10,252 Pages

• Mensors, tensors with tensor dimensions

September 20, 2018 by Ubersketch

In my last post I mentioned something called mensors. This is a concept I came up with after reading Bowers' Hypernomial page and writing tensors like so: [a,b[2]d,e[3]f,g[2]h,i]  Mensors are basically tensors except they use tensor dimensions and mensor dimensions. For example, a mensor could be like [a,b[c,d]e,f] I don't know what this would be used for but its something I'd like to mension.

• My unnamed function is finished?

September 20, 2018 by Ubersketch

So this is pretty exciting, since I've officially found semi-formal rules for my function I explained a few days ago. Here are the rules. 1. f(a){[1]b} = f(f(... where the function is applied b times to a.

2. f(a){[b]c} = f(a){[b-1]a[b-1]a...} where there are c as.

3. f(a){[b]c[d]e...} is evaluated like so: f(a){[b]c} = f(a) in f(a){[d]e} = f(a) in... until it is fully evaluated.

f(a) can be any unary function, but to be self-contained, a+ = the successor of a.  For example: 2+{[1]3[1]2} = f(2){[1]2} = 8

I have plans to getting around to mensor function inputs and mensor seperators/limit ordinal seperators.

• C function

September 19, 2018 by Ynought

imagine a map that can be coloured with n colors, given a set of rules that give the smallest non infinite amount of maps coloured that doesn’t violate any of the rules

• I need some help with this unnamed notation.

September 18, 2018 by Ubersketch

So the other day I posted this to reddit.
I haven't well defined it, and I haven't even made any remotely formal rules. Here goes nothing. Only the first entry can be followed with a +.
{a} {a+} = a+1
{a+[1]b} = {a+}+}...} with b pairs of braces. {a+[1]b[1]c} = {a+[1]b} c times (i.e. {a+[1]b[1]c} uses {a+[1]b} as the input for a c times)
{a+[2]b} = {a+[1]a[1]a...} with a [1]s.  @a = {a+[2]b} {a+[3]b} = {@...a} with b @
This goes on for a long (uotc) while. I'd love it if you guys pointed out and flaws, undefinedness, and left ideas for extensions.

I suspect this function has the level of $${\varepsilon_0}$$

• A dump of my (on hold) shot at well-defining pi definition

September 16, 2018 by Syst3ms

I'll be straightforward : I don't know how to actually formalize stuff, so I won't bother saying this is formal, since it is extremely far from it. I'm just dumping this here so that it doesn't get lost in some Discord channel somewhere. This is definitely better than "transfinite FSes", however.

• Infinite-order Xi Function

September 10, 2018 by ChromaticiT

The ITTM Busy beaver is to the Busy beaver as the Infinite-order Xi function is to the Xi function.

We define a combinatory language with countably infinite symbols such that there is a bijection between these symbols and the natural numbers (including 0).

0x := $$\iota$$ x as in Chris Barker's iota combinator. Nxyzw := Where N is a natural number denoting a symbol, $$\Omega$$xyz (as in the oracle combinator) if x is well founded, w otherwise.

Now that we have our language, let's just reuse the definition for the Xi function.

If we start with a string of n symbols and we beta-reduce it, the largest possible finite output is called $$\Xi_{\infty}$$(n)

• intervalux

September 6, 2018 by Grand fuzili

pense assim:

g Meameamealokkapoowa oompa = J

J2 = JJ

J3 = JJJ

INTEVALUX = Jg Meameamealokkapoowa oompa

grande numero

September 4, 2018 by DWither

The busy beaver function is uncomputable because it's impossible to make a program that will evaluate BB(n) in a finite time, right? And this is because the computer will never know if it's going 1to halt until it halts. Well, I'm pretty sure that someone has to have asked this before, but why can't you just check for loops in the behavior of the turing machine to know if it's going to keep going forever?

I thought of a step by step method to calculate BB(n) that definitely seems computable to me. I don't know enough about any programming language to directly show code though, so sorry about that. Tell me what I'm missing in all of this that makes my step by step method useless.

