On Mar 24, 1:44 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > On Mar 24, 12:22 pm, Virgil <vir...@ligriv.com> wrote: > > > > > > > > > > > In article > > <d28c00b4-cbad-4067-a2b1-cf63a83f9...@g4g2000yqd.googlegroups.com>, > > > WM <mueck...@rz.fh-augsburg.de> wrote: > > > On 24 Mrz., 03:01, Virgil <vir...@ligriv.com> wrote: > > > > > > In set theory we can construct the set of all elements that have a > > > > > certain property. Does that mean that the property vanishes if too > > > > > many elements belong to the set? > > > > > It only vanishes from among non-members of that set. > > > > Correct. But here I use only members, namely FISONs. > > > > If you subtract a finite number from an infinite number, nothing > > > happens. > > > > If we remove a finite line from our list > > > > 1 > > > 1,2 > > > 1,2,3 > > > ... > > > > the union of all lines is not changed. > > > Except that that list, which does not have a last member, cannot be one > > of WM's lists. which are all required to have a last, though evanescent, > > member. > > > > We can even repeat this procedure for an infinite number of times > > > Not in Wolkenmuekenheim, in which nothing can be done infinitely many > > times, even theoretically, so Zemo's [sic] riddles prevent all motion there. > > Are you saying that lim_n->oo Sum_i=1^n 1/2^i =/= 1, or, that > Sum_i=1^oo 1/2^i = 1? > > Then, where the lines are n-sets: what is the union of the lines? > Via induction, any element of N is in line n and all following. There > does not exist n e N s.t. not exists m>n-set contains n, and for all m> n. If the union of the lines is not a line, then only the union of > > finitely many lines is a line, transfer doesn't hold. Yet, there are, > or aren't, infinitely many lines. > > Basically this gets into the difference between the _definition_ of > successor (axiomatization), and successor as a _structural > consequence_, of variety (deduction). > > The sum of all the finite numbers (natural integers): isn't a finite > number, and for no finite number is it their sum. Yet, addition is a > closed operation in the integers. That gets into the difference > between operations that are closed for finitely many, and unboundedly > many, and infinitely many applications of the operation, here > addition. The transfer principle, that which is so for each is so for > all, can be further refined to bounded and infinite transfer. And, it > should be. Then there is reasoning as to separate the notions of the > quantifiers for each / for any / for every / for all into various > categorizations of application, that the "universal" quantifier > correctly reflects the existence of transfer. > > Then, just as there are Euclidean and non-Euclidean geometries of > reasonable import, there are Archimedean and non-Archimedean natural > integers, not just potential and complete, but along those lines. > > In the consideration of the application of the operations, and Mazur's > swindle as it were or the well known result of 1-1+1-1..., in the > telescopic it is seen that addition and its complement as subtraction, > is finite-closed and bounded-closed, and may be infinite-closed, _for > the range of the inputs_, that the set isn't just defined by its > elements, _but all operations contingent upon it_. And, for some > collections, their elements are defined by their access, here the > continuum of real numbers: sensitive in their ordering. > > Then, where the limit is the sum, or regardless of that and the limit > exists, EF sweeps [0,1], BT the CIBT, the list as 0, 1, 2, ..., > prefixed by the radix: has for the only element different from each, > that each is only different from each other. > > To each their own. And, all for one and one for all, as it were. >
It is much as the consideration of convergence (rel. divergence), here of the convergent series or infinite product. Addition is closed for the convergent series to exist. It only need be bounded-closed for the standard results to hold. Yet, the limit, _is_ the sum, else, the sum is partitioned into its pieces, again the limit _is_ the sum, as we know Zeno's arrow arrives. Here, then in what today we call a single reference frame, Zeno's arrows arrive because _all_ of them arrive. Arithmetical scalar properties of non-zero convergent series exist because they share the free variable.