There's an infinite number of types of infinity. Basically it turns out that the cardinality (size) of the set of subsets of a set cannot be the same as the cardinality of the set itself - there's a theorem called Cantor's Theorem that proves this (it has a really neat, intuitive proof).
So if you take the natural numbers N = {1,2,3,4,5,6,...} and their set of subsets P(N), also known as their "power set" where P(N) = {{1},{1,2},{3,59,23},{any other group of integers}}, roughly, then the size of N is definitely not the same as the size of P(N).
The size of P(N) is referred to as "aleph one" and the size of N is referred to as "aleph zero". If you consider the set of subsets of P(N), P(P(N)), you get "aleph two", and so forth, which is an infinite sequence of infinities of increasing infiniteness :-)
Oh yeah, from memory the countable infinity of the rationals is the same size as the countable infinity of the integers. Two sets are the same size if you can define a one-to-one mapping from one to the other. Any two countable items you just count them both, mapping first to first, second to second, etc., so all infinite countable sets are the same size.
Nope. You can count the rationals various ways, so there are the same number of them as there are natural numbers (and also integers).
Another interesting fact about numbers (well, I think it's interesting): for every real number, even irrational numbers such as pi, there is a countable sequence of rational numbers that approaches the real number arbitrarily closely. The real numbers are the analytical closure of the rationals. Another metaphor ripe for the picking :-)
Hmm. OK, help me out here. I thought that, for example, 0.25 (or 1.25 or whatever) was a rational number (because it can be expressed as a fraction) but not a natural number (because it's not a 'whole'). So by those rules there are more rational numbers than natural ones. Is that all wrong?
So if you take the natural numbers N = {1,2,3,4,5,6,...} and their set of subsets P(N), also known as their "power set" where P(N) = {{1},{1,2},{3,59,23},{any other group of integers}}, roughly, then the size of N is definitely not the same as the size of P(N).
The size of P(N) is referred to as "aleph one" and the size of N is referred to as "aleph zero". If you consider the set of subsets of P(N), P(P(N)), you get "aleph two", and so forth, which is an infinite sequence of infinities of increasing infiniteness :-)
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Another interesting fact about numbers (well, I think it's interesting): for every real number, even irrational numbers such as pi, there is a countable sequence of rational numbers that approaches the real number arbitrarily closely. The real numbers are the analytical closure of the rationals. Another metaphor ripe for the picking :-)
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