Step 1:

Take all the possible combinations of states you can ha…

• H-Boogol-Boogol Bit by bit: PART 2: Higher Order Logic

September 4, 2018 by DontDrinkH20

This is a series of blog posts I will make to try to generalize the understanding of H-boogol-boogol to the general public of this wiki. It's quite a mind boggling concept, but the kind of person on this website is likely the kind of person who can figure it out if they just read a couple different detailed descriptions. So just believe in yourself, and feel free to get out a notebook and start taking notes.

In the last blog post, Model Theory 101, we looked at systems of mathematics, and formalized the idea of one. We also formalized the idea of "this universe obeys the system of mathematics...." This idea is known as model theory. If you haven't read that yet, go read it now!

In the last blog post, you may remember that I kept putting (fi…

• Math number

September 2, 2018 by MilkyWay90

Let's create a "math language" called N.

In N, you can: Add (F(x, y) = +(x, y))

Subtract (F(x, y) = -(x, y))

Use natural numbers (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...)

Recurse (F(x, y) = F(x, -(y, 1)), F(x, 0) = +(x, x))

Variables (a0, a1, ..., b0, b1, ..., z0, z1, ..., aa0, aa1, ...)

Other functions (A(x) = x, F(x) = A(x + x))

Equality (equal sign, but no greater than or less than or not equal)

To define a function, do Function(v0, v1, v2, ..., vn) = (expression)

Note: Addition and Substraction can only hold 2 arguments. Use +(2, +(2, 2)) instead of +(2, 2, 2)

Now let's get into symbols!

F(x) = +(x, x) has 6 symbols because F(x) is 2 symbols (F, and x), = is one symbol, +(x, x) is 3 symbols (+, x, and x)

So Functions, natural nu…

September 2, 2018 by Nedherman1

I don't speak english.

Hello, today I showed my new notation that I call Arrow Destruction!
(I was inspired by Eners' notation)

Basic:

we can put more than two entries:

if we want, we can put more entries, which will follow this same logic

We can make this notation more powerful in this way:

we can also put more than two entries that also follow the same logic

However, it does not end here: ​​​​

I explained in two ways and I showed an example, if I did not understand, see [a]^n is different from [a]^[n], since in the first example "n" is not between [ ], which means that we will use the rule shown above to solve, in the second example, "n" is between [ ], that is, first we will solve using the basic rule and, after we know the result of [n], we wil…

• H-Boogol-Boogol Bit by bit: PART 1: Model Theory 101

September 1, 2018 by DontDrinkH20

This is a series of blog posts I will make to try to generalize the understanding of H-boogol-boogol to the general public of this wiki. It's quite a mind boggling concept, but the kind of person on this website is likely the kind of person who can figure it out if they just read a couple different detailed descriptions. So just believe in yourself, and feel free to get out a notebook and start taking notes.

If you know enough about mathematics folklore, you would know that Godel's incompleteness theorem shows that there are some things which just cannot be proven by a working system of mathematics; one that doesn't imply 0 = 1.

In model theory, a theory $$T$$ is that system of mathematics, aided by a language and a formal logic (but those ar…

• Gleive's U Notation - Part 1 of ? Parts

August 30, 2018 by Gleive

I originally planned to release this on September 1st, but meh.

While going through this notation, you might notice that despite being finished in 2018 (bruh), it doesn't really grow as spectacularly fast or introduces some amazing new stuff. This is just a normal googologist's normal work, and I just want to say that I've tried many thing in googology and this is just a milestone I want to keep.

Anyways, enough semi-emotional stuff, let's get into the notation.

I mentioned in my previous post that this notation was refined and revisited several times. The original version was an extension on down-arrow notation (Maybe because BEAF is based on up-arrows?), the second version used my own functions, and the third version meddled with polygonal …

• Some ZFC-Definable very big LCOs - uncomputable?

August 30, 2018 by DontDrinkH20

This time I'm going to be defining some really big ZFC-definable countable ordinals (some even provably exist). These are based on the principle of making ordinals which are not definable within another very "wide" structure, like a transitive model of ZFC or $$V_\kappa$$ for large enough $$\kappa$$.

Let $$\varphi$$ be a formula and let $$x$$ be a set. $$\varphi:x$$ if and only if $$\varphi(A)$$ if and only if $$A\in x$$ for all $$A$$.

