On 21 Jan., 19:07, Zuhair <zaljo...@gmail.com> wrote:
> Doesn't that say that mathematics following ZFC is only grounded in > Mythology driven principles! > > Doesn't that mean that ZFC based mathematics is too imaginary that > even if consistent still it is based and rooted in fantasy that cannot > really meet reality!
ZFC is not consistent unless inconsistencies are defined to be no inconsistencies, distinctions need not be distinguishable, incomletenesses need not be incomplete, and so on.
Consider, for instance, all terminating binary fractions b_n 0.0 0.1 0.00 0.01 0.10 0.11 0.000 where some numbers are represented twice (in fact each one appears infinitely often). Constructing the diagonal d we find that d differs from *every* b_n *at a finite place*.
Since the above list is complete, which is possible because all terminating fractions, as a subset of all fractions, are countable, it is impossible that the diagonal differs from all entries b_n at a finite place. If this was possible, the list would have a gap, namely a finite initial segment of d. That means, the diagonal up to every bit can be found in the list. And after every finite place there is nothing that could distinguish two numbers.
Therefore the diagonal does not increase the cardinal number of the listed entries b_n.
The diagonal may be infinitely long. But what does that mean? Every given number of bits is surpassed. But the same holds for the entries of the list. The only difference could be a bit of the diagonal that has no finite index. But such bits are not part of mathematics and of Cantor's argument.