I'll throw up another example, along the same lines, of why infinity is weird. Plus, it's pretty neat topic.
Take the set of natural numbers:
0,1,2,3,4,5,6, ......... (going out to infinity)
And take the set integers:
(going out to minus infinity) ......... -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6, ......... (going out to infinity)
Now, which set has more elements in it?
Intuitively, you would want to say integers, but that would be wrong. It can be shown that a one-to-one correspondence can be set up between the two sets. That means for every element in the integers there is a unique element in the set of natural numbers. They have the same "number" of infinite elements - or in mathematical speak, they have the same cardinality. This cardinal "number" gets a special name, which if I remember correctly is called "chi" (insert greek letter here)
Some sets have a larger cardinality than the natural numbers, like the real numbers that I mentioned above, and they get a different symbol. Any set with the same cardinality as natural numbers is considered countably infinite, for obvious reasons. Any set with a larger cardinality than the natural numbers is considered uncountable.
Here's an excerpt from WIKI, which is okay, for more info:
A set is a collection of elements, and may be described in many ways. One way is simply to list all of its elements; for example, the set consisting of the integers 3, 4, and 5 may be denoted {3,4,5}. This is only effective for small sets, however; for larger sets, this would be time-consuming and error-prone. Instead of listing every single element, sometimes an ellipsis ('…') is used, if the writer believes that the reader can easily guess what is missing; for example, presumably denotes the set of integers from 1 to 100. Even in this case, however, it is still possible to list all the elements, because the set is finite; it has a specific number of elements.
Some sets are infinite; these sets have more than n elements for any integer n. For example, the set of natural numbers, denotable by , has infinitely many elements, and we can't use any normal number to give its size. Nonetheless, it turns out that infinite sets do have a well-defined notion of size (or more properly, of cardinality, which is the technical term for the number of elements in a set), and not all infinite sets have the same cardinality.
To understand what this means, we must first examine what it does not mean. For example, there are infinitely many odd integers, infinitely many even integers, and (hence) infinitely many integers overall. However, it turns out that the number of odd integers, which is the same as the number of even integers, is also the same as the number of integers overall. This is because we arrange things such that for every integer, there is a distinct odd integer: … −2 → −3, −1 → −1, 0 → 1, 1 → 3, 2 → 5, …; or, more generally, n → 2n + 1. What we have done here is arranged the integers and the odd integers into a one-to-one correspondence (or bijection), which is a function that maps between two sets such that each element of each set corresponds to a single element in the other set.
However, not all infinite sets have the same cardinality. For example, Georg Cantor (who introduced this branch of mathematics) demonstrated that the real numbers cannot be put into one-to-one correspondence with the natural numbers (non-negative integers), and therefore that the set of real numbers has a greater cardinality than the set of natural numbers.
In mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.
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