Then, $$\theta_\alpha$$ is defined as the smallest ordinal $$\beta$$ such that there is no formula $$\varphi$$ such that $$\varphi^{V_\alpha}:\beta$$. That is, the smallest ordinal which is not definable within $$V_\alpha$$. Given a transitive structure $$\mathcal{M}=(M;\in)$$, $$\theta(\mathcal{M})$$ is the a…

• BM2 doesn't terminate.

August 28, 2018 by Bubby3

There is an infinite descending chain at (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,2,0,0)(5,1,1,1)(6,2,1,1)(7,3,1,1). Here is to show it is standard. The result is here. It is actually the last example on the calculator

• (0,0,0,0,0)(1,1,1,1,1)
• = (0,0,0,0)(1,1,1,1)(2,2,2,2)(3,3,3,3)...
• > (0,0,0,0)(1,1,1,1)(2,2,2,2)
• = (0,0,0,0)(1,1,1,1)(2,2,2,1)(3,3,3,1)...
• > (0,0,0,0)(1,1,1,1)(2,2,2,1)...
• = (0,0,0,0)(1,1,1,1)(2,2,2,0)(3,3,3,1)...
• > (0,0,0,0)(1,1,1,1)(2,2,2,0)
• = (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,4,1,1)(5,5,1,1)...
• > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,4,1,1)
• > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,4,1,0)
• > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,4,0,0)
• > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,3,1,1)
• > (0,0,0,0)(1,1,1,1)(2,2,1,1)(3,3,1,1)(4,3,1,0)
• > (…
• An important note to everyone: I never EVER create subpages for my user page

August 28, 2018 by PsiCubed2

There's a guy here who seems to be fond of spamming my personal space here with nonsense. He does this both by spamming the comment sections of my blog posts, and by creating fake subpages to my user page.

The first kind of abuse was already taken care of, by disabling the comments on all my blog posts.

As for the second kind - I'm taking care of it with this post, by announcing the following statement:

I never EVER create subpages to my username. Never did, and never will. So if you see such a page, you can be 100% certain that it is a fake. If you're an admin, please delete it. If you're a regular user, please ignore it because it most certainly isn't my work.

Thank you.

• Analysis DAN up to Z2

August 28, 2018 by Scorcher007

My analysis is based on the following assumption:
1) DAN is limit of П1n-CA0
{n,n(1,,1,2)2} ~ П11-CA0
{n,n(1,,,1,2)2} ~ П12-CA0
{n,n(1,,,,1,2)2} ~ П13-CA0
e.t.c.
2) Hypcos hypothetical analysis, that:
{n,n(1(1(1(1...(1,,2,,)...2,,)2,,)2,,)2)2} = {n,n(1(1',,2,,)2)2} ~ KP+Пn

The analysis was conducted on the basis of pattern in the stable ordinals, which are similar to smaller ordinals
Stable ordinals notation:
S[a+n] = La1La+n
S[a+S[a+1]] = La1La+Lb≺Lb+1
S[Ωa+1] = La1Lωa+1CK = (+)-stable (zoo 2.8)
S[Ia+1] = La1LIa+1 = inaccessibly-stable (zoo 2.11)
S[Ma+1] = La1LMa+1 = Mahlo-stable (zoo 2.12)
S[a2+1] = La1Lb1Lb+1 = doubly (a+1)-stable = (zoo 2.13)
S[aω+1] = ω-ly (a+1)-stable = nonprojectible (zoo 2.15)
S[aα+1] = α-ly (a+1)-stable
S[I-aα+1] = 1st inacces…

• Help

August 25, 2018 by GdawgGamerTV

I did coin the numbers in my blog posts but somehow they are getting turned into pages and it says I did it. Will somebody fix this?

• Heres my attempt to beat Utter Oblivion

August 25, 2018 by GdawgGamerTV

Heres my attempt at beating Jonathan Bowers' Utter Oblivian.

I call it Oblivous Plex

Utter OblivianUtter Oblivian * Graham's Number * (Bisuperiorgraham * Bisuperiorgraham * BIG FOOTGoogolplexianth * Oblivian * Oblivian * Oblivian * Googolplexian * 100Bisuperiorgraham * Bisuperiorgraham * Utter Oblivian

• Updating my superior series of numbers.

