On 17 Apr., 17:17, fom <fomJ...@nyms.net> wrote: > On 4/17/2013 9:57 AM, WM wrote: > > > > > > > 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. > > http://planetmath.org/basis > > not quite
Absolutely quite. Of course AC is true and every vector space has a basis. Alas there is nothing uncountable.