On 21 Feb., 21:51, Virgil <vir...@ligriv.com> wrote:
> > Or consider the union of natural numbers in a set B while there > > remains always one number in the intermediate reservoir A. > > > A B > > --> 1 -->{ } > > --> 2,1 -->{ } > > --> 2 -->1 > > --> 3, 2 -->1 > > --> 3 -->1, 2 > > --> 4, 3 -->1, 2 > > --> 4 -->1, 2, 3 > > ... > > --> n -->1, 2, 3, ..., n-1 > > --> n+1, n -->1, 2, 3, ..., n-1 > > --> n+1 -->1, 2, 3, ..., n-1, n > > ... > > > One would think that never all naturals can be collected in B, since a > > number n can leave A not before n+1 has arrived. > > > Of course this shows that ZF with its set of all natural numbers is > > contradicted. > > WM's A and B are not sets but sequences of sets, so if WM wants to > consider a limit to any such sequences, he must first define what he > means by such a limit, as there is no universal definition for "the" > limit of a sequence of sets.
By definition of A we know it is never empty. That implies that B never contains all natural numbers. B always has a last element, but we cannot know it, because if we say n, then n+1 is as well in B.
That is the property of infinity. I am not responsible for that behaviour, I only recall what our ancestors knew.
