(Disclaimer: All that you are about to read is a mathematical axiom, of a bored finitist)

So then, what is the biggest number that you can name? Maybe a million, or a septillion? Maybe even a googol (10^100), or even a googolplex(10^(10^100))? Maybe if you are that knowledgeable, a milli-million(10^3003), or if you are that intelligent, it could even be an oplex, which is 10^(10^3003)? Or what about ∞?

So, is ∞, the biggest number that you are able to think of? I mean it has to be, it is the biggest thing, that nobody can count up to it, or higher than it, or near to it, if they had millenniums on millenniums on millenniums. However, ∞ isn't actually a number. ∞ is instead, the name of the group of every countable number, in existence, all of the way from 1 up to 10^10000................ You get the idea.

However, there are also a multitude of ∞s, for example there is the 'Countable ∞' which is referring to every integer or natural number, that exists. There is also the 'Uncountable ∞' which refers to every real number, or decimal that exists, which is bigger than the 'Countable ∞'. So now I am sure you are wondering what the biggest countable number actually is, right?

First things first, I'd like to explain that the term for every countable amount, for example the amount of apples, or the amount of years until the universe ends, or the amount of hours NightBot spends outside (it includes 0), is called, a 'Cardinal number'. Therefore, what is the highest Cardinal number? The answer to that, is Aleph Null, which, to many mathematicians who aren't finitists, it is the smallest infinite Cardinal number, which I will explain later on.

Aleph Null, is a number that is SO BIG, that it is uncountable, completely. It's size includes the amount of square numbers that exist, the amount of prime numbers, the power set of a googol P(10^100), and so on.

(By the way for anybody, who doesn't know, a power set, is the amount of sets, or combinations, you can get without an amount of numbers, for example, lets look at if you had 1,2,3 and 4. You could have no combinations, 1 on its on, 2 on its own, 3 on its own, 4 on its own, 1 and 2, 1 and 3, 1 and 4, 2 and 3, 2 and 4, 3 and 4, 1 and 2 and 3, 1 and 2 and 4, 1 and 3 and 4, 2 and 3 and 4, or 1 and 2 and 3 and 4. The way to work out without doing it like that is with the formula 2^n, where n is the amount of integers, so for this it would be 2^4, so 16.)

However, what about, if you did the P(Aleph Null) or the P(P(Aleph Null)), or maybe even the P(P(P(P(P(P(Aleph Null)))))). Surely, it would be bigger, than Aleph Null, wouldn't it?

In order to go to a higher number, we have to first understand a different type of number, the ordinals. Which are used in order to describe the order of something. If you came Aleph Nullth in a place, you would be in position ω0 (Omega null). So if you came in this position, in a race, you would be saying, that there has been an Aleph Null amount of people who finished before you. Due to them being linked but not completely linked, due to them being different types of numbers, we can use the equation Aleph Null ~ ω0.

What would happen, if you finished, one place AFTER that? You would be in position ω + 1. That's all well and done, but due to the equation, you also have to add 1 to Aleph Null, which gives you Aleph Null again. So what if it was 1000 places, after ω, you would be in ω0 + 1000, or Aleph Nullth place. You probably see the problem by now, but just go with it, until the end.

(So you could keep on going forever, ω + 10000, ω + 1234567, ω + 10^100. However, if it went on forever, you would eventually get to ω0 + ω, or 2ω. Therefore a problem arises, it is impossible to add ω to Aleph Null, they are completely different number types. Just ignore this for now, it will make sense later.)

You could of course, keep on going forever, that is true getting to a number such as ω^ω^ω^1000. That is possible, and the second highest infinity in this case, for ordinals, is ε0, and once again you could do the same thing to ε0, so ε0 + 1, ε0^ε0^ω. Eventually, you would get to, ω1. However, due to Aleph Null ~ ω0, then you can also say, Aleph 1 ~ ω1, right? Surely? It has to be, hasn't it?

That is incorrect though, Aleph Null, is a number which you can't count to, and so is Aleph 1, so isn't there a chance that Aleph Null could be equal to Aleph 1??? That can't be possible. Then Aleph 1 would be Aleph Null, so there couldn't be an Aleph 1.

But to now continue on with Ordinals, you would get ω1 * 10, ω1 + 1000, and so on, and so forth.

There would eventually be names for it, but what if there was a number, which you couldn't add 1 to for Ordinals aswell, you couldn't multiply it, you couldn't square it, or do anything to it, whatsoever, so you reached the highest number possible for something, ever. This is called, an 'Inaccessible number'. A number which you can never ever get to, no matter what. Coincidentally, we have seen a number like this already, in a different number sequence, which would be Aleph Null. Therefore, it is very safe, to implement an axiom, which would be:

The highest Ordinal Number ~ Aleph Null

So then, what is the biggest number that you can name? Maybe a million, or a septillion? Maybe even a googol (10^100), or even a googolplex(10^(10^100))? Maybe if you are that knowledgeable, a milli-million(10^3003), or if you are that intelligent, it could even be an oplex, which is 10^(10^3003)? Or what about ∞?

