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.
The description of it would be significantly different.
A non-standard explanation for this relationship between bases and the axiom of choice would involve a class of lattices sometimes called matroids. Among the lattices that satisfy the definition of matroids are the partition lattices or equivalence lattices. In other words, there is an identity criterion involved with using vector spaces to represent "reality" (it would be more precise to say that representative something is a little more algebraically complex -- a Euclidean point space defined over an inner product space). One of the motivating principles for the study of matroids had been the idea of studying linear independence without the numerical coefficients. So, there is a confluence between the mathematics of vector spaces and the logic of equivalence classes in these structures.