August 25, 2018 by GdawgGamerTV

My old coined numbers, a small Superior series, I will be adding onto it. This may contain very VERY large numbers, such as Bisuperiorgraham and Bisuperior-Rayo

Superior-quattordecillion =  QuattordecillionQuattordecillion

Superior-quindecillion = QuindecillionQuindecillion

Superior-sexdecillion = SexdecilionSexdecillion

Superior-septendecillion = SeptendecillionSeptendecillion

Superior-octodecillion = OctodecillionOctodecillion

Superior-googol = GoogolGoogolGoogol

Superior-googolplex = GoogolplexGoogolplexGoogolplex

Superior-rayo = Rayo's NumberRayo's Number

Bisuperior-million = MillionMillionMillion

Bisuperior-billion = BillionBillionBillion

Bisuperior-trillion = TrillionTrillionTrillion

Bis…

• I coin these 20 new numbers.

August 25, 2018 by GdawgGamerTV

Yearillion, as 10100,000,005

Gamma-raytillillion as 10150,762

Yihuh as 2,121,887,952

Yiplex as 999,057,912

Yifufi as 987,638,996

Yinin as 819,276,947

Yidid as 813,996,888

Yikin as 804,887,450

Yimen as 801,086,337

Yiunu as 67,956,111

Yicuc as 67,894

Yidud as 3,195

Yipup as 894

Yimum as 847

Yibase as 5.92342345666645584884585847645

Yiutu as 1.4

Short Yi-bucket as 0.0000000000000000000000000000007

Long Yi-bucket as 0.00000000000767

• How do I get past this?

August 25, 2018 by MachineGunSuper

Hello. I have made many ordinal functions that reached the Omega Fixed Point, but what's next? How do you get past the Omega Fixed Point?

August 25, 2018 by MachineGunSuper

I've got a few things to cover in this blog bost:

I already know what they mean, but in case you don't, it means doing an infinite amount of stuff in a finite amount of time.

A great example is with a cake:

Cut a cake in half, then wait 30 seconds before cutting the half in half. Then wait 15 seconds and cut again. and if you keep cutting into half and each time wait half as much, you would in a finite amount of time cut and infinite amount of slices and place them in a finite space.

My favourite examples are these:

If we built a machine that could display the digits of pi like this: Start with the 3, then in 30 seconds the 1 then in 15 the 4 and so on, we would display all the digits of pi. What would be on the screen now?

Second, if on a monit…

• Hierarchy of ordinals Z 2. Help me to understand

August 25, 2018 by Scorcher007

Help those who understand the hierarchy of ordinals to the level of Z2. Have I built the list of ordinals correctly?
(zoooo #) means number of ordinals in Madore's Zoo of ordinals

countable ordinal ~ equiv. cardinal ~ PTO
1• ω1CK = 1st admissible = П10-comprehension = (zoooo 2.1) ~ ω1 = 1st uncountable = 1st 1-regular ~ OCF→PTO ACA0
2• ω2CK = 2nd admissible ~ ω2 = 2nd uncountable = 2nd 1-regular ~ OCF→KPω
3• ωωCK = ω-th admissible = П11-comprehension = (zoooo 2.2) ~ ωω = ω-th uncountable = ω-th 1-regular ~ OCF→PTO П11-CA0
4• I ~ recursively inaccessible ~ weakly inaccessible = (zoooo 2.3) ~ OCF→PTO KPi
5• M ~ recursively Mahlo ~ weakly Mahlo = (zoooo 2.5) ~ OCF→PTO KPM
6• K ~ П3-reflection = 2-admissible = (zoooo 2.6) ~ П11-indescribable = 2-regular = weakl…

• Coining a new number

August 25, 2018 by GdawgGamerTV

I hereby state that I shall coin these three new numbers. King's dozen as 23, King's gross as 223 and Kingstillion as 1010001000050000450

• hi

August 24, 2018 by GdawgGamerTV

hello, everyone, I am Sven and I like numbers k  I also claimed some new numbers, wilkillion, welkillion, welkillillion and more, you can see them here https://sites.google.com/site/numberwikipedia/the-wilkinson-array

