On 9 Dez., 00:09, Virgil <vir...@ligriv.com> wrote:
> > 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.
But it takes either an infinite sequence of decimals (which cannot be given by giving its terms) or a finite definition of the infinite sequence in order to define the real number such that it can be distinguished from every other real number. And that must be possible, because otherwise you would not know how to find every finite sequence of decimals. > > > > > 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?
My argument shows that there is a contradiction in assuming bijections as determining cardinal numbers of actually infinite sets.