[RBJamez]

[forebirdo]

Why not

Why don't we coin VERSE (typed all in caps) as "everything"

[CefGemYAffews]

The thing you're talking about is called the Cardinality of a set, and is a pretty important part of mathematics. In layman's terms, it's "How many elements does a set contain?" It's also as hard as hell. A member on another math-and-other-sciences-forum once said

It was only about a year ago that this was introduced to me in a formal setting, I remember the day before my lecturer advised us all to get a good nights sleep and make sure we'd had enough breakfast because understanding cardinalities is not easy.

On that same forum, I asked an interesting question that was bugging me for quite some time. For every real number that's larger than one, you have it's reciprocal that's smaller than one. From that, you can safely assume that the number of reals between one and infinity is the same as between zero and one. That in itself is pretty disturbing, but it get's better. Add one to the reciprocal of any real number larger than one, and you'll get a number between one and two. This means that the number of reals between zero and one is the same between one and two AND it's also the same as the number of reals between one and infinity. Strange?

I also made up a metaphor that asks the same thing and is probably more understandable to most people: Say we have a house and in it an infinite amount of rooms. Each room takes up the same amount of space, and each room takes up the same amount of space as all the others put together. How can this be?

As it turns out, real numbers don't follow the rules of common sense, which is why this is very much possible.

Anyway, you have three types of sets: finite, countably infinite (isn't that just weird?) and uncountable. A finite set is any set who's cardinality can be expressed by a natural number, eg.: 308. The official definition is "If the cardinality of a set S (denoted |S|) is smaller than the cardinality of the set of Natural Numbers (denoted |N|) it is considered finite. Symbolically, if |S| < |N|, then it is finite". A countably infinite set is a set who's cardinality is equal to that of natural numbers, ie.: |S|=|N|. The cardinality of that set is denoted by the symbol (read Aleph Null). Now, you might ask how can one tell if a set has the same number of elements as the Naturals. They both have an infinite number of elements. Well, there's a neat trick here. Imagine a set as a big bubble, with little bubbles inside representing the individual elements. If you had two of these bubble, and you could draw lines between them connecting two elements in such a way that no bubble is connected to two other bubbles and no bubbles are left out, then you could say that the two sets have the same ammount of elements (bubbles) right? Here's an illustration to help:



How can you do this in math? Well with functions of course. If there exists a function that will "assign" an different element from B to every element in A, we can say that they have the same cardinality, right? The official way of saying this is "If there exists a bijection (bijective function) between A and B...". In this manner, we can prove that the set of Even Naturals is the same as the set of all Naturals, since there exists a bijection between them: the function f(x)=2x. In the same way we can say that the cardinality of the set of Prime numbers is Aleph Null, since we can index them.

As for uncountable sets, this is a little beyond me. I know that the cardinality of the set of reals, denoted |R|, is equal to the power set of the Naturals, which is (two to the power of Aleph Null). As to why this is so, and what is the bijective function between the two, I have no idea, although the OP sort of explains that. If anyone wants, they can PM me and I can send them the original topic on that forum, where the guy explains it to me. I don't want to post it here to avoid advertising.

Anyway, I love infinity, and hope my post brings you closer to this state of utter speechlessness in the face of such beauty . And I'd like to thank the OP for taking the time to share this with us.

Cheers,

Gabe