Infinity

Also known as Aleph-Null. But it is called "Aleph Null" when you mean the cardinality of the number. In an informal sense all of these concepts are the same, but there are important technical distinctions to be made. Infinity in calculus refers to a real quantity which increases without bound. It is not so much a number, as a way of expressing the behavior of a limit.omega refers to the order-type of the set of non-negative integers. Aleph-null on the other hand, is defined as the cardinality of the set of positive integers. In plain speak Aleph-null is the "number" of numbers. The problem with this is that the set of positive integers is suppose to represent all things that we might wish to count. It however, can not count itself. So is the "number" of numbers, even a number then? In some ways we can treat aleph-null as a number, in that we can compare it to other numbers and determine which is larger. Using the concept of one-to-one correspondence, we can rationally say that aleph-null is larger than any positive integer, even though the previously prevailing wisdom was that infinity was not a number and could not be compared in this way. But accepting this view leads to some mind bending anomalies. Using one-to-one correspondence we can show there are just as many even numbers, squares, cubes, etc. as there are positive integers, despite that fact that these are all subsets of the positive integers. This violates the principle that the "whole is always greater than any proper part of the whole". So aleph-null is a number such that a proper part of it is still just as large... baffling. When working with finite numbers we implicitly understand the exclusivity of "larger" vs. "equal". A number can not be both. Hence when one particular correspondence shows that one finite set has more than another finite set, we know that no correspondence can exist which shows they are equal. Not so with infinite sets! Even if we have a correspondence which shows one is larger than the other, it doesn't necessarily mean that a correspondence doesn't exist showing they are equal. in order for an infinity to be truly larger than another infinity it must be shown that "there does not exist ANY one-to-one correspondences". Since there must be an infinite number of such correspondences, checking each one individually is not an option. It is necessary to come up with a proof which shows the impossibility of such a correspondence. It might be assumed that all infinities are essentially the same and can be put in one-to-one correspondence with each other. However, there are infinities which could not be put in one-to-one correspondence with aleph-null. There is no one infinity ... but an infinity of infinities ... (See aleph-one). This is infinity paradise, or nightmare, depending on your perspective.