On 14 Apr., 21:46, fom <fomJ...@nyms.net> wrote: > On 4/14/2013 2:23 PM, WM wrote: > > >>> 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. > > Actually, there is. But, since your general > purpose is to use quantifiers ambiguously, > you would respond as you have. > > > For every finite natural number we have FIS(n) of the first column = > > line(n) of T. > > Simply repeating what you said incorrectly in the > first place will not make it correct.
But looking at T will convince every mathematican.
> >>> 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.) > > Then, why did you use the term "majorant"? The internet search > turned up a bunch of European hits. That's fine. The English > equivalent is "upper bound".
No, the majorant (a_n) of a sequence (b_n) is a sequence (a_n) such that for every n >= n_0 : b_n =< a_n or b_n c a_n. > > > Fact is, that the majorant criterion holds for every finite natural > > number. And there are no other elements in |N.
For every n: FIS(n) of the first column is a subset of line n.
Claiming that the first column contains more than the lines is false in mathematics. None of the lines contains an actually infinite set.