Question I've just read about Cantor'd diagonal argument, and I have some questions about it... Let's say we want to map every real number between 0 and 1 to natural numbers. If I'm not mistaken, that can be done this way: If we have number of form 0.abcdef... (letters stand for decimals, but only some are shown since there is infinite amount of them), then we can produce number N which equals 2^a * 3^b * 5^c * 7^d * 11^e * 13^f * (next prime)^(next decimal). For example, number 0.12 equals to 2^1 * 3^2 (* 5^0 * 7^0 * ...) = 18. Given any natural number N, we can easily determine which real number it represents (if any). My first question is: is all this consistent with Cantor's diagonal argument? (Can both be true at the same time?) Cantor proved there is no one-to-one mapping (not just any mapping), is that important for his result? If yes, it somehow seems intuitive to me, at least at the first sight, that one-to-one mapping can be achieved by simply removing natural numbers that don't represent any real number between 0 and 1, and thus we could say "n'th representing number is number x (which is equal or bigger than n, because x = n + number_or_removed_numbers), which decoded, stands for real number y". Now, this would produce a list vulnerable to original Cantor's diagonal argument. I don't understand why... could it be that "normal" mapping (the one which just turns real numbers to natural, not one-to-one mapping) works, but there is something illegal when that mapping is converted (at least using described "technique" of removing certain numbers) to one-to-one mapping?

Accepted:

November 17, 2010