Date: Mar 18, 2013 3:02 AM
Author: fom
Subject: Re: Matheology § 224

On 3/18/2013 1:39 AM, Ross A. Finlayson wrote:
>
> So, the question is, if Virgil says there exists


First, although I have not read that signature
element too carefully, I doubt Virgil is claiming
that anything exists. WM made a claim. Virgil
is demanding proof of the claim according to the
standard meaning of the terms.

> 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).


You are making a mistake in these equations.

The multiplication in the definition,

f(x+y) = f(x) + f(y)
f(ax) = a*f(x)

is a scalar multiplication.

Of course, so much mathematics is done in
familiar number systems where the scalar
domain is related to the arithmetic of the
additive abelian group that one does not
think twice about it.

This is different. There is no definition
for multiplication of sequences from the
binary tree as sequences in the binary tree.

Linear mappings in this sense are not
immediately about continuity. Continuity
is a topological property. Linearity in
this sense is an algebraic property.


>
> 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.)
>


Virgil is not making a claim. He is asking
that a claim be substantiated.

> Then, about compact admissibility

<snip>

It is a pretend construction that was
done to give you an idea of what might
be required for WM to meet Virgil's
expectations.

I would be required to do some work
to make it something you could put
on a wikipedia page....