On 1 Feb., 12:25, forbisga...@gmail.com wrote: > On Friday, February 1, 2013 12:12:59 AM UTC-8, WM wrote: > > On 1 Feb., 06:01, forbisga...@gmail.com wrote: > > > Yes it does, under the assumption that the continuation > > > of all of your initial segments is not the repeating sequence > > > . If it is then I will > > > If you could determine it by nodes or digits, you need not make > > provisions. So you cannot determine by nodes whether 1/3 it there or > > not. QED. > > I can determine at the finite initial segment 0.01 that either > 1/3 is not there or 1/4 is not there because one has the continuation > [01} and the other .
It is not the question what is "not there", but the only question is what is there! If you want to apply a number in mathematics, then you have to uniquely distinguish it from all other numbers, i.e., from the complete Bibary Tree. You show this already when you talk about 1/3 and 1/4. That are (abbreviations of) unique definitions of numbers.
> Each of those paths are distinguishable from each other
And without a finite definition you could not even say what paths you are talking about, because you cannot write an infinite sequence.
That is the basic error of matheology: From every finite definition F of a number we can obtain an infinite sequence of digits S. F ==> S But now you reverse this implication and claim that an infinite sequence could be used to define a number. S ==> F. It is really a shame to see that stupidity with mathematicians who, by the general public, are considered to be intelligent people. But many of them make this blatant error, similar to concluding from "all inhabitants of Boston live in America" onto "all americans live in Boston".
A Cantor does not construct a diagonal, unless every line is defined. But then also the diagonal is defined and, therefore, belongs to a countable set.