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. > 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. >>
Your arguments are based on the structure of triangular numbers.
It is "obvious" by the same geometric reasoning you hope will convince others that classical mathematics in its modern conception is flawed.
So, in your denial you choose another geometric intuition. Just for kicks, here is the same link about vector spaces I just provided for the earlier remarks on AC
And, just for more kicks, here is what Leibniz had to say about the principle of identity of indiscernibles along with a definite interpretation of an individual in a metric space according to Cantor:
> "What St. Thomas affirms on this point > about angels or intelligences ('that > here every individual is a lowest > species') is true of all substances, > provided one takes the specific > difference in the way that geometers > take it with regard to their figures." > > Leibniz > > > > "If m_1, m_2, ..., m_v, ... is any > countable infinite set of elements > of [the linear point manifold] M of > such a nature that [for closed > intervals given by a positive > distance]: > > lim [m_(v+u), m_v] = 0 for v=oo > > then there is always one and only one > element m of M such that > > lim [m_(v+u), m_v] = 0 for v=oo" > > Cantor to Dedekind >
I keep trying to tell you not to use that darn equal sign unless you really do believe in infinity.
Actually, that goes for names too.
> "All existential propositions, though true, > are not necessary, for they cannot be > proved unless an infinity of propositions > is used, i.e., unless an analysis is > carried to infinity. That is, they can > be proved only from the complete concept > of an individual, which involves infinite > existents. Thus, if I say, "Peter denies", > understanding this of a certain time, then > there is presupposed also the nature of > that time, which also involves all that > exists at that time. If I say "Peter > denies" indefinitely, abstracting from > time, then for this to be true -- whether > he has denied, or is about to deny -- > it must nevertheless be proved from the > concept of Peter. But the concept of > Peter is complete, and so involves infinite > things; so one can never arrive at a > perfect proof, but one always approaches > it more and more, so that the difference > is less than any given difference." > > Leibniz >
Unlike your hero Newton who let the mathematics be his philosophy, Leibniz actually had enough respect for others to explain himself.
(In Newton's defense, gravity is just a little bit hard to explain.)