In Mathematics, set-theoretic topology is a subject that combines set theory and general topology.It focuses on topoogical questions that are independent of Zermelo-Fraenkel set theory as know as ZFC.
In the Mathematical field of general topology we have something called Dowker space which is a topoogical space that is T4 but not countably paracompact.
Dowker conjectured that were no Dower spaces and conjecture was not resolved until M.E Rudin constructed one in 1971. Rudin’s counterexample is a very large space of cardinality.Basically we say numbers have the same cardinality when they have the same number of things.

For instance 4 Bananas and 4 apples. Bananas and apples are different, however they have the same cardinality. Because we can match each banana with each apple.We have for instance a list of ten first cardinals: 0,1,2,3,4,5,6,7,8,9.But how many natural numbers are there? It cant be a number inside this set, because we always can add one after it. However actually in set-theoretic theory using dowker spaces we have a name for it: Aleph which we can represent as: ℵ
The first letter of the hebrew alphabet.We will see more about Aleph soon(next article).
But now, let’s come back and talk about countable numbers. Last article we saw about TREE(3) which was pretty big, right? We already saw how Graham’s number(g64) puts Googolplex in your pocket and after that TREE(3) puts g64 in your pocket too.
Now it’s the time to surpass TREE(3)!
You are about to see the biggest known countable numbers…

Subcubic Graph Theory

Let’s start with SGC(13) which is already bigger than TREE(3).
Subcubic graph theory are the outputs of a fast-growing combinatorial function.They were devised by Friedman, who showed that it eventually dominates every recursive function and itself.
One output of the sequence, our little boy SGC(13), is a subject of extensive research. It is known to surpass TREE(3).
Definition: Given an integer k, suppose we have a sequence of subcubic graphs(G1, G2…) such that each graph Gi has at most (i + k) vertices and for no i < j is Gi.
So, for each value of k, there is a sequence with maximal length. We denote this maxial length using SGC(k).
It is possible to show that SCF(0) = 6. The first graph is one vertex with a loop, the second is two vertices connected by a single edge, and the nexy four graphs consist of 3,2,1 and 0 unconnected vertices. All maximal sequences will peak and decline this way, as you can see in pic bellow:

The following bound SCG(1) is already bigger than a Googolplex.SCG(2) is already bigger than Graham’s Number(g64). Well, SCG(3) We cannot know for sure, but SGC(13) is already much bigger than TREE(3)!
But how big is SCG(13)? Ok, it’s bigger than TREE(3), but SCG(3) might be bigger too.What about SCG(13)? How can we measure this?

Well, SCG(13) is bigger than TREE(↑↑↑↑…(TREE(3) of UpArrows)…↑↑↑↑) .
That’s insane. We have a TREE(3) up to a power stack of TREE(3), TREE(3) times!However, I must tell you that SCG(13) is theoretically a computable number…
Wow, SCG(13) still computable? Is there a number that represents the maximum of a computable number(using modern computers)?
Of course! It’s Loader’s Number!

What is Loader’s Number?
Loader’s number is the output of a computer program named loader.c
Made in C Language by Ralph Loader that came in first place for the Bignum Bakeoff constest.
Whose objective was write a C program(in 512 characters or less) that generates the largest possible output on theoretical machine with infinite memory. It is among the largest computable number ever devised.The Program diagonalizes over thet Huet-Coquand calculus of constructions, a particularly express lambda calculus. Is defined as:
D⁵(99) = D(D(D(D(D(99))))) where D(k) is an accumulation of all possible express provable within aproximately log(k) inference steps in the calculus of constructions(encoding proofs as binary numbers and expressions as power towers).
David Moews has shown that D(99) is larger than 2↑↑30419, and the even D²(99) would be much larger than F27(10⁶) in the fast-growing hierarchy.(it means a stack tower of 30149 of 2’s.
Something like: 2²^²^²…²
The final output of D⁵(99) is much larger than TREE(3) and SCG(13). It is overpowered by finite promise gams and greedy clique sequences. Loader’s function is computable so any number beyond that is uncomputable.
I’m graduated in Systems Analysis by Fatec(Santos, Brasil), and in the future I pretend to write an article about the code of loader.c(which in free and you can find it on web).

