In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 22 Dez., 00:34, William Hughes <wpihug...@gmail.com> wrote: > > > Yes it is definable. It has been defined. Nevertheless it does not > > > exist, because the sets do not exist. > > > > It is definable which means that WM can use it to prove that the > > bijection exists, but it does not exist. > > The bijection of all finite words with all natural numbers has been > defined in binary: > > 0 > 1 > 00 > 01 > 10 > 11 > 000 > and so on.
While I can see some binaries, I do not see any words being paired off with them. Thus WM's claim fails. > > From this definition the natural number belonging to any desired > finite word can be obtained. It can easily be translated into any > other language. Or do you need some help?
What is the natural number of the word "help"? > > Nevertheless there is no set of all natural numbers and no set of all > infinite words.
In standard mathematics, like in ZFC, for example, there is a set of all natural numbers, or at least a set having all the properrties one wants or needs for a set of natural numbers. > > It is the same with pi. The (potentially) infinite string of digits of > pi can be defined. In fact there are (potentially) infinitely many > definitions. Nevertheless there is no actually infinite string > expressing pi.
On the other hand, the ratio of the circumference of a circle to its diameter is perfectly well defined. > > Yes, I know that is not easy to understand.
WM certainly illustrates his own difficulty in understanding quite clearly! --