On 4/15/2013 6:52 AM, WM wrote: > 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. >
So, you have made yourself clear on how a majorant is different from an upper bound (???).
Until such time as you provide axioms, definitions, and a deductive calculus to prove your statements I will continue to interpret your statements mathematically, when possible.
So, I repeat:
The fact is that for any *given* natural number one can find another different natural number whose existence derives from the axiomatically *given* successor function and whose property of being an upper bound derives from the order relation on the domain that is determined by the axiomatically *given* successor function.