Uncomputable Numbers

Now, lets check what we have beyond computable numbers.
At this point we are near in the realm of mathematical theories numbers, which is becoming abstract math concepts. If your brain is already collapsing I advise you to stop here.We are about to see the Top 5 Biggest FINITE known Numbers(until January 2019).

TOP 5
Well let’s start the Top 5 with Rayo’s Number, the smallest positive integer bigger than any finite positive integer. Named by and expression in the language of first order set theory with a googol symbols or less. Rayo’s Number = Rayo(10¹⁰⁰).
Rayo’s function is very quicky fast-growing function, Rayo(n).This Number is uncomputable, that is impossible for a Turing Machine(any modern computer) to process this value. Only FOOT(n) function(we are goint to see latter) surpasses it in strength. Rayo’s number was honored as the largest named number until 2014 when BIG FOOT was defined, using a non-naive express of n-th order set theory.

Top 4

BIG FOOT number, larger than Rayo’s number and it’s difficult to describe…is defined as FOOT¹⁰(10¹⁰⁰).Basically Big Foot is a counterpart of Rayo’s Number based on a extended version of the language of first-order set theory. As a result it is among the largest named number. It was defined in October 2014 by “Wojowu”.
Big Foot extends first-order set theory by making use of unique domain of discoure called oodleverse, using a language called first-order oodle theory(Foot) and generalizing n-th order set theory of arbitrarily large n.Letting FOOT(n) denote the largest natural number uniquely definable in the language of FOOT in at most n-symbols, we define BIG FOOT as FOOT¹⁰(10¹⁰⁰) where FOOT^a(n) is FOOT iterated a times(recursion) BIG FOOT is equals to: FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(FOOT(10¹⁰⁰))))))))))

TOP 3
The Top 3 finite Number is Little Bigeddon.
Little Bigeddon is a googolism based on a extension of language of set theory. It was defined on the 5th January 2017 by “Emlightned”. The Creator of Big Foot even admitted that the number might be greater than BIG FOOT.By definition Little Bigeddon is the largest number such that there is some unary formula in the ς = {∈, T} of quantifier rank <= 12↑↑12.

TOP 2
The Top 2 is for Oblivion.
Oblivion is a large googolism coined by Jonathan Bowers.It is defined as “the largest number defined using no more than a kungulus symbols in some K(gongulus) system”Wait what is a kungulus symbol?
Well, the kungulus is equals to a gaggol, the explanation of a Gaggol is:
Let N1 be 10.
Let Nx be 10¹⁰^¹⁰…¹⁰ (power tower of Nx-1 tens)
A Gaggol is N100. (insane right?, but N100 is not bigger than graham’s number)
Now, a Gongulus is a 10¹⁰^¹⁰… we have a power tower of 10’s a Googol of times. It’s speculates that Little Bigeddon is something like K(10,000) systems. Oblivion is allegedly the second-largest finite number up until now. Only smaller than its cousin: Utter Oblivion.

TOP 1

And the Larger finite known Number is… Utter Oblivion!
Utter Oblivion is allegedly the largest finite number, coined by Jonathan Bowers. It is definite as “The largest finite number that can be uniquely defined using no more than an oblivion symbols in some K(Oblivion) system.
So we have: Utter Oblivion = {K(Oblivion), K2(Oblivion), K3(Oblivion)…
K4(Oblivion)… K Oblivion(Oblivion)}.
And that’s the Biggest Finite Known Number.

Conclusion

Well, in this article we went to a totally diferent level.
If when you started reading my articles you thought that Graham’s number(g64) was the biggest number at all, now you have realized that you were wrong. WELL WRONG.

