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. > > Zuhair
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 large.
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.