• my array notation

August 24, 2018 by DatoXx8

{n}=n^[n]n         n^[n]n=n^^^^...n arrows…^^n

{n,m}= {n,m-1} with {n,m-1} sets of { } brackets around n

{n,m,o} =}}}}}

{n,m,o,p}= }...{n,m,o,p-1}...}}}

n_0= n^[n]n!!!!!.....n^[n]n!!!!!.....n^[n]n!!!!!..............

itterating n^[n]n!!!!...n !s…!  =

n_1 is ((((((n_0)_0)...n_0 times…._0)))

n_2 is ((((((n_1)_1)...n_1 times…._1)))

….

a_b_c= a_(a_(a_…a_b_c-1 times…(a_b))))))....)

a_b_c_d= a_b_(a_b_(a_b_(......a_b_c_d-1 times….a_b_c_d-1)))).....)

….

=n[m]= n_n_n_n…….n[m-1] times…._n

n[m{a}]=n[n[n[n[n[.....n[m{a-1}] times……n[m{a-1]]]]....]]]

n[m{a\b}]=    n[m{a^[n[m{a]]b}]=(step 1)

step 2

step 1 but a replaced with step 1  2 times

step 3

step 2 but a replaced with step 2 3 times

….till step b

n[m{a\b#c}] =

n[m{a\b}] but b replaced with n[m{a\b#c-1}]  …

• How do you come up with growth rates for functions?

August 23, 2018 by BlauesWasser

Just wondering... Because, you know... I'm a "noob" so.. Yeah, idk, just wondering..

Let's take my recently created function, and find the growth rate. (Another user said that the limit to this function is F Omega+1(n) in the FGH)

• FGH Array Extension Examples

August 23, 2018 by IAmNotATRex

I have decided to write an entire blog post of examples using my FGH extension to (hopefully) give people a greater understanding of it. To understand this blog post, you need to know the definition of my extension, which can be found here: https://googology.wikia.com/wiki/User_blog:IAmNotATRex/Array-Based_Extension_to_the_FGH. I advise that you keep two tabs open, one with this page and one with the definition of my extension, in case you ever get confused.

I'll start off with a simple example: $$f[0, 0](2)$$.

If you also understand my old extension, know that this is equal to $$F1_{0}(2)$$ using my old extension. However, that isn't very important, considering the fact than my new extension is significantly more powerful than my old extens…

• A function I have created (still unnamed)

August 23, 2018 by BlauesWasser

So... I have created  new function... Tell me if it's ill-defined..  Hopefully it's clear (I'm technically still a beginner, so.. yeah)

{a,b} = $$a^b$$

{a,b,c} = $$a↑^c b$$

{a,b,c,d} = $$a↑^{(c^d)}b$$ or {a,b,{c,d}}

{a,b,c,d,e} = $$a↑^{(c↑^ed)}b$$ or {a,b,{c,d,e}}

etc, etc... There can be more entries, just use the other letters in the alphabet; it's just the same logic..

Here are some examples:

{3,3} = $$3^3$$

{3,3,3} = $$3↑^3 3$$

• Array-Based Extension to the FGH

August 23, 2018 by IAmNotATRex

After creating my own extension to the FGH, I have decided to create a more powerful extension using arrays. My previous extension can be found here: https://googology.wikia.com/wiki/User_blog:IAmNotATRex/My_Extension_to_the_Fast_Growing_Hierarchy.

For this blog post, the notation for arrays I will use is $$[a_{1}, a_{2},\ldots, a_{k-1}, a_{k}]$$, where $$a$$ is the name of the arrays and $$k$$ is its length. For the sake of clarity, $$[\langle a\rangle]$$ will be equal to the array $$a$$ instead of $$a$$ nested inside a one-element array.

I will first define a less powerful function, $$f[\langle a\rangle](n)$$, and then a more powerful function, $$F_{a}[\langle b\rangle](n)$$. If I made a mistake in my logic or a typo anywhere, please let m…

• Quick poll

August 23, 2018 by Alemagno12

survey

How exactly do each of you guys come up with your results for a growth rate of a notation? (Not how you prove it, i.e if you either do analysis tables or normal justifications, but rather how you come up with it)

• Gleive's U Notation - Almost finished!

