Date: Mar 18, 2013 2:39 AM Author: ross.finlayson@gmail.com Subject: Re: Matheology § 224 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).

So from an apocryphal comment that there is a linear mapping and thus

vector space and field over [0,1], I wonder how Virgil backs this

claim, as I well imagine it's not a linear function with f(0) = -oo

and f(1) = oo. (And it is.)

Then, about compact admissibility, yes there are general notions that

if N and R are compactified it's with points at infinity, then about

the form and product you mention, there is not an inverse of the

product, and I don't see it defined for all the elements of the CIBT

or Cantor set. Please feel free to further explain that.

Regards,

Ross Finlayson