On 2009-06-13 14:29:05 -0400, WM <email@example.com> said:
> We agree that in Cantor's diagonal argument, applied to real numbers, > the numbers are represented and identified solely by their digits. No > further information is available. > > We assume that a real number p can be distinguished from a set Q of > real numbers q by general considerations, for instance, if p is a > transcendental number and Q consists of rational numbers q only. > > Of course it would be impossible to distinguish p from all q, because > for every digit d_n of p, there is a number q that shares all digits > up to d_n with p.
Wait, so you're arguing that you can't distinguish 1/pi (roughly 0.31830988...) from a set containing all rationals in [0, 1] because, for example, 3/10 is equal to 1/pi to the first decimal place, 31/100 is equal to the second, and so on? What about the non-zero difference between any one rational number and 1/pi?
|1/pi - 3/10| is greater than the rational 1/50. |1/pi - 31/100| is greater than the rational 1/125. In fact, for each rational q in [0, 1], 1/pi differs from q by an amount greater than some other rational r in [0, 1], so we can distinguish 1/pi from every rational in [0, 1].