Date: Mar 24, 2013 5:07 PM Author: ross.finlayson@gmail.com Subject: Re: Matheology § 224 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:

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> > 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.

Can't have a diagonal without a square.

Regards,

Ross Finlayson