On 1/5/2013 6:19 PM, Ross A. Finlayson wrote: <snip>
> The irrationals are the rationals' complement in the reals. Each > rational has a particular representation as a reduced fraction. I > agree that's a general development.
But, what you are not seeing is that the reals within set theory are built up from the finite ordinals initially using the axiom of pairing. The order relation in that construction yields the identity criterion of some set as a "real" number. On the basis of set theory, it is not even clear that the reals are a set, although one would have expected a contradiction to arise. What I mean by this is that forcing admits models where the cardinality of the real numbers can be just about any cardinal number.
> The rationals and irrationals > each have the properties of being dense in the reals and not > satisyfing continuity as the reals do, though some naive definitions > see the rationals meet the definition. The (standard) reals as set > setminus the (set of) rationals is the irrationals, and vice versa: > the irrationals' complement in the reals is the rationals. > > I can see that each point in the space R has uncountably many > neighborhoods (in ZFC), and that for each neighborhood, as open, it is > covered by a union of elements from a countable collection of open > subsets of R, so it is first-countable, with only countably many > neighborhoods with that basis.
Almost. Compact sets (closed and bounded) are usually spoken of in terms of coverings by open sets.
Relative to basis sets, every open set is a union of basis sets.
> > Ah, then for r_beta, here r_gamma is a next lesser element in the > normal ordering from R, from a new well-ordering of the radii less > than r_beta. Choice has that for each r_beta, from the uncountably > many r < r_beta, there is quantifiable one of those as r_gamma, > leaving uncountably many less than r_beta and r_gamma in the normal > ordering. In that sense it scatters the order topology, of a well- > ordering, from the existence of all the others, where the well- > ordering of the reals isn't unique. Yet, then arranging that into a > transfinite induction schema and finding more than countably many > r_alpha for ordinal alpha, that yields a contradiction that between > any r_beta and r_gamma there's a distinct rational. That would yield > a contradiction: there are and aren't uncountably many distinct > neighborhoods of a point in R, each with correspondingly distinct > elements of Q, and P.
Ok. Since you are ignoring logic, you need to explain how you are mapping a transfinite sequence to the open disks. Just as with Skolem's criticisms of Zermelo, the function has to exist in whatever model you are taking to correspond with truth conditions.
If, in fact, you are trying to map that transfinite sequence to irrational numbers in such a way as the open disks have decreasing diameters, then you are mapping the transfinite into the order relation of the real numbers through which the rationals are understood as dense. You cannot map an uncountable transfinite order monotonically into any order of real numbers that satisfies the completeness axiom.
No such function can exist in any consistent model.
> > Mutual consistency would have that _all_ the properties hold _all_ the > time. > When they don't then yes that would be -Con(ZFC). > > The irrationals are the rationals' complement in the reals, and the > rationals are dense in the reals. > > Then, you seem to imply that the sets' elements: are dependent on > their order. Would you expand on that?
The identity criterion is based on the order of the natural numbers as the construction builds pairs. The fact that two sets having the same elements in set theory are the same does not place a real number in the universe of sets. The construction I outlined above does. It takes the finite ordinals. It generates equivalence classes of pairs that correspond to integers...
At each step the identity criterion of the number system follows from the order relation from the previous step.
You seem to be thinking about real numbers as if they are urelements.
> > http://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te > > Viete as algebraic and Descartes as geometer: mutual consistency would > have _all_ the properties hold _all_ the time. I'd be interestes to > know what you saw as their liberties which gave us mathematical > progress.
WM is whining about Cantor, but it was Descartes that put nondenumerably many names on a geometric form
Viete began using variables and parameters. In so doing, he conflated the distinction between monadic numbers and geometric magnitudes that existed prior to his time. The Greek mathematicians did not treat numbers in the way that we do now. But, we end up with infinities to explain it all.