On 17 Apr., 16:27, fom <fomJ...@nyms.net> wrote: > On 4/17/2013 2:45 AM, WM wrote: > > > On 16 Apr., 22:45, Virgil <vir...@ligriv.com> wrote: > > >> It is not clear to me, or to anyone sensible, that the entire sequence > >> of all naturals in A, which has no maximal member, is "in" any line of > >> naturals that has a maximal member, and it is equally clear that every > >> line does have a maximal member. > > > Is there any number of A that is not in at least one line? > > Responding to a statement saying "It is not clear..." > with a question. How typical.
A rhetorical question. Even Virgil knows that the answer is "no".
> > For all n : (1, ..., n) of A is in line n of B. > > You really need to become more consistent concerning > when a number corresponds with a mark ('1') and when > a number corresponds with a finite initial segment > of the natural numbers *given* by the axioms.
Neither of them has to do with axioms. Those are merely applied to confuse newbies. If AC is accepted, the reals can be proven to have a well-ordering. If AC is true, the reals can be well-ordered. If AC is not accepted, reality is the same as if AC is accepted. A lot of trash has infiltrated mathematics. For the value of axioms compare Lesniewski (§ 251). > > > For all n : line n ob B is in (1, ..., n) of A > > This current mode of argumentation corresponds to the > subsequence of triangular numbers
My argument it is based on the truths that FISONs (1, ...,n) are identified (not only enumerated) by their last numbers n like polar vectors are identified by the points they point to. n = n is an identity. > > > So we have an identity. There is no actually infinite line in B, so > > there is no actually infinite sequence A. (Of course A is potentially > > infinite as is B.) > > No. What we have it one of your particular 'thought > experiments'
All mathematics is thought experiments. In order to maintain your silly opinion, you would need two different values for n in (1, ..., n) and as n. To refute that "idea" does not require a big thought experiment (though every kind of thinking can be called a thought experiment), but your argument will not attract but repulse any mathematicians.
> When the ancient Greeks actually considered what is > involved with *calculation*,
namely practical experiments with stones (limestones, calcis or chalculi) or thought experiments describing such calculations.