In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 8 Dez., 10:41, Zuhair <zaljo...@gmail.com> wrote: > > > I will give two answers, the first one depends on the finity of the > universe, the second one does not. > > > Mathematics is "discourse > > and discourse needs the tools available to us. They all are finite. > > > > How those forms are known to us? the answer is through their > > exemplification as part of the discourse of consistent theories about > > form. > > Consistent means that discourse about all elements is possible. > We know that all finite words belong to a countable set. We know that > no infinite word can be mentioned without having a finite name. > > Therefore this silly argument is really silly: > "I can decide for a real number x whether a real number y deviates in > its decimal (or any other) expansion from that of x." > The complete infinite expansion of x is never known, not even in an > infinite universe, but only the finite formula allowing expansion to > any required level.
It only takes finitely many decimal places to determine that the decimals for two real number differ. > > So the argument is silly that there are uncountably many x because x > has an infinite expansion. No x is known without a finite formula, > name, word, ...
The definition of a set being uncountable was that there cannot be any surjection from the naturals. The set of all reals provably satisfies this requirement. Does you argument prove such a surjection does exist?