> 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.
This is your proof that ZF is inconsistent, is it? > > The diagonal may be infinitely long. But what does that mean?
It means that d is not a terminating fraction, you moron, so d is not part of the set of all terminal fractions and hence you haven't shown that *this* enumeration is not surjective, much less that the set of terminal fractions is uncountable.
> 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.
You are incapable of very basic mathematical reasoning. You are a shame to your school.
-- "And yes, I will be darkening the doors of some of you, sooner than you think, even if it is going to be a couple of years, and when you look in my eyes on that last day of work at your school, then maybe you'll understand mathematics." -- James S. Harris on Judgment Day