Date: Dec 9, 2012 5:20 PM
Subject: Re: Mathematics in brief
On Dec 9, 12:24 pm, Zuhair <zaljo...@gmail.com> wrote:
> On Dec 9, 10:59 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> > The real numbers as we teach them are at issue. Everything else may be
> > left to the "experts" or fools of matheology.
> > Regards, WM
> The real numbers you teach can be presented by countable models, and
> also can be proved uncountable in other models.
Measure theory, that we use for all standard results, has yet
countable additivity. The partition of the segment is to countably
many partitions, adding those back up gives us the integral, the area
under the curve.
That there are even non-measurable sets of reals, is from Vitali's
argument that there would be some infinitesimal constant, Vitali's c,
the sum of which over the naturals is two (or between one and three).
Measure theory doesn't need re-Vitali-ization to see that the results
are courtesy: countable additivity. As well Banach and Tarski's ball-
doubling might see much more realm for application, given only
slightly different first principles.
The paths of the tree are of the nodes, with the rationals being quite
Yes, the structure of transfinite cardinals is a mathematical
abstraction tractable to our devices of reason, but, nobody's
discovered applications for them yet, for real analysis (or physics).
And: dependence on them as the foundation: closes the door on any
consideration of considering the points: falling in a line, in their
natural order, for what they do.
Simply enough, Cantor's results are true in that the line can't be
drawn, in the graphical and the intuitive sense and as a plain
projection of the space, without drawing them (its points) in order.
And, they're important in allowing to mathematics the infinite and
transfinite ordinals, and relevant transfinite induction, for the
ordinals besides the cardinals. And, they do establish a relevant
ordering of infinite digital sets, but not the only one nor for that
matter one unavailable to the construction in the ordinals, simply a
more direct one. (Half of the integers are even.) And, they're
important as a part of the historical development, yet another chapter
in the discussion since antiquity, of the realm of thought.
The Universe as it exists would be its own powerset, and Cantor did
see an Absolut infinity in his Mengenlehre, and ZFC is post-Cantorian
with infinite ordinals. Yggdrasil: there's an eagle on top, and the
eagle on top of it. There's nothing to contain the universe but
itself (re Kant's the Ding-an-Sich). And Cantor, Georg, also wanted a
completed infinity he could count from, toward the origin, not just
to, from the origin, in his own words. And: it's turtles.
Most all our stories start with nothing and go from there. That
answers a real deep question.
And then there was light.