> > Proof: Consider the table T of numbers: > > > 1 > > 2, 1 > > 3, 2, 1 > > ... > > n, ..., 3, 2, 1 > > ... > > > > 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. > > Do you not mean for "every *given* natural number"?
There is no need of giving numbers. For every finite natural number we have FIS(n) of the first column = line(n) of T. > > > 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. > > There is no "*given* natural number" that is an upper bound > for "*every* finite initial segment of natural numbers" from > A. >
That is not necessary. (In addition, there is no upper bound of lines.) Fact is, that the majorant criterion holds for every finite natural number. And there are no other elements in |N.
Therefore it does neither help to quote the infinity of |N nor the tha axiom of infinity. Mathematical fact is simply that every number in |N is finite and subject to law and order of mathematics, namely
FIS(n) of the first column = line(n) of T. (*)
The number of numbers does not play a role because (*) in no way depends on the number of numbers. You could with same right argue, that the invention of artificial electricity causes the first column to surpass every line.