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Re: Matheology § 224
Posted:
Mar 18, 2013 2:39 AM
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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
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