With this article I conclude finite numbers, but I introduced you on set order thoery at the beginning of this article because will be useful on next article when we will start seeing Infinty Numbers. Now, you might be thinking that Infinity is the largest number, right? Think again.
We have just seen a small sample of a infinite number, ℵ (Alef-Number). Which represents in set order theory a SET of all Natural Numbers.
ℵ = {0,1,2,3,4,5,6,7,8,9,10…googol…graham’s…TREE(3)…Rayo’s…Utter Oblivion…}
Everything we’ve seen so far is within this set.
If we were explorers we would be leaving the earth right now. Ready to expore the solar system.
Are you ready for this?
I’ll See you in the next article!

References

Set-Theoretic Topology: https://bit.ly/2DhhmDo
Subcubic Graph Number: https://bit.ly/2RvvSQN
How big is Loader’s Number: https://bit.ly/2ARFUBv
RAYO’s Function: https://bit.ly/2HgLnaz
BIG FOOT Function: https://bit.ly/1RFruCp
Little Bigeddon Number: https://sites.google.com/site/largeordinums/numbers/little-bigeddon
Oblivion and Utter Oblivion: https://bit.ly/2TYn9mY

Set-Theoretic Topology was originally published in An Introduction on Googology on Medium, where people are continuing the conversation by highlighting and responding to this story.

Untill now the biggest number we have seen was Graham’s Number.
If you don’t remember or are too tired to read my second article about this number I will explain quickly for you how big this number is.
In the last article I wrote about UpArrow Notation, which is a way of representing large numbers more effective than potentiation notation.
For instance 2²= 4
UpArrow notation we have: 2↑2 = 4

The second level of Up arrow notation is: with two up arrows(↑↑)
3↑↑3 =3 ³^³ which is 3²⁷, which is equals to aprox 7.6 Trillions
So yes, The Up Arrow Notation is pretty good to create huge numbers!
The Third level is like: 3↑↑↑3 = This number has a stack of aprox. 7.6 Trillions threes. 3 to power of 3 to the power of 3 … 7.6 trillion times.
But that’s not the end, we have the Last level of Up Arrow Notation: 3↑↑↑↑3
this number is the result of 3↑↑↑3 plugged into a power of threes stack
is like: 3 to power of 3 to the power of 3… (3↑↑↑3 times!!)
The interesting part about it is this 3↑↑↑↑3 is named as g1.
g2 = 3↑↑↑↑↑(the number of up arrows is g1!)… 3
That’s massive, really massive right?
g3 = 3↑↑↑↑…(the number of up arrows is g2!)… 3
And we do that until g64.
g64 is the Graham’s Number, which is a Extreme-Large-Number.

As you can see Graham’s Number is massive and the UpArrow Notation is very effective to represent large-numbers. But we need something that grows faster to demonstrate numbers beyond g64.
For instance I can create a number right now, is a useless number without any applicational use, I will call this number “Dummie-Massive-Number” which is 10↑↑↑↑Googolplex.
This number is not bigger than Graham’s Number but interesting to think of.
We are going to the same problem we encountered using potentiation!
The Knuth Notation was created to be simple and easy to understand.
But at this point is getting confuse and complex.
We can also think about something like: Googolplex↑…(googolplex of up arrows)…↑Googolplex
Its huge, but how can we compare to Graham’s Number?
Is it big or not? We need something more simple, faster and good enough to compare monstrous-numbers.
Now we are good to talk about The Fast-Growing Hierarchy.

The Concept

The Fast-Growing hierarchy works with this general-formula: fa (n) = b
we can demonstrate the most dummie level of this function as: f0(n) = n+1
for instance: f0(3) = 3+1 = 4
Working with zeros at botton seems good but now another example:
At base one:
f1(3) = f0(f0(f0(3))) which is 6.
f1(5) = f0(f0(f0(f0(f0(5))) which is 10.
Following the general formula: f1(n) = 2n

Now let’s to the second level:
f2(3) = f1(f1(f1(3))) which is 24
The General formula is f2(n) = 2^n X n
Then, f2(3) =2³ X 3 = 24

