The problem of "infinite" in versus debating

This blog is not meant as an attack against any person, or any verse, or any site, though it may be taken as an attack against any of them and is not intended as one of them. This is a plea for a mathematically consistent view of infinity.

There seems to be a misunderstanding that infinity x infinity = something else. In some cases this is hidden more. “A multiverse contains infinite universe, and a megaverse contains infinite multiverses so therefore a megaverse is infinitely larger then a multiverse and a megaversal is infinitely stronger then a multiverse” or “An infinite 5-dimensional cosmos is infinitely larger then an infinite 4-dimensional cosmos and so a 5D being is infinitely stronger then a 4D being”

Ok, let’s go over some basic terms first. “Infinity” is not a number. “Infinity” is an abstract concept denoting a never-ending nature. There is a set of numbers called infinite numbers, which are defined as being larger then any finite number. Next, let me talk about the concept of cardinality. Take the set of [Banana, Apple, Orange]. This set has 3 items and so it has a cardinality of 3. Take the set of [John, Tim, and Drake]. This set also has 3 items. Set Theory also says that two sets with same cardinality are the same size. You can also have cardinality in a set of numbers. [1, 7, 9] has a cardinality of 3. [1, 2, 3] also has a cardinality of 3.

Now let’s take the set [All Counting Numbers] or in other words the set of [1,2,3,4,5…]. What is the cardinality of this set? The cardinality is what is called Aleph-Null. Aleph-Null, or “countable infinity” is the set of “and so on”, the set of things that you can order forever. While it’s endless technically, you can go to any arbitrarily high amount you want in theory. Aleph-Null is in fact a number. It can be used in arithmetic formula, though it has some very unintuitive results. To explain this let me introduce my good friend Hilbert.

Hilbert has a hotel. This hotel has Aleph-Null rooms and every single one is full. Let’s say 1 person comes by and wants a room. Hilbert says “no problem” and moves every single person in every room up 1 and puts the new person in room 1. Aleph-Null + 1 = Aleph-Null. As you might imagine same works with subtraction. If that person leaves everyone is moved to their room down 1.

Now let’s say a bus comes in. This bus is an “Infinity” brand bus. “Infinity” brand Buses boast an Aleph-Null seats, and each one on this bus is full. Hilbert sees this and moves everyone to their room times 2 and puts each person in the odd-numbered rooms. Aleph-Null times 2 = Aleph-Null. This shows that the cardinality of all even numbers is equal to the cardinality of all numbers. I know this is highly counterintuitive. But it is objectively true. Aleph-Null times 2 = Aleph-Null.

What if Aleph-Null of “Infinity” buses comes in, each one full? Surely Hilbert’s Hotel cannot possibly accommodate this right? Well Hilbert, being a clever man numbers each bus and numbers each seat in each bus. Let’s call the Bus number B, and the Seat number S. He then moves each person to room 2^B * 3^S. Each number has a unique prime factorization, a set of prime numbers that multiplies to reach it. Because of that each possible combination gives a unique room number. In conclusion Aleph-Null * Aleph-Null or Aleph-Null^2 equals Aleph-Null.

Since there are Aleph-Null prime numbers you can raise Aleph-Null to as high a power as you want. 2^A * 3^B * 5^C or 2^A * 3^B * 5^C * 7^D etc.

In other words, Aleph-Null^Aleph-Null = Aleph-Null.

There are ways to create higher infinities, but that is not it. Cardinalities don’t work get bigger by adding or multiplying like that.

The thing that took me forever to learn is that space CAN’T be higher cardinalities. Aleph-null is the biggest a physical space can be. If something is arranged into a coordinate of points, then by necessity it is ordered and countable thus Aleph-0. The whole idea of megaverses of infinite multiverses of infinite universes or of dimensions being infinitely larger just doesn’t make sense.

Now I know some smart guy is probably thinking of “what about the ordinals”, fine fine clever guy though if you know about ordinal numbers you really should already know the problem with that.

Ordinal numbers are basically taking numbers that don’t add on to a set because they are not part of a set. It’s like taking the set of [Counting Numbers, Pi]. Counting numbers are an Aleph-Null sized set but Pi isn’t part of it. This is designed as Omega + 1. You can also make other Omega Sets such as Omega + 2, Omega + 3, Omega times 2, Omega^2, Omega^Omega etc. And these are all technically different numbers. That said while there are different NUMBERS they all have the same CARDINALITY, the same VALUE, which is Aleph-Null.

So let’s take the final problem presented, which is that “sure it doesn’t TECHNICALLY make mathematical sense, but come on, fiction breaks rules all the time, how is this different then people moving faster than light?”

To explain that I need to explain the difference between irrational and illogical. “Irrational” means contradictory to reality, while “Illogical” means contradictory to itself.

Fiction is very often irrational, and we can accept that just fine. See the thing is versus debates are hypotheticals, they aren’t arguments about what are but about what if this hypothetical reality was existent.

Something being Illogical is something you absolutely can not discuss. Nothing, not even a hypothetical can contradict logic, because contradicting logic means not meaning anything at all. If you grant an illogical statement then you can prove literally anything:

A logician was famously asked “given 0=1, prove you are the pope” and he said “add 1 to both sides and you get 1=2, the pope and I are two people, therefore we are one person. I am the pope.”

Saying an illogical phrase like “0=1” or “a rock so big an omnipotent can’t lift” or “a triangle circle” isn’t saying anything at all. They are as meaningful as saying glxtbnm. It’s a string of sounds but doesn’t refer to any hypothetical concept. Saying a number is greater then itself is illogical since it contradicts it’s own definition.

Saying that the space this character destroyed is aleph-null times larger then aleph-null is like two characters being able to release 1 megaton of force, but the story trying to insist one megaton is somehow larger. At best it’s unquantifiable and at worst it’s unusable.

If a character destroys an infinite area, then you should be very cautious that it is actually infinite and not a hyperbole because an infinite area of space is as large an area of space as you can possibly have.