Date: Mar 18, 2013 3:21 AM Author: Virgil Subject: Re: Matheology � 224 In article

<ce69aaf4-ebe0-4085-8306-966697313da2@8g2000pbm.googlegroups.com>,

"Ross A. Finlayson" <ross.finlayson@gmail.com> wrote:

> On Mar 17, 10:28 pm, fom <fomJ...@nyms.net> wrote:

> > On 3/17/2013 7:11 PM, Ross A. Finlayson wrote:

> > >

> >

> > > A simple and trivial

> > > continuous mapping was noted.

> >

> > > Regards,

> >

> > > Ross Finlayson

> >

> > That is not enough Ross.

> >

> > By definition, a linear map must satisfy

> >

> > f(x+y) = f(x) + f(y)

> > f(ax) = a*f(x)

> >

> > So, the domain must at least have the

> > structure of a module since it needs

> > to have an abelian addition of domain

> > elements and a map from the domain

> > into itself with a scalar multiplication.

> >

> > Furthermore, it is unlikely that one

> > could take the scalar multiplication

> > to be the Galois field over two

> > elements since multiplication by

> > zero would be the zero vector and

> > multiplication by one would be

> > the identity map.

> >

> > A morphism with that scalar field

> > could not reasonably be expected

> > to have a linear map with a

> > system of real numbers.

> >

> > In order to build a scalar that

> > could even possibly serve this

> > purpose, given WM's claims related

> > to various finite processes, one

> > would have to invoke compactness

> > arguments involving completed

> > infinities.

> >

> > For example, for any non-zero

> > sequence of zeroes and ones

> > that becomes eventually constant

> > with a trailing sequence of zeroes,

> >

> > 1001101000......

> >

> > we can replace that sequence with

> > a trailing sequence of ones,

> >

> > 1001101111......

> >

> > We want to use these forms because

> > of the products

> >

> > 1*1=1

> > 1*0=0

> > 0*1=0

> > 0*0=0

> >

> > Then, coordinatewise multiplications

> > along the trailing sequence of ones

> > retains a trailing sequence of ones.

> >

> > In addition, on the interval

> >

> > 0<x<=1

> >

> > we can associate 1 with the constant

> > sequence,

> >

> > 111...

> >

> > Given these facts, we can now say that

> > a collection of infinite sequences is

> > "compactly admissible" if for every

> > finite collection of those sequences

> > coordinatewise multiplication yields

> > a sequence different from one

> > consisting solely of an initial

> > segment of zeroes followed by

> > an initial segment of ones.

> >

> > In other words, even though

> >

> > 000000111...

> >

> > may be representationally

> > equivalent to

> >

> > 000001000...

> >

> > for some purposes, compact

> > admissibility has to ignore

> > what happens in this conversion.

> > The situation above is

> > interpreted as corresponding

> > with a non-compact set of

> > sequences.

> >

> > Given this, sequences like

> >

> > 1000...

> > 11000...

> > 110000...

> > 1101000...

> >

> > yield

> >

> > 1111..

> > 11111...

> > 110111...

> > 1101111...

> >

> > whose coordinatewise product

> > is

> >

> > 1101111...

> >

> > So that the original sequence

> > is compactly admissible.

> >

> > Given a construction along these

> > lines, one could then think of

> > compactly admissible collections

> > as possibly forming a sequence space

> > as described here

> >

> > http://en.wikipedia.org/wiki/Hilbert_space#Second_example:_sequence_s...

> >

> > Obviously, the compactly admissible

> > collections are not defined as

> > converging in the sense of a sequence

> > of partial sums.

> >

> > Equally obviously, I have not done

> > all the work necessary to decide

> > whether or not this would work.

> >

> > My purpose here is to explain that

> > the scalar multiplication would

> > require a construction along these

> > lines just to even begin to talk

> > about whether or not WM could

> > do what Virgil is asking.

>

>

> I looked to it that a linear mapping would need a vector space over a

> field. Then basically it was found various magma(s), those being a

> set equipped with an operation closed in the set, using addition being

> the integer part of natural addition and multiplication the integer

> part of natural multiplication. But that is not a field because it

> lacks distributivity, and multiplicative inverses. Then there's the

> notion to define addition-1 being the non-integer part of natural

> addition, and addition-2 being the non-integer part of natural

> addition, that equals one if the non-integer part is zero, so there

> are two operations with that are associate, transitive, have inverses

> in the field and distinct identities, but addition-2 isn't

> distributive.

>

> So, the question is, if Virgil says there exists a field over [0,1],

> or the elements of the CIBT or Cantor set, there would be a continuous

> function f: R_[0,1] <-> R that had a (+) b = f^-1(f(a)+f(b)) and a (*)

> b = f^-1(f(a)f(b)), and that f(ab) = f(a)f(b) and f(a) + f(b) = f(a

> +b).

A linear mapping between linear spaces need not be in any sense a

continuous mapping or involve any continuity at all, as no topological

structure is required of linear spaces in general.

So Ross is, as usual, off on an irrelevant tangent again.

--