Date: Mar 18, 2013 3:21 AM
Subject: Re: Matheology � 224
"Ross A. Finlayson" <firstname.lastname@example.org> 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
> 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
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.