Quote:
Originally Posted by zoncxs
There are more real numbers between 0 and 1 than there are whole numbers from 0 to positive infinity.
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On one hand, there are as many rational numbers between 0 and 1 as there are whole numbers between 0 and infinity. But on the other hand, there are more real numbers between zero and one than there are rational numbers between zero and one. (See Cantor's diagonal proof.)
Long ago, when I was taking real analysis, I asked my instructor why Cantor's argument didn't apply equally to whole numbers. He answered that rational numbers, in decimal representation, could have an infinite number of digits, but whole numbers could not. So using an apeirohedral die would give you many numbers with infinite digits; in fact I think that statistically that would be the expected result, and perhaps getting a number with finite digits might be so unlikely as to have effective probability 0. A sword of infinite length would be unwieldly. But that problem doesn't need real numbers; you can get it already just with rational numbers.