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 So... do I have this right? Posted By: MrHen Date: 12/16/06 6:35 a.m. In Response To: Re: hehe, Infinity (treellama) : The only way the set of rational numbers could be "bigger", is if : you could find some that you don't have enough natural numbers to assign : to them. But you can't: if you use Cantor's ordering you have a natural : number for every rational number you can think of. Now I think I vaguely remember why the reals are "larger". Does this sound right? When looking at and comparing reals and integers there is no one-to-one correspondence (bijection?) between them. Looking at the integer "1" we can say that the next number is "2". There is no such "next number" on the real line. The number after "1" is... what? The rational number have a strict definition involving integers on integers which gives them an ordered progression. The reals have no such thing. There are an infinite number of numbers between "1" and "2" and trying to find them all would be an utter waste of time. The key is not which "numbers" are in the set but the size of the set. I vaguely remember trying to think of it as how fast the set is growing instead of how many numbers are in the set. The infinite set of reals grows infinitely faster than the infinite set of integers. The latter grows at the same speed as the rationals. Does that sound right?
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