Leave these fields empty (spam trap):
Name
You can leave this blank to post anonymously, or you can create a Tripcode by using the format Name#Password
Comment
[i]Italic Text[/i]
[b]Bold Text[/b]
[spoiler]Spoiler Text[/spoiler]
>Highlight/Quote Text
[pre]Preformatted & Monospace Text[/pre]
[super]Superset Text[/super]
[sub]Subset Text[/sub]
1. Numbered lists become ordered lists
* Bulleted lists become unordered lists
File

Sandwich


Harm Reduction Notes for the COVID-19 Pandemic

Ordering of large finite numbers

Reply
- Fri, 10 Jan 2020 12:02:42 EST zCQgmjjm No.79738
File: 1578675762414.jpg -(72414B / 70.72KB, 620x466) Thumbnail displayed, click image for full size. Ordering of large finite numbers
I'm looking for some sort of ordering or classification of finite numbers.
Not infinite numbers, I know about ordinals but I find them kind of boring by themselves.

So any of these numbers would be strictly smaller than the first infinite ordinal.
For instance

  1. Trivial numbers like 1, 42, 4354353843745897394759347592347 that can easily notated using a number base
  2. Numbers that can be represented by scientific notation like 2^256, 10^100, etc...
  3. Using power tower (up arrow notation) like 2↑↑↑↑3, grahams number (g64)
  4. Chained arrow notation like 2 → 3 → 4 → 5 → 6
  5. Numbers that can not be notated explicitly but can be referenced by their fast growing function representation like TREE(3)

is this possible ???
6. Something that has to come after that, like numbers that are too large to be referenced by a function representation but can be deduced by other means like a process of describing a function like using self modifying code...


Will be positing about these notations and functions once I understand them well enough.
>>
trypto - Fri, 10 Jan 2020 19:02:29 EST dEGML8x1 No.79739 Reply
>>79738
Good question. I only know about up to #3, and only from a numberphile vid

It sounds like what you're asking is:
Is there a limit on the size of a finite number that can be represented through some notation.

Is that kinda what you're asking? Or just, what's after #5?

I didn't know about the tree function, but just looked it up. That's pretty wild. I don't understand yet.
>>
Nathaniel Sublinghare - Fri, 10 Jan 2020 22:11:03 EST zCQgmjjm No.79741 Reply
>>79739
I learned about this from Numerphile too, they also have a video about TREE(3) and even talk about things like TREE(g64)

I'm not quite sure what I am asking perhaps if numbers that are too large to be useful to be used. I guess every finite number that can be reasoned with must have a function representation (if you know where is the proof for that)

Perhaps the 6th category on my list would be numbers for which it's simplest function representation is too complex.
For instance take the closest prime number to TREE(3), or almost any other number with the same magnitude.

What's also interesting if there is any way to compare numbers in the latter categories. For instance can we come up with some numbers that serve as a lower and upper limit for TREE(3)?
>>
Nathaniel Sublinghare - Fri, 10 Jan 2020 22:30:23 EST zCQgmjjm No.79742 Reply
Sort of answering my own questions here except that the links here go a bit over my head.
https://googology.wikia.org/wiki/Subcubic_graph_number
> Adam P. Goucher has shown that SSCG(2) << TREE(3) << SSCG(3).

and
https://googology.wikia.org/wiki/Transcendental_integer
>If n is an integer then we call it transcendental iff the following holds: let M be a Turing machine, such that there is proof in ZFC of length at most 2^1000
showing that M halts.
This seems a little arbitrary though.
>>
trypto - Fri, 10 Jan 2020 22:59:10 EST dEGML8x1 No.79743 Reply
>>79741
>numbers that are too large to be useful to be used
Used how? I think even a number like 10^5000 is nonsense in any physics application. Whether you're talking about time or number of particles, exponents get to nonsense pretty quickly (I think 10^50 is also mostly useless, I just bumped it up a couple orders of magnitude to be sure).

>Perhaps the 6th category on my list would be numbers for which it's simplest function representation is too complex.
>For instance take the closest prime number to TREE(3), or almost any other number with the same magnitude.
Oh, OK. That's cool. "Too complex" has some implications, though. With infinite amount of space and brain, I'm sure you could prove that no finite number is too complex. But for normal people and a brain, maybe there's a limit?