The third level:
f3(3) = f2(f2(f2(3)))
f3(3) = f2(f2(24))
Which F2(24) is equals to 2²⁴ X 24. That’s something big, that’s equals to 402.653.184,
f2(402653184) which is 2⁴⁰²⁶⁵³¹⁸⁴ Times 402653184
That’s massive, that’s already bigger than a Googol, let’s convert to base
10²⁴⁰²⁶⁵³¹⁸⁴ equals aprox 10¹⁰^⁸

We can demonstrate the Fast-Growing Hierachy levels as:
f0(n) = n + 1,
f1(n) = 2n,
f2(n) = 2^n,
f3(n) > 2↑↑n
f4(n) > 2↑↑↑n
fm(n) > 2↑m-1 n

So now let’s demonstrate Graham’s number with fast-growing hierarchy:
The First Layer(g1)
g1 = 3↑↑↑↑3 = f5(3) which is f4(f4(f4(3) (which is BIG, really big)
what about g2? Well, g2 uses g1 number of up-arrows, right? Then: g2 = fg1(3) which is F(f5(3))(3).
So, g64(which is G) is equals Fg64(3).
Now, we are ready to talk about more extreme-large-numbers.

TREE(3)

Tree sequence is a fast-growing function arising out of graph theory, devised by mathematical logican Harvey Friedman. Friedman proved that the function eventually dominates all recursive functions provably total in the system A C A0 + π¹ 2 — BI

The smallest non-trivian member of the sequence is the famously TREE(3), notable because it is a number that appers in serious mathematics that is larger than Graham’s Number.

The definition is:
1. Each tree Ti has at most i vertices.
2. No tree is homeomorphically embeddable into any tree following it in the sequence.

The First two values are TREE(1) = 1 and TREE(2) = 3.
The Next value, TREE(3) is famously very large.
Using the Fast-Growing hierarchy Graham’s Number we can also wrote as: FA(64)(4)
Using the Fast-Growing hierarchy Tree(3) is: fA(187196)(2)
where A(187196) is tree^tree^tree^tree^tree⁸.
Wait? What is A(n)?
A(n) comes from Ackermann’s function which is a fast-growing function too!
We can also represent as A(x,y).

We can define it, for instance as:
A(4,2) = (2↑↑5) -3
If A(n) then A(n,n) which is (n↑^n+1) -3.
So now we found a number bigger than graham’s number, TREE(3)

Conclusion

Fast-growing hierarchy is a certain hierachy mapping ordinals to functions.
It is particularly notable in googology not only in the growth rates of the numbers but also it’s very simple formal definition.
it’s more effective use Fast-Growing hierarchy than UpArrow Notations at this point.
However I will still using UpArrow Notation for demonstrative and comparative purposes.
In this article I have shown how they work and the importance of those Fast-Growing Funtions.
At this point I have already proved you that a Googol or even Googolplex are not big at all.
But we will keep going up untill where? You may be thinking that infinity is the limit.
Well, maybe for cardinal numbers but not for ordinal numbers.
Where we will get to our first infinity? Soon, I promise.
Now we are ready to look beyond these extreme-large-numbers and search for infinite numbers, and even beyond.
In the Next article I will talk about another bunch of extreme-large-numbers and their applications.

Thank you!

References

Fast-Growing Hierarchy: https://bit.ly/2SNYj9i
Ackermann’s Function: https://bit.ly/2GZt9Kx
TREE(3): https://bit.ly/2FjrIRo

The Fast-Growing Hierarchy was originally published in An Introduction on Googology on Medium, where people are continuing the conversation by highlighting and responding to this story.

