On Oct 26, 5:20 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 26 Okt., 22:07, William Hughes <wpihug...@gmail.com> wrote: > > > On Oct 26, 2:47 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > On 26 Okt., 18:42, William Hughes <wpihug...@gmail.com> wrote: > > > > > Incorrect. All other elements are sets with > > > > a largest element. b_oo is a set without a largest element. > > > > Sets are defined by elements, not by missing elements. > > > It is trivially obvious that a set without a largest > > element cannot have the same elements as a set with largest > > element. > > > Thus there is no i in N such that b_i = b_oo > > It is pretty obvious that a set like b_oo cannot have any element at > all. If b_oo existed, it would be constructed as the limit of the the > infinite union of the b_i
not quite: as the limit of the b_i or the union of the b_i
> - in the same way as |N is constructed as > the limit of the union of all natural numbers
again limit of the union is redundant. limit of all natural numbers, or union of all natural numbers.
b_oo is so constructed. Indeed b_oo is N.
The important point. Let the set of column indexes be C (a subset of N) Then n in C iff n in some element of (b_1,b_2,b_3,...) We immediately have C=N=b_oo. Let the set of line indexes be L (a subset of N) Then n in L iff n in some element of (d_1,d_2,d_3,...,d_oo). We immediately have L=N=d_oo so number of lines = number of columns