August 22, 2018 by Gleive

Yes, we all know that the initials are GUN.

I've been working on this notation for quite some time. Ever since I picked up googology this is the most refined work I've done. Most others are (Not naive, but not that significant either) extensions on existing notations or other projects that ended up being salad notations or not getting past linear beaf level.

This will probably be finished a few days later. Will upload the contents to both my website and to here.

• I've been doing googology for a long time, but I still feel like a beginner

August 22, 2018 by BlauesWasser

I still feel like a beginner, I don't believe I know enough googological stuff.. For example, I don't understand fgh.. What are some links to help get started? (again)

• I'm ready for the easiest OCF!

August 20, 2018 by MilkyWay90

I am ready to start learning the easiest OCF!

I have learned about finite parameter multi-variable Phi function (phi(1, 1, 0) = the limit of gamma_n), and the definition of an uncountable ordinal (an ordinal where there is no function f(n) where for every element in the ordinal (y), there will be a value f(x) = y where x is a natural number)

So I am ready to start learning (the easiest) OCFs!

So can somebody give a link/definition in the comments to whichever OCF is easiest to learn in their opinion?

• My Extension to the Fast Growing Hierarchy

August 19, 2018 by IAmNotATRex

I don't know much about googology or ordinal numbers, but I thought it would be fun to try to create my own extension to the fast growing hierarchy. I don't know if this has been done before or how this compares to other notations or functions, so I'd appreciate it if someone could compare this to other forms of notation or other functions.

I don't have much experience formatting with LaTeX, so please forgive me if something doesn't look good.

I haven't really thought of a good name or notation, so I'll just use temporary notation for this blog post and refer to this as "my extension."

My new extension uses the following format: $$Fn_a(b)$$, where $$F$$ is the name of the function.

For $$n=0$$, $$F$$ simply reduces to the fast growing hierarch…

• Ultrafinitism and Poincaré recurrence time

August 19, 2018 by Arasena

The longest time, after which everything repeats, is Poincaré recurrence time for the Universe, which is around $$10^{10^{10^{10^{2.08}}}}$$ (approximately a googolplexian) Planck times. Thus, it will have exactly the same state as for now. Is this time the largest number?

If the Universe is infinite, does Poincaré recurrence time exist? Or if sub-atomic particles can be divided infinitely?

• Categorizing of the rule sets for all sub versions of bashicu matrix

August 18, 2018 by Koteitan

This is the table of the rule sets of (almost) all sub versions of Bashicu Matrix invented in 2015-2018.

The aim of this entry is the comparison of the rule sets of them. The all sub versions can be categorized according to the difference of the three micro rules; Bad root searching rules, Bad part ascending rules and Ascension modification rules. In this article, the all sub versions are re-written in the mathematical definition with the same format. So that the comparison become easier. Please use this for your proof of the termination or analysis.

Rule sets
Authors

BM1
Bashicu
21,August'15

• BASIC pseudo code

Left met…

• H-Boogol-Boogol: Hopefully a very big non-salad number

August 17, 2018 by DontDrinkH20

WARNING!!! You should be well-equipped with model theoretic terminology and computability theoretic terminology before you work with this paper, as well as some basic set theory. Feel free to grab a notebook and take some notes as you go along. You might need them; this one is a real doozy.

There was originally going to be another subclass of Type C called the representable googolisms. These would be a weakening of the expressible googolisms where "metamathematical" was replaced by a less expressive type of language. However, I then realized that there was no known concise definition of "a natural number can be expressed by a language" so I decided to make one.

It turns out that it doesn't require just a language, but rather a structure (whic…

• A Formal Definition of Pair Sequence System using BM1 Rules

August 14, 2018 by Rpakr

In this blog post, I will define BM1 pair sequences formally without using things like ... which are ambiguous, and defines BM1 number.

In this section I will define some functions, sets, etc. I will be using in the definition. When I use quotation marks, it means the string inside them should be treated as strings and not as numbers or expressions.

$$\mathbb{N}_0$$ is the set of all nonnegative integer.

$$\mathbb{N}_0=\mathbb{N}\cup\{0\}$$

When A and B are strings, $$con(A,B)$$ is defined to be the string that is the concatenation of A and B.

