Re: cantor's theorem - the pieces of the puzzle

On Nov 16, 9:09 pm, Virgil <Vir...@xxxxxxx> wrote:
In article
<13a11e4d-d1dc-42ea-bb7a-72bf667be...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

Virgil,

That those "countable additivity" properties hold over the summation
of the infinitely many modelled by the iterated finite in the analog
does not moderate that the countably many additive partitions or
summands characterizes lines completely in terms of their content as
real numbers. There are countably many on the line and none other
that would disturb the power series with trichotomy of their
exponents. Close the vector space, infinity is in it.

Math content zero.



Ah, that's excellent. Why should I waste it on a reply to you? In
your face, Hancher! It's countably additive! The orthogonal power
series are unique component-wise. Integration does happen over the
infinitesimal. So, take a uniform distribution over the unit
interval. As was recently under discussion here, each interval of the
same length has the same probability of being a sample, b-a you know,
in the degenerate a single point [a, a]. So then each real number is
random. The function is defined for each finite difference.

The content of the above is totally disconnected.
The content that was below, I has snipped as being of even less
interest.

Virgil,

Typically from you, that's disputable.

No, that makes sense. It's meaningful using the terms correctly.

It is a coherent explanation of there only being countably many
elements in the integration, and that is enough to totally
characterize for example the area of the shape defined by the same
function and boundaries, in the radius of convergence of the
expansion.

Compare the definite integral, defined by a function, to for example
the particular geometric shape, composed of smaller geometric shapes,
that has as its area the same value as the integral.

Area(f, boundary}_b^d = Int_boundary fn d_boundary

The function is a boundary, generally replacing constant boundaries in
the lower dimensions. In generalized coordinates derived by linear
transforms, everything is quite general and in terms of content of the
system, countable, in that it only takes particularly countably many
to extract "true" (obviously expressible classically) results, for
example those that correspond with area and under natural translations
that preserve area in geometry remain constant. The differential
areas that compose the area are real-valued, if infinitesimal, and the
operation on them _is_ the sum. Then, the difference is not between
naturals and reals, but between unit interval of reals and infinite
interval of reals. Then, in the cycloperiodic, which I use to
describe reduction systems, standard analytical results hold
(countably).

With suitable matching conditions the integral over the shape is the
same thing as the area of the shape. You can't divide the shape into
uncountably many partitions, for example dividing the line into
uncountably many (disjoint) partitions, because then each of those
would have a rational. So, the shape is divided into countably many,
and that's adequate to describe it, in terms of constants, and
particularly few of those. The shape can be defined by its points in
set theory, those that exist for linear transformations, at least in
terms of all linear transforms. Yet, via subset and well-ordering the
reals, not all point subsets exist, or: those subsets do exist, as
they do by axiom, and the system is countable.

Ross

--
Finlayson Consulting
.

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