The Problem With Potentiation

Potentiation in Math is usually raising something to a power, for instance if X is potentiated by Y then you have X^Y
For instance: X² = X times X.
So if we have 2 in the power of 2 we have:
2 in which it’s 2 times 2, then 2² = 4.
The general formula is: a^n

Potentiation is enough to non-large numbers such as in scientific notation. We have for instance avogrado’s number: 6 x 10^-23.
In my last Article I wrote about the number of stars in observable universe and I also wrote something about 10¹⁶, which is ten Quadrilions.
Even Googol or a Googolplex can be demonstrated by potentiation:
Googol = 10¹⁰⁰
Googolplex = 10¹⁰^¹⁰⁰
Size of a Proton = 10^-15 of a meter
Plank Lenght: 6 x 10^-35 meters
Weight of Earth in Kilograms = 5.9722 x 10²⁴ Kg
We also have Millillion: 10³⁰⁰³ (thats huge but not bigger than a googolplex)
Or even Milli-Millillion: 10³⁰⁰⁰⁰⁰³.

So, as we can see for Scientific notation, Potentiation is enough to demonstrate how big numbers can get, yes? Well, NO.
When we talk about really huge numbers such as skewes’s number we realize that potentiation begins to hamper our calculations!
Skewes’s Number was know as the biggest useful number know until 80s.
Skewes’s Number is: 10¹⁰^¹⁰^³⁴.
Ok, thats a large number.
Its bigger than a Googolplex.
But where did this number come from?

Well, the Skewes’s number is the number Sk1 above which π(n) < li(n) must fail assuming that the Riemann hypotesis is true where π(n) is the prime counting function and li(n) is the logarithmic integral.
Isaac Asimov featured the Skewes’s number in his science fact article Skewered! (1974)
In 1912, Littlewood proved that Sk1 exists(Hardy 1999, p17) and the upper bound was subsequently found by skewes(1933).
Looking for a number bigger than Skewes? Well we have Skewes two(Sk2): 10¹⁰^¹⁰^⁹⁶⁴.
In Physics the larger applicable number is know as “Longest Time”
(poincare recurrence) which is the time it will take to the Universe reset it self!
PR = 10¹⁰^¹⁰^¹⁰^¹⁰^¹.¹

As you can see we are getting more and more potentiations as we go through large numbers.
The Point is that Potentiation is useless when we are introduced to EXTREME-LARGE-NUMBERS so we need a new system of notation, right?
So now lets talk about UpArrow Notation. Get ready because I’m going to a totally new level.

Knuth Arrow Notation

Let me introduce you a new kind of notation: The UpArrow Notation, also know as Knuth Arrow Notation, was introduced by Donald Knuth(1976)
Donald Knuth is a computer scientist, he invented the UpArrow Notation.
As we saw previosly we are having some issues to work with potentiation notation for really large numbers.
So Knuth invented a system of notation not-complex and easy to understand that grow fast to measure really large numbers.
So for instance we have A and B
If we want to represent A in the power of B:
A^B
Using UpArrow Notation:
A↑B
Then, A^B = A↑B
2² = 2↑2
General Formula is:
A↑(n) = A↑(n-1) times A↑(n-1) times A↑(n-1)… B times
For instance:
3↑2 = 9 ’cause 3² = 9
3↑3 = 27 ’cause 3³ = 27

But now comes why Knuth Notation is better than Potentiation Notation:
We, for instance, can do this:
3↑↑3 = ?
Okay, what’s happening here?
Basically this: 3↑↑3 = 3↑(3↑3)
Yeah, things start to change at this point.

Let’s see what that means:
(3 to power of 3) to the power of 3.
So: 3↑↑3 = 3^(3³) = 3²⁷ = 7.6 X 10¹² (7.6 Trillions)
If we for instance: 3↑↑4 = 3↑3↑(3↑3)
so we have a stack of 4 threes: 3³^³^³

Now, what if we add one more arrow?
3↑↑↑3 = 3↑↑(3↑↑3)
Which (3↑↑3) is: 3↑(3↑3), right?
So: 3↑↑↑3 is equal to 3↑↑(7.6 trillions)
Humm, things are getting interesting here.
So now we have a tower of 7.6 trillions of Three.
Something like: 3³^³^³^³^³^³^³^³^…(7.6 X 10¹²)
The size of this stack of threes is 7.6 X 10¹².
Can you imagine something like this?
The fun fact about 3↑↑↑3 is that if each three has a meter we will have a tower of three that would reach the surface of the SUN!
The distance between the earth and our star is like 149.600.000 Km
in meters = 149.600.000.000 (Billions).
Yeah that’s how big this number is!

