On 14 Jun., 17:31, Owen Jacobson <angrybald...@gmail.com> wrote: > On 2009-06-13 14:29:05 -0400, WM <mueck...@rz.fh-augsburg.de> 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].
If 1/pi exists as an actually infinite digit sequence.
There are two statements: 1) Path p can be distinguished from every path of the countable set P that is used to construct the tree. 2) Path p cannot be distinguished from every path of P.
The first is assumed to be correct before P was used to construct the tree. Above you are just arguing in favour of (1). The second is assumed to be correct after P was used to construct the tree.
One of them is false, unless it is a magic tree where something happens during construction. But I do not believe in magic, least in mathematics.
The second statement can be proved to be correct, because in fact you are not able to distinguish p from P (by means of digits) when the tree has been constructed.
Therefore the first statement is falsified. This means that there are no actually infinite digit sequences, but only potentially infinite digit sequences. For *every* digit d_n of 1/pi there is a terminating path that is identical to the digit sequence d_1 to d_n and even until d-(n^n^n...^n). Therefore your argument above is not sufficient to save set theory.