I knida doubt there's a good answer. It's like a more complex version of 'what's the biggest number you can name" and then you take that number and add one. But in this case, it's a function that you're gradually improving.

> Adam P. Goucher has shown that SSCG(2) << TREE(3) << SSCG(3).
woah

>If n is an integer then we call it transcendental iff the following holds: let M be a Turing machine, such that there is proof in ZFC of length at most 2^1000

Looked at the link. don't understand this 1 fucking bit lol. Keep at it, amigo
>>
Lydia Drorryledging - Wed, 29 Jan 2020 18:27:08 EST zCQgmjjm No.79754 Reply
>>79743

>If n is an integer then we call it transcendental iff the following holds: let M be a Turing machine, such that there is proof in ZFC of length at most 2^1000


>Looked at the link. don't understand this 1 fucking bit lol. Keep at it, amigo

As I understand it it's simply a number that is larger than a turing machine of 2^1000 elements can run for.
A Turing machine is essentially an academic version of a computer.
And that number would be the longest running (finitely running) program that finishes inside such a machine with 2^1000 elements of memory.

This is not really precise as it's non trivial to determine the longest running program for any computer but largely irrelevant for the discussion as a turing machine with 2^1000 "memory" or a equivalent computer can never exist in physical space.
The main point here seems to be that every turing machine of X length must have some finite amount of maximum steps Y it can take.
Consider for a minute that we had a magical mechanism that gives you that program. Such a longest running program (or set of programs) must exist for every turing machine with finite length.

Anyway what kind of bugs me is that they picked 2^1000 for the memory cutoff but not something more rooted to something else.
Like the amount of atoms in the observable universe (10^82) or something.


Kind of weird that 10^82 is so minuscule even in category 2.
>>
Lydia Drorryledging - Wed, 29 Jan 2020 18:40:15 EST zCQgmjjm No.79755 Reply
Another thing would be what's the largest quantity that can be constructed in physics.
In the regular universe maybe it's the number of plank seconds from the big bang to the heat death.
From google the seconds till heat death is ~10^100 and a plank second is 5.3*10^44 so that will be 1.85*10^143.

Quite minuscule but the multiverse is a weird place.
AFIK the amount of universes created is growing more than exponentially from each plank second to the next and it is my hope that numbers that can not be notated with scientific notation might occur there.
>>
Phyllis Grandcocke - Fri, 31 Jan 2020 04:00:03 EST wrOD+yZL No.79756 Reply
Okay so picture a number line on an orbit, spiraling around a central point [0,0,0,0,0]
[distance from center,change in angle,total prime factors,rotations,bearing]

so 1 is:
1 from center
1 change in angle
0 prime factors
0 rotations
2 bearing

which gives [1,1,0,0,2]

and n+1=n etc
[1,1,n+1,n-1,n+2]
>>
Nicholas Hedgewater - Wed, 05 Feb 2020 15:38:36 EST aTetnkWi No.79766 Reply
>>79756
>n+1 =n

> n + 1 = n
>-n + (n + 1) = -n + n
> 1 = 0

Uh... I think I broke my number line. Guys help.
>>
Augustus Drenkinlet - Mon, 10 Feb 2020 00:15:26 EST wrOD+yZL No.79768 Reply
>>79766

its more of a number spiral.

>-n + (n + 1) = -n + n

your problem was taking n as a constant but not assigning a value, its n0, n1, n2 per iteration.

you're trying to cancel infinity with negative infinity, which when you normally do it also gives you the result 1=0
>>
Jack Sossleway - Tue, 25 Feb 2020 18:32:47 EST PGs4WbUU No.79772 Reply
>>79768
Oh are you guys inverting circles? Is that what this is?
>>
Isabella Nobblechark - Tue, 03 Mar 2020 21:50:52 EST SEEbWuZe No.79787 Reply
There are some uncomputable functions, i.e. functions that grows faster than any Turing Machine can.
https://en.wikipedia.org/wiki/Busy_beaver
This means there are number of this sequence for which it is impossible to prove an upper bound, for any logical system.
>>79755
If you assume that there is no super Turing machines in the physical world, then uncomputable functions are literally bigger than the largest quantity that can be constructed using physics.

Report Post
Reason
Note
Please be descriptive with report notes,
this helps staff resolve issues quicker.