Talk:Archie Sonic-Verse VS Paper Mario-Verse/@comment-26374068-20160416190323/@comment-26374068-20160416234508

This one's gonna get complicated, & probably not make much sense, alright? Let's-a go!

First, before I start the debate, lemme explain how you count past infinity. some infinities are, bigger than others. Cardinality is a word to explain all natural numbers. Count as high as you want. Those are cardinal numbers. To count all cardinal numbers, is the FIRST infinity, Aliff Null. This is the first letter of the Hebrew Alphabet, also being the first, & smallest, infinity. We can count past this. Using our friend Supertasks, we could draw lines next to the next line, a fraction of the same line. The fraction of each, last line. We can fit an unending number of lines into a finite space. The number of lines are equal to the number of cardinal numbers there are. The two can be matched, one-to-one. These two sets both have the cardinality, Aliff Null. But what happens when I add an extra line to that Aliff Null? Aliff Null plus one? No, because that's not the same as finite amounts. The more I add, I always end up with Aliff Null things. But hold on a sec, clearly an Aliff Null made of two Aliff Nulls HAS to be bigger, right? If it's not the amount, then what is it? Let's go back to having only one line after an Aliff Null. Instead of counting them, why don't we label them in order? In the realm of infinity, labeling is different from counting. That extra line doesn't contribute to the total, but now we need a number of labels that extends past the naturals. We need ordinal numbers. The first trans-finite ordinal is Omega, the lower-case letter Omega. This isn't a joke or a trick, it's literally just the next label you'll need after using the infinite collection of every counting numbers.

Having Omega after Aliff Null, is like saying an infinite amount of people finished a race, then you did. After Omega is Omega plus one, then Omega plus two, Omega plus three, etc. Ordinal numbers, labeled in order. Ordinals aren't about how many there are, no, instead, they tell us how all those things are arranged. Their, order type. The order type of a set is just the first Ordinal Number now labeled to put every number in the set, in order. For finite numbers, ordinality, & order type, are the same. The order type of all the naturals, is Omega. The order of the sequence to come is Omega plus one, & then, Omega plus two. No matter how long an arrangement becomes, as long as it's well-ordered, as long as every part of it contains a beginning element, the WHOLE thing, describes a new ordinal number. After you reach the Aliff Null of Omega, it becomes Omega plus Omega. Always. Remember this, it'll be important as this explanation goes on. Omega plus one isn't. bigger, than Omega, it just comes after it. But, Aliff Null isn't the end of infinity, either.

Why? Cause it can be shown, that there are infinities, bigger than Aliff Null. That LITERALLY contain, more things. One of the best ways to do this is with Cantor's Diagonal Argument. But, we'll use that later, & first focus on the Power Set of Aliff Null. The Power Set of a set, is the set of all the sub-sets that you can make from it. For example, from the set of 1 & 2, I can make a set of nothing, or one, or two, or even, one & two. The set for 1, 2, & 3, are nothing, 1, 2, 3, 1 & 2, 1 & 3, 2 & 3, & 1, 2, & 3. As you can see, a Power Set contains many more numbers than the original set. 2^how ever many numbers the original set had, will make more. So, what's the Power Set of ALL the naturals (Aliff Null)? Well, let's see:

Imagine a list of every natural number. Cool. Now, the sub-set of say, even numbers would look like this:

0=y, 1=n, 2=y, 3=n, 4=y, 5=n, and so on. The sub-set of all odd numbers would be the opposite of what I just put down. Now if we put only the sub-set of every number, except five? It'll be where every number =y, while 5=n. No number except five? 5 is the only number TOO =y. Obviously, this list of sub-sets will be infinite. But imagine them all one-to-one with a natural. Doing that, now there's even more sub-sets than even listed before. Now the set has more members than there are natural numbers.

A BIGGER infinity than Aliff Null. Move diagonally down through your sub-set. 0 is a member, so we will NOT use it. Same with 1. 2 is NOT in the sub-set, so it will be in ours, & so on. Now we are making sub-sets different from every sub-set on this Aliff Null-sized list. Even if we put this new set back in, diagonalization can still be done. The Power Set of the naturals will always resemble a one-to-one correspondance with the naturals. It's an infinity, bigger, than Aliff Null. Repeated applications of Power Set will produce sets that can't be put into one-to-one correspondance. This is a quick way to keep making bigger, & bigger infinities. The point is there are more cardinals than Aliff Null. Aliff Null is a good infinity to represent 6th Dimensional Beings, & Omnipotent beings, just like the Ancient Walkers. Slap 6 Power Sets onto it & you've got yourself Paper Mario. Now that that explanation is outta the way, I'll make my next comment dedicated tpo our original argument.