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.
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> > > Correct. But here I use only members, namely FISONs.
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> > > If you subtract a finite number from an infinite number, nothing
> > > happens.

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> > > If we remove a finite line from our list
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> > > 1
> > > 1,2
> > > 1,2,3
> > > ...

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> > > the union of all lines is not changed.
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> > 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.

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> > > We can even repeat this procedure for an infinite number of times
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> > Not in Wolkenmuekenheim, in which nothing can be done infinitely many
> > times, even theoretically, so Zemo's [sic] riddles prevent all motion there.

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> 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.
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> Basically this gets into the difference between the _definition_  of
> successor (axiomatization), and successor as a _structural
> consequence_, of variety (deduction).
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> 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