> >Please be aware, that I require mathematics, not matheology. In the > >latter case the faithful believe that A contains only all finite > >natural numbers and that B also contains all finite natural numbers, > > A "finite natural number" is the same as a natural number.
Absolutely true, but it has turned out to be useful, sometimes, to recall this fact.
> So asking for a number in A not in B makes no sense, there > are no numbers in B in the first place. B is a list of sets > of numbers.
Yes, B is a list or sequence of sets of numbers.
> A is a set of numbers not in B. Answering > your question.
You claim that B does not contain the set of numbers A. This claim is false.
Proof: Consider the table T of numbers:
1 2, 1 3, 2, 1 ... n, ..., 3, 2, 1 ...
Every line is a set of numbers, every columns is a set of numbers too. (I drop the curly brackets.) The lines are the same as my set B, the first column is the same as my set A.
You claim that A = |N but that no term of the sequence B is |N.
It is easy to see that for every *finite* natural number n, there is a term of B that has the elements 1 , 2, ..., n.
Therefore B is a majorant of the finite initial segments (FISs) of A *for all n* - and there is nothing else in A, by definition. This cannot change by unioning its elements or FISs, because the union does not change the elements and does not add anything that has been missing before.
So your claim is false, unless you think that after all finite natural numbers n something unexpected and infinite happens. But that is matheology. And even if thousands of seemingly intelligent people think so, there is no more reason for anybody to accept that than to accept astrology or comparable nonsense.