm**Location:**Getting a snacker from R-Bee's

### Re: Ordinals in googology

∂ |-> 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.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?

### Re: Ordinals

What if I...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`

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
```

Code: Select all

```
x = 12, y = 12, rule = AnimatedPixelArt
4.P.qREqWE$4.2tL3vSvX$4.qREqREqREP$4.vS4vXvS2tQ$2.qWE2.qREqWEK$2.2vX
2.vXvSvXvStQtL$qWE2.qWE2.P.K$2vX2.2vX2.tQ2tLtQ$qWE4.qWE$2vX4.2vX$2.qW
EqWE$2.4vX!
```

- testitemqlstudop
**Posts:**1241**Joined:**July 21st, 2016, 11:45 am**Location:**in catagolue-
**Contact:**

### Re: Ordinals in googology

Any limit ordinal ∂ is the supremum of all elements in its fundamental sequence?

- testitemqlstudop
**Posts:**1241**Joined:**July 21st, 2016, 11:45 am**Location:**in catagolue-
**Contact:**

### Re: Ordinals in googology

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.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.

- Moosey
**Posts:**2789**Joined:**January 27th, 2019, 5:54 pm**Location:**A house, or perhaps the OCA board. Or [click to not expand]-
**Contact:**

### Re: Ordinals in googology

You can't just assign arbitrary values unless you follow the such that.testitemqlstudop wrote: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.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.

Of course it won't work for ∂ = w2

By definition, yes.testitemqlstudop wrote:Any limit ordinal ∂ is the supremum of all elements in its fundamental sequence?

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

- BlinkerSpawn
**Posts:**1929**Joined:**November 8th, 2014, 8:48 pm**Location:**Getting a snacker from R-Bee's

### Re: Ordinals in googology

Of course w2 doesn't work, it's not of form w^(wa).testitemqlstudop wrote: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.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.

- Moosey
**Posts:**2789**Joined:**January 27th, 2019, 5:54 pm**Location:**A house, or perhaps the OCA board. Or [click to not expand]-
**Contact:**

### Re: Ordinals in googology

BlinkerSpawn wrote:Of course w2 doesn't work, it's not of form w^(wa).testitemqlstudop 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.

Moosey wrote:You can't just assign arbitrary values unless you follow the such that.

∂ |-> 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.

Of course it won't work for ∂ = w2

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

- Moosey
**Posts:**2789**Joined:**January 27th, 2019, 5:54 pm**Location:**A house, or perhaps the OCA board. Or [click to not expand]-
**Contact:**

### Re: Ordinals in googology

Does this notation I created do what I want?

It's sm but the g analogue preserves iteration

Remember:
Speaking of which, I should demonstrate the awesome power of sm:

sm_w(3)=
So yeah...
while f_w(3) = (2^402653184)*402653184. When you have limit ordinals, sm becomes much much

Here's a stronger sm analogue:

smĘ_m_n(x)=
You could call it the "hardy strong-moose Ę hierarchy"

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
```

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
```

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!
```

*much*more powerful than the fgh, even though it behaves the same for finite numbersWell donePkmnQ wrote: ↑September 27th, 2019, 10:20 amWhat if I...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`

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`

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
```

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

- Moosey
**Posts:**2789**Joined:**January 27th, 2019, 5:54 pm**Location:**A house, or perhaps the OCA board. Or [click to not expand]-
**Contact:**

### Re: Ordinals in googology

Anyone care to evaluate

smĘ_w_w(1)?

smĘ_w_w(1)?

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

- Moosey
**Posts:**2789**Joined:**January 27th, 2019, 5:54 pm**Location:**A house, or perhaps the OCA board. Or [click to not expand]-
**Contact:**

### Re: Ordinals in googology

Here's a new thing

The decord (

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:

Obviously this leads to a new function:

the term in the decord sequence for the ordinal a at which it becomes zero = dco(a) (

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

The decord (

**dec**reasing**ord**inal) 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) (

**d**e**co**rd)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

Last edited by Moosey on October 5th, 2019, 9:08 am, edited 1 time in total.

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

### Re: Ordinals in googology

I will, after I see what I can do with

Moosey wrote: ↑October 4th, 2019, 6:31 pmHere's a new thing

The decord (decreasingordinal) 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

Code: Select all

```
x = 12, y = 12, rule = AnimatedPixelArt
4.P.qREqWE$4.2tL3vSvX$4.qREqREqREP$4.vS4vXvS2tQ$2.qWE2.qREqWEK$2.2vX
2.vXvSvXvStQtL$qWE2.qWE2.P.K$2vX2.2vX2.tQ2tLtQ$qWE4.qWE$2vX4.2vX$2.qW
EqWE$2.4vX!
```

### Re: Ordinals in googology

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

₩(w) = 0

₩(w+1) = 1

₩(w^2+14233221) = 14233221

n < w dco(n) = n+1

w < n < w2 dco(n) = 2₩(n+1)+1

Code: Select all

```
x = 12, y = 12, rule = AnimatedPixelArt
4.P.qREqWE$4.2tL3vSvX$4.qREqREqREP$4.vS4vXvS2tQ$2.qWE2.qREqWEK$2.2vX
2.vXvSvXvStQtL$qWE2.qWE2.P.K$2vX2.2vX2.tQ2tLtQ$qWE4.qWE$2vX4.2vX$2.qW
EqWE$2.4vX!
```

### Re: Ordinals in googology

Moosey wrote: ↑October 3rd, 2019, 4:55 pm

smĘ_m_n(x)= (m and n)You could call it the "hardy strong-moose Ę hierarchy"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`

Code: Select all

```
x = 12, y = 12, rule = AnimatedPixelArt
4.P.qREqWE$4.2tL3vSvX$4.qREqREqREP$4.vS4vXvS2tQ$2.qWE2.qREqWEK$2.2vX
2.vXvSvXvStQtL$qWE2.qWE2.P.K$2vX2.2vX2.tQ2tLtQ$qWE4.qWE$2vX4.2vX$2.qW
EqWE$2.4vX!
```

- Moosey
**Posts:**2789**Joined:**January 27th, 2019, 5:54 pm**Location:**A house, or perhaps the OCA board. Or [click to not expand]-
**Contact:**

### Re: Ordinals in googology

what is the definition of ₩?

My CA rules can be found here

Also, the tree game

Bill Watterson once wrote: "How do soldiers killing each other solve the world's problems?"

### Re: Ordinals in googology

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

Code: Select all

```
x = 12, y = 12, rule = AnimatedPixelArt
4.P.qREqWE$4.2tL3vSvX$4.qREqREqREP$4.vS4vXvS2tQ$2.qWE2.qREqWEK$2.2vX
2.vXvSvXvStQtL$qWE2.qWE2.P.K$2vX2.2vX2.tQ2tLtQ$qWE4.qWE$2vX4.2vX$2.qW
EqWE$2.4vX!
```