In article <5acbaafd-203e-4fef-8d81-be285ceca8e8@y9g2000yqg.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> 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.
Before the tree is built, every path that exists so far is, by necessity, one of those to be used to build the tree, so is in P.
> 2) Path p cannot be distinguished from every path of P.
Whatever countable set of paths, P, may be used to build a maximal infinite binary tree, there are too many paths in the resulting tree to be contained in the original P.
