On Thursday, 31 July 2014 13:30:35 UTC+2, Michael Klemm wrote: > WM wrote; > > > > > For every k in |N there is n_0 in |N such that for n >= n_0: (n-k, n] c > > s_n, i.e., an interval that does not contain rational numbers indexed by n > > or smaller naturals. > > Why so complicated? For any finite number of reals there are always gaps > between any two of them.
There are always gaps between real numbers. Impossible to close them.
But here we have the overwhelming evidence that the rationals cannot be enumerated by the naturals. The number of unit intervals, each one containing infinitely many rationals without index =< n, increases infinitely, i.e., beyond any upper bound. Of course a matheologian will brush this aside by the standard blether "cardinals are not continuous". But everybody with a critical intellect will ask *why* he should believe this. The honest answer is: it is forced by the claim that infinity can be finished. Who would give a dime for that? Exept matheologians with their well-known brain defect, carefully implanted during the study.