Define $$Pair(a,b)$$ to be the string “(a,b)” when $$a$$ and $$b$$ are nonnegative integers.

$$Pair(a,b)=P\Leftrightarrow (a\in\mathbb{N}_0∧b\in\mathbb{N}_0∧P=con(“(“,con(a,con(”,”,con(b,”)”)))))$$

$$Pair(A… Read more > • First post August 14, 2018 by DatoXx8 I got a few functions that grow fairly fast but sadly i only got them on google docs so.... here are the links smallest: https://docs.google.com/document/d/1PQ08ZFbEu8g1qsqm4u8HCJfJp7PPhZ3aKrZsoa9EmgU/edit?usp=sharing medium: https://docs.google.com/document/d/1E18XQOEI-HKNFuEKsDrNIo7yZU9MvuwqTjcspl0XkfI/edit?usp=sharing giant: https://docs.google.com/document/d/14y_YlzB_i2I3njz5DrGOLBFwpy8HXPQPwmOIEdkW8sk/edit?usp=sharing but they arent well defined and they have tons of opertunety`s for salad numbers Read more > • Expanded Googological Hierarchy August 14, 2018 by DontDrinkH20 In my previous blog post, I rattled on for several paragraphs about my "Type A, B, and C" classifications of googological specimen. In this post, I will go into detail on the expanded full hierarchy I have since completed. The motivation behind these classifications is the fact that there are obvious differences in the ways certain googologies are defined, and usually those differences can show up in their size as well. The set of imaginable googolisms is the smallest type of googolism. These specimen can be easily imagined as quantities in the brain. For example, the amount of letters in a novel is imaginable. These are usually not considered true googolisms. Here are some examples of imaginable numbers: • \(6$$
• $$100$$
• $$1532$$
• $$2167$$
• Kaprekar's…

• New notation idea?

August 14, 2018 by Eners49

Here is the basics:

• {a} = a!!!...!!! with a nested factorial signs. Pretty basic for 1-entry arrays.

That's already a really powerful function. Just a single factorial sign is powerful by the layman's standards, but if we have a bunch of them, that's going to be amazing!

• {0} = 0
• {1} = 1! = 1
• {2} = 2!! = 2! = 2
• {3} = 3!!! = 6!! = 720! = 10^1,746.42
• {4} = 4!!!! = 24!!! = 10^10^10^24
• In general, {a} is about 10^^a

With one-entry arrrays, we already have created a strong function, but we're only in between f2(n) and f3(n) in the fast-growing hierarchy, so we need to keep going. Let's see what two-entry arrays look like:

• {a, b} = }...}}}, b-1} where there are {a, b-1} brackets. We've created a pretty strong recursion so far.
• {a, 1} = {a}. Just like BEAF, …

August 13, 2018 by Gleive

Okay so this just made my damn day.

Warning : The comment in this image may be offensive for some people.

August 13, 2018 by P進大好きbot

Hi. I am studying the strength and the termination of pair sequence system, and I need your help!

Could you tell me the pair sequences of standard form corresponding to the following ordinals?

1. $$\psi_0(\psi_1(\psi_2(\psi_3(0))))$$
2. $$\psi_0(\psi_1(\psi_2(\psi_3(\psi_4(0))))$$
3. $$\psi_0(\psi_3(0))$$
4. $$\psi_0(\psi_4(0))$$
5. $$\psi_0(\psi_3(\psi_2(0)))$$
6. $$\psi_0(\psi_3(\psi_2(\psi_3(\psi_2(0)))))$$
7. $$\psi_0(\psi_3(\psi_3(0)))$$
8. $$\psi_0(\psi_3(0)+\psi_2(\psi_3(0)))$$
9. $$\psi_0(\psi_3(0)+\psi_3(0))$$

Here, $$\psi_n$$ denotes Buchholz's OCF. (I know that I listed up the same ordinal more than one time.) Since they are smaller than $$\psi_0(\Omega_{\omega})$$, they are strongly believed to be uniquely presentable by pair sequences of standard form. Please tell…