An uncountable countable set
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From: Virgil on 3 Oct 2006 02:37 In article <4521fa43(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > In IFR we compare sets of reals mapped from a common set over a common > range. In discussing "well orderings", we are free to arrange things as > we wish, are we not? In discussing any orderings, we are free to arrange things as we wish, and even to disregard all orderings. > > > > >> On the other hand > >> I don't know why I said "neither can the reals". In any case, the only > >> way the ordinals manage to be "well ordered" is because they're defined > >> with predecessor discontinuities at the limit ordinals, including 0. > >> That doesn't seem "real" > > > > In what sense of "real". There are subsets of the reals which are order > > isomorphic to every countable ordinal, including those with limit > > ordinals, so until one posits uncountable ordinals there are no problems. > > > > In the sense that the real world is continuous We do not know that the real world is "continuous". > The real line is a line, with each point touching two others. In a mathematical "real: line, no points can "touch" since there is always others (uncountably many others) between them. > >> and the axiom of choice aside, I don't see > >> there being any well ordering of the reals. > > > > The point is that no one can see it even if, given the AC, it is there. > > It is not proven, from what I understand, that one cannot discover an > explicit well ordering of the reals. Then by all means do it yourself.
From: Virgil on 3 Oct 2006 02:39 In article <4521fad8(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <4521128b(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> They are saying that the vase empties because every ball inserted is > >> removed. They agree that this does not occur before noon, when there are > >> always balls in the vase, but by noon the vase is empty. But they cannot > >> say that, even though there are balls before noon, and none at noon, > >> that the vase "became empty" at noon, because they are claiming "the > >> limit doesn't exist". So, don't ask me what they mean. I can't figure it > >> out. > > > > > > The lack of any "limit" and the emptyiness at noon are only two of many > > quite straightforward things that TO cannot figure out. > > Define "straightforward". Compare and contrast with "assbackwards". "Straightforward", at least in this context, means following from some specific set of axioms. "Assbackwards", in this context, means TO's way of doing things.
From: Virgil on 3 Oct 2006 02:43 In article <4521fc40$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <1159797618.679513.221400(a)i3g2000cwc.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >> Virgil schrieb: > >> > >> > >>>> By the only meaningful and consistent definition: A n eps |N : > >>>> |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. > >>>> Do you challenge its truth? > >>> I challenge the "truth" of its being the ONLY meaningful and consistent > >>> definition. > >>> > >> If we insist on unique results in mathematics, then this definition and > >> the bijection exclude each other. > > > > Uniqueness of results depends on what one is doing. > > If one is considering finding a possible order relation on a set of two > > or more elements, would "Mueckenh" insist that there is a unique result? > > > > Does "Mueckenh" claim that there is only one function mapping > > {1,2,3,...} to {2,4,6,...} or vice versa? > > There is only one natural order-isomorphic relation, defined by y=2x. > The inverse clearly indicates the second set is half the first, whether > subset or no. That presumes, among other things, that a proper subset must be in every sense smaller than its superset, but that is not true, as the bijection y = 2x proves..
From: Tony Orlow on 3 Oct 2006 02:47 Virgil wrote: > In article <45216360(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <1159784963.257471.99490(a)i42g2000cwa.googlegroups.com>, >>> Han.deBruijn(a)DTO.TUDelft.NL wrote: >>> >>>> Virgil wrote: >>>> >>>>> The problem as I recall it was this: >>>>> >>>>> Given an infinite set of balls numbered with the infinite set of >>>>> naturals and an "infinitely large" initially empty vase, and a positive >>>>> time interval in seconds, t, and a small positive time interval in >>>>> seconds, epsilon ( much smaller than t/2). >>>>> (1) At time t before noon balls 1 through 10 are put into the vase and >>>>> at time t - epsilon before noon ball 1 is removed. >>>>> (2) At time t/2 before noon balls 11 through 20 are put into the vase >>>>> and at time (t - epsilon)/2 before noon ball 2 is removed. >>>>> ... >>>>> (n) At time t/2^(n-1) before noon balls 10*(n-1)+1 through 10*n are put >>>>> in the vase and at time (t-epsilon)/2^(n-1) before noon, ball n is >>>>> removed. >>>>> ... >>>>> >>>>> The question is what will be the contents of the vase at or after noon. >>>> There is no noon in this problem. >>> There is in mathematics. The problem does not exist in physics at any >>> time. >> So, the limit "exists". > > There is a function, f, from the reals, R, representing time in suitable > units to the set of von Neumann naturals,N = {0,1,2,3,...}, such that > f(t) = the number of balls in the vase at time t, for all t. Yes, f(t)=9*floor(-log2(noon-t)), in minutes. > This function has an isolated jump discontinuity at each time at which > any balli is inserted or removed from the vase and and at noon there is > a condensation point of such discontinuities and the function is > unbounded to the left in every neighborhood of noon, but is still > right-continuous at noon. > There is no jump. That's a leap of faith. > > > Is that what you're saying? There's "the >> infinite case"? > > There is the function case, as described above. "I can neither confirm nor deny." Thanks. > >>> Everything mathematical is purely mental, so can only match "reality" by >>> some sort of analogy, not by any actuality. >> "Purity" is velocity of c, or zero volume for finite substance. It's >> unattainable, practically. > > Whoever said math had to be practical? Einstein, for one, expressed > great surprise at how practical something so inherently impractical as > mathematics kept turning out to be. Einstein built on many concepts regarding the practical description of the universe.
From: Virgil on 3 Oct 2006 02:48
In article <4521fcd5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <1159798407.217254.275710(a)m7g2000cwm.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >> Tony Orlow schrieb: > >> > >> > >>>> This is an extremely good example that shows that set theory is at > >>>> least for physics and, more generally, for any science, completely > >>>> meaningless. Because the numbers on the balls cannot play any role > >>>> except for set-theory-believers. > >>> Yes. I was flabbergasted by this example of "logic". The amazing thing > >>> is, set theory is supposed to apply where we know nothing except for the > >>> membership status of each element in a set, and yet, here is applied > >>> this property of labels that set theorists claim is crucial to answering > >>> the question. Set theory in the finite sense is a fine thing, but when > >>> it comes to the infinite case, set theorists don't even know anymore > >>> what they're TRYING to do. > >> One should think that set theory, if useful at all, should be capable > >> of treating problems like this. But here we see it fail with > >> gracefulness and mastery. > >> > >> Set theorists always see only the one ball escaping the vase but not > >> the 9 remaining there. So they can accept that there are as many > >> natural numbers as rational numbers. I cannot understand how this > >> theory could invade mathematics and how I could believe it over many > >> years without a shade of doubt. > >> > >> Regards, WM > > > > If we alter the problem to start with all the balls in the vase and > > remove them according to the original schedule then every ball spends at > > least as much time in the vase as before, but not everyone will see that > > the vase is empty at noon. > > > > Those who argue that having the balls spend less time in the vase leaves > > more of them in the vase as noon, have some explaining to do. > > No, if you start with all the balls in the vase, it clearly becomes > empty, but when you are adding more balls per iteration than you are > removing, the eventual emptying of the vase is impossible. Thus TO claims that by putting balls into the vase earlier, but removing them as before, one will find fewer of them in the vase at noon. I do not see the logic of that argument. |