But what’s the value of 3↑↑↑3???
Well, I don’t know.
But it’s already bigger than a googol, googolplex, googolduplex…
You may be thinking that this number is large right?
Well we can still add the last arrow:
3↑↑↑↑3 = 3↑↑↑(3↑↑↑3)
We know that (3↑↑↑3) is equal to that crazy tower of threes, but what 3↑↑↑↑3 is equals to ????
3↑↑↑(3³^³^³^³^³^³^³^³^³^… (7.6 X 10¹² times).
It’s like you try to perform (3↑↑3 … 3↑↑3) and this space between is that crazy tower. So you would have to perform 3↑↑3 for 7.6 X 10¹² times.
So basically now we have 3 up to a insane power of threes…now we are starting to measure really large numbers…

We usually don’t use 5 UpArrows(something like: 3↑↑↑↑↑3)
Cause it would be insane and would lose the sense of having a simple notation. After 4 UpArrows the Knuth Notation would result in the same problem we had with potentiation, right?
So to go further we will need a new notation…

Graham’s Number

Graham’s Number is an enormous number that arises as an upper bound on the answer of a problem in the mathematical field of Ramsey Theory. It was discovered by Ronald Graham(1977)
The Ramsey Theory is: “How many elements of some structure must there be to guarantee that particular property will hold?”.
Ron Graham describes Ramsey Theory as a “Branch of combinatorics”.
The Graham’s Extreme-Large Applicable Number.

It’s such a large number that if you could imagine how big that number would be, your brain would be such a massive cause of synaptic connections and would become a Black Hole.

We would need a Googolplex of Universes to write each digit of this number in a single atom and maybe it wont be enough…
But can we measure this number using potentiation? Well, not if you wanna see how big this number is this century… however using the Knuth Notation we might have an interesting measurement(and it wont take a few centuries at all, just a few minutes)
We already had a tiny-tiny conception of how big this number actually is.
I wrote about 3↑↑↑↑3.

This number is a “crazy tower of threes” 3³^³^³^…(7.6 X 10¹² size) plug into 3↑↑↑3^(This “crazy power of threes”)
That’s huge, that’s insane!
Even for a normal-computer it would take a few hours to process 3↑↑↑↑3. Maybe Super-Computers would take a few minutes or not even it.
But this value is astronomical, right?
Well however 3↑↑↑↑3 is not Graham’s Number at all.
Let’s call 3↑↑↑↑3 = g1.
Well if this insane number is g1 what is g2?
g2 = 3↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑…(g1 Times)…↑↑↑↑3
No words can express how big this number is.
The Number of arrows is g1.
So is g2 equals to Graham’s Number? No! NOT EVEN CLOSE!
What about g3 = 3↑↑↑↑…(g2 Times)…↑↑↑↑3
Still not even close to Graham’s Number.
And not even g4, g5, g6, g7…
at g12 we have “Little Graham’s Number”.
But not even g20 is good enough.
Not even g32 or g48 but then we reach g64.
g64 = 3↑↑↑↑…(g63 times)…↑↑↑↑3
and g64 is equal G, and G is the Graham’s Number.

Conclusion

As we can see here to reach really-large numbers we needed a new kind of notation called Knuth Notation. At this point Potentiation is obsolete.
In the future we will see that even Knuth Notation will become obsolete and I’m gonna show you extremely-larger-applicable-numbers.

Graham’s Number has turned into pop culture and it’s mainstream now.
However, there are more underground numbers that would make Graham’s Number(g64) look small.

References

Skewered Number: https://goo.gl/cN9tqd
Knuth Notation: https://goo.gl/21W19f
Graham’s Number: https://goo.gl/23M464

The UpArrow Notation was originally published in An Introduction on Googology on Medium, where people are continuing the conversation by highlighting and responding to this story.

