On 26 Jan., 23:19, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> > It is unclear why you apparently are unable to understand, that we are > > working in the set of terminating decimals. Therefore the diagonal > > cannot be actually infinite, although there is no last digit. > > Let me ask you a very simple question. > > Is 0.777.... a terminating decimal representation or a > non-terminating decimal representation?
That depends on the domain where you work in. We have started to work in the domain of terminating decimals. Since the diagonal consists only of (changed) digits of these decimals, it is obviously a terminating decimal. Now, to answer your question: You did not say where you take 0.777... from. And obviously that cannot be determined from the digits, as I jusr explained.
I could answer: You can look whether there is a digit of 0.777... that is not in a (in fact in infinitely many) finite initial segment(s). Then you have a proof that 0.777... does not belong to the set of terminating decimals. But it is clear that you cannot find such a digit. Therefore you can only decide your question by defining where 0.777... has been talen from. The reason for this uncertainty is the fact, that the Binary Tree constructed by all finite paths cannot be distinguished by digits (i.e. without further definition) from that Binary Tree that contains all infinite paths too.
> > Does the real number corresponding to 0.777.... have a terminating > decimal representation? > > Much thanks for answer what is surely a trivial question.
You are in error. The question unfortunately is far from being trivial. It has only been overlooked that the sets F of all finite decimals and R of all infinite decimals cannot be distinguished other than by a finite definition.
Would be nice if you really tried to understand that, although it requires a complete change of your habbits of thinking. But if you consider the different Binary Trees, you should come to the correct result.
Here a trivial example: The set of all finite initial segments of the infinite path of 1/3 = 0.010101... contains all nodes that belong to that path. So you cannot distinguish *by nodes* whether you are working in the set F or the set R. You need additional information. But that informtation does not matter in Cantor's argument.