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
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....