Date: Mar 24, 2013 4:44 PM
Subject: Re: Matheology § 224

On Mar 24, 12:22 pm, Virgil <> wrote:
> In article
> <>,
>  WM <> wrote:

> > On 24 Mrz., 03:01, Virgil <> 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.


Ross Finlayson