On 4/14/2013 2:23 PM, WM wrote: > On 14 Apr., 19:55, fom <fomJ...@nyms.net> wrote: > >>> 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.
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.
What you mean to say is "for every *given* natural number".
>> >>> 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".
> Fact is, that the majorant criterion holds for every finite natural > number. And there are no other elements in |N. >
Until you clarify your use of the term, 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.