Do not get confused with googlology (the study of the Googol Search Engine).

The antithesis to googology is called ultrafinitism, which states that the set of natural numbers is finite and at some point numbers become large enough and they simply (some how) cease to exist.

Ultrafinitism is an extreme form of finitism philosophy, which states that infinite objects such as transfinite ordinals are nonexistent.

Basically ultrafinitsts state that the natural numbers have an ending.

The term “Googology” was coined by Andre Joyce, combining googol which is a classic large number with the word from greek suffix “logos”, that means “study”.

Joyce’s googology involved devising a system of names for numbers based on wordplay and whimsical extrapolation.

Some Huge Numbers

Googol is a classic large number that’s equal to ten to the power of one hundred = 10¹⁰⁰.
Can you imagine a number like this? It’s ten followed by a hundred zeros.
Something like img 001:

As you can see here a googol is a pretty large number.
Scientists know about one hundred bilions stars only in our galaxy.
And there are about one hundred Bilions Galaxies in observable universe.

One hundred billions = 10⁸
Then, 10⁸ X 10⁸ = 10¹⁶
10¹⁶ = Ten Quadrilions (10.000.000.000.000.000)
A Googol is far distance from 10¹⁶.
Even multiplying this number of stars by one hundred Bilions again we still not even close from a Googol.
But how many atoms are in the observable universe?
Well Scince belives that there are about 10⁸¹ atoms.
Still neither near a Googol.
Now you might have a sense of how big this number is.

Should i tell you that a Googol is not a huge number at all?
We remain infinitely distant from a really large number.
Let’s talk about Googolplex as know as Googolplexian.
A Googolplex is ten to the power of a GOOGOL!
Yes, 10¹⁰^¹⁰⁰!
Milton Sirotta originaly defined it as “one, followed by writing zeros until you get tired”
If you take a second to write every zero, you would take 100 seconds to write a Googol.
But you would take a Googol seconds to write a Googolplex.
How many is a Googol seconds???
Well, is something really, really big.
A Year has 31.536.000 seconds. It’s like 3,15 X 10⁷.
The Universe has about fourteen billions year so even if we multiply fourteen billions year by thirty one milions seconds you still not even close from a Googol
So if you start right now writing a Googolplex zero by zero you would take literally an eternity.
Or not, because there wouldn’t be enough atoms to write this number.
To have a ink or graphite you need atoms and every atom would have finished before you would have written a Googolplex.
Scince believes that proton’s decay is about 10³⁴ years it’s huge but still nothing near a Googolplex.

Can we go further? Well of course just implementing +1 and you get a number bigger than previous number before.
Well, there are many numbers bigger than a Googolplex. But they are so huge numbers that it simply becomes difficult to write them with only scientific notation using potentiation.
You have for instance Googolduplex (10¹⁰^¹⁰^¹⁰⁰) but as you can see not even potentiation is good enough to take care of really large numbers.
I honestly must tell you that these numbers I have presented so far are not that big…

Conclusion

The Numeric system is extremely interesting. Here I just wrote about natural numbers.
But yes, we can go even further. Very, very further. Even beyond infinite.
Natural numbers can still get a complex discussion about how large it can get.
How ever you might be thinking about huge mathematical applicable numbers, they exist?
Yes, of course but I intend to right more articles in the future about them.
This Article was just a brief introduction on this area of study so interesting called googology.
If you liked please let me know. Follow me and check the links on “references” section.
Come be a part of this community.

References

Googology Wiki: https://goo.gl/PV9gaZ

Googol Number: https://goo.gl/fLgRsW

Googolplex Number: https://goo.gl/Dn4ZqK

Googolduplex Number: https://goo.gl/ARZSh6

How many stars and galaxies are out there: https://goo.gl/GYovbs

Proton Decay: https://goo.gl/NU8kDn

What is Googology? was originally published in An Introduction on Googology on Medium, where people are continuing the conversation by highlighting and responding to this story.