So, is ∞, the biggest number that you are able to think of? I mean it has to be, it is the biggest thing, that nobody can count up to it, or higher than it, or near to it, if they had millenniums on millenniums on millenniums. However, ∞ isn't actually a number. ∞ is instead, the name of the group of every countable number, in existence, all of the way from 1 up to 10^10000................ You get the idea.

However, there are also a multitude of ∞s, for example there is the 'Countable ∞' which is referring to every integer or natural number, that exists. There is also the 'Uncountable ∞' which refers to every real number, or decimal that exists, which is bigger than the 'Countable ∞'. So now I am sure you are wondering what the biggest countable number actually is, right?

First things first, I'd like to explain that the term for every countable amount, for example the amount of apples, or the amount of years until the universe ends, or the amount of hours NightBot spends outside (it includes 0), is called, a 'Cardinal number'. Therefore, what is the highest Cardinal number? The answer to that, is Aleph Null, which, to many mathematicians who aren't finitists, it is the smallest infinite Cardinal number, which I will explain later on.

Aleph Null, is a number that is SO BIG, that it is uncountable, completely. It's size includes the amount of square numbers that exist, the amount of prime numbers, the power set of a googol P(10^100), and so on.

(By the way for anybody, who doesn't know, a power set, is the amount of sets, or combinations, you can get without an amount of numbers, for example, lets look at if you had 1,2,3 and 4. You could have no combinations, 1 on its on, 2 on its own, 3 on its own, 4 on its own, 1 and 2, 1 and 3, 1 and 4, 2 and 3, 2 and 4, 3 and 4, 1 and 2 and 3, 1 and 2 and 4, 1 and 3 and 4, 2 and 3 and 4, or 1 and 2 and 3 and 4. The way to work out without doing it like that is with the formula 2^n, where n is the amount of integers, so for this it would be 2^4, so 16.)

However, what about, if you did the P(Aleph Null) or the P(P(Aleph Null)), or maybe even the P(P(P(P(P(P(Aleph Null)))))). Surely, it would be bigger, than Aleph Null, wouldn't it?

In order to go to a higher number, we have to first understand a different type of number, the ordinals. Which are used in order to describe the order of something. If you came Aleph Nullth in a place, you would be in position ω0 (Omega null). So if you came in this position, in a race, you would be saying, that there has been an Aleph Null amount of people who finished before you. Due to them being linked but not completely linked, due to them being different types of numbers, we can use the equation Aleph Null ~ ω0.

What would happen, if you finished, one place AFTER that? You would be in position ω + 1. That's all well and done, but due to the equation, you also have to add 1 to Aleph Null, which gives you Aleph Null again. So what if it was 1000 places, after ω, you would be in ω0 + 1000, or Aleph Nullth place. You probably see the problem by now, but just go with it, until the end.

(So you could keep on going forever, ω + 10000, ω + 1234567, ω + 10^100. However, if it went on forever, you would eventually get to ω0 + ω, or 2ω. Therefore a problem arises, it is impossible to add ω to Aleph Null, they are completely different number types. Just ignore this for now, it will make sense later.)

You could of course, keep on going forever, that is true getting to a number such as ω^ω^ω^1000. That is possible, and the second highest infinity in this case, for ordinals, is ε0, and once again you could do the same thing to ε0, so ε0 + 1, ε0^ε0^ω. Eventually, you would get to, ω1. However, due to Aleph Null ~ ω0, then you can also say, Aleph 1 ~ ω1, right? Surely? It has to be, hasn't it?

That is incorrect though, Aleph Null, is a number which you can't count to, and so is Aleph 1, so isn't there a chance that Aleph Null could be equal to Aleph 1??? That can't be possible. Then Aleph 1 would be Aleph Null, so there couldn't be an Aleph 1.

But to now continue on with Ordinals, you would get ω1 * 10, ω1 + 1000, and so on, and so forth.

There would eventually be names for it, but what if there was a number, which you couldn't add 1 to for Ordinals aswell, you couldn't multiply it, you couldn't square it, or do anything to it, whatsoever, so you reached the highest number possible for something, ever. This is called, an 'Inaccessible number'. A number which you can never ever get to, no matter what. Coincidentally, we have seen a number like this already, in a different number sequence, which would be Aleph Null. Therefore, it is very safe, to implement an axiom, which would be:

The highest Ordinal Number ~ Aleph Null