An uncountable countable set
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From: Tony Orlow on 3 Oct 2006 01:39 Ross A. Finlayson wrote: > Tony Orlow wrote: >> Ross A. Finlayson wrote: >> >> Hi Ross - >> Nice to see you. I hope you don't mind my adopting the Finlayson Numbers >> in my IFR sort of way. Cheers. >> >>> Tony Orlow wrote: >>>> Virgil wrote: >>>>> In article <45201554(a)news2.lightlink.com>, >>>>> Tony Orlow <tony(a)lightlink.com> wrote: >>>>> >>>>>> Virgil wrote: >>>>>>> A set is a container, and is not one of the objects that it contains. >>>>>> It is nothing more or less than its contents. >>>>> It is determined uniquely and entirely by its contents, as stated in the >>>>> axiom of extentionality. >>>> So we agree. There is nothing besides the members. >>> Also, the members are only sets. >> Each member is a set? Well, yes, a set of property values which >> distinguish the members. >> >>> I feel that I have been fair presenting construction of reason that >>> enable a rational person to make their own decision that, for example, >>> forceing uses a universal ordinal, >> I don't know much about "forcing", but it sounds kind of aggressive and >> not very nice. ;) >> >> the generic extension of N contains >>> no elements not in N yet bijects to R, >> Sequential form can be applied, I think, to any structure, if "forced". >> (Is that what that means?) >> >> the Borel/Combinatorics impasse, >>> and those as basically non-controversial. >> The impasse being over...what? Thanks in advance. :) >> >>> I see some truths in some of the posters with alternative opinions, >>> also, I'm quite familiar with the standard viewpoint. I use my >>> viewpoint, which I develop in part myself. >> How dare you even feign that right? The nerve! >> >>> These "infinities", as they are, are on the one hand various, as >>> infinite more various than any finite set could ever be, on the other >>> hand they each share certain properties and it seems at root they are >>> each irregular. >> Not just regular old numbers? Well I suppose not... >> >>> Then, where some "least" infinite set has a given construction, there >>> are what are to most implicit aspects of that construction, implicit in >>> the sense of being true and not just unstated but known, to somebody, >>> thus true thus implicit. >> Like the set of multiples of neN, as n->oo? That would be one definition >> of the least, beyond the simplest. >> >>> There is no universe in ZF where there are only sets, and not all of >>> them. There is no set of all sets, nor set of all ordinal (nor >>> cardinal) numbers as sets, no collection of a variety of other things >>> that are necessary for the establishment of certain formal arguments, >>> in ZF. >> There are objects, properties, and relations. Is there else? >> >>> If you talk about and use those things, as most do with for universal >>> quantification over sets, then thus necessarily ZF is not sufficient >>> and is at once contradictory. >> With self, or reason? >> >>> Oh, I'm not a crank. >> I am. :) >> Does that mean I'm wrong? >> Define "crank". >> I think it means someone who makes the entrenched "cranky". >> >> I think Goedel tells you the null axiom theory is >>> the only possible theory, where if it's inconsistent or incomplete it's >>> not A theory. >>> >>> Ross >>> >> According to most, that would mean, in the "light" of Godel, that "there >> is no theory". Is there a spoon? >> >> Have nice continuum, >> >> Tony > > Oh, I don't mind. > > You mention of Inverse, some years ago I was trying to figure out if a > system like ZF had something along the lines of axiom of inverse. You > might want to frame your inverse function rule in terms of a space of > functions, for example the Hilbert space leading to the L^infinity with > linear and non-linear differential operators and so on. Perhaps. I see the definition being that there exists a mapping from a standard set, the naturals or reals, defined by a formula f with an inverse g, such that f(g(x))=g(f(x))=x, and such that f(x)<f(y) -> g(x)<g(y). Then, over any range [a,b], the set contains floor(g(b)-g(a)+1) elements. Isn't that sufficient? > > In representing numbers as sets, or for that matter other mathematical > constructs as sets, if in the set theory there are those objects there > are all the rules about them mechanistally as sets. There are only > sets in set theory, if it's not a set, it's not in the set theory. What's a set? That's the question. Is a set equivalent to a property of its elements? > > It's easy to say, "here's a set of numbers, here's how they're defined > and operate", but in a set theory that entire self-contained > description must be in the set theory, for example along the lines of a > type theory, where various items fit into various categories. So, > having the empty set be zero might be very useful transitively but the > number zero is not the empty set. In some cases considering that it is > leads to some useful results from vacuity, almost all of which are > meaningless. I rather see 0 as the origin, a point in a quantitative space, relative to which all other points are specified. A set is a discrete collection, and tying together the notions of collection and measure is kind of what mathematics is all about. So, sets are a part of that, but not the whole picture. It's like trying to pretend the universe is made of particle, and waves don't exist. :) > > (Quantify over sets: not a set in regular "set theories".) > > Set theory as applied to the finite is almost totally > non-controversial. It is only in the infinite that for whatever reason > various researchers and discutants have mutually controversial and > various interpretations of those systems interpreted as formal > mathematics. Of course. We have no direct evidence to back up any of our claims. The merits of one approach or another can only be measured by their consistency with known finite examples. May the best generalization win. > > For example, half of the integers are even. Well-order the reals. If the H-riffics can't do it, well, I just think it's impossible. Prove me wrong. :) > > Oh, forceing/forcing is just a word that basically means fiat, to > basically axiomatize a completed infinite set. Borel-vs-Combinatorics > refers to Borel's set being almost everywhere in the reals, and > Combinatoricist's set being almost everywhere in the reals, yet they > are
From: Tony Orlow on 3 Oct 2006 01:53 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".
From: Tony Orlow on 3 Oct 2006 01:59 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. :)
From: Tony Orlow on 3 Oct 2006 02:02 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. If I rub the vase, will a genie come out?
From: Tony Orlow on 3 Oct 2006 02:06
Virgil wrote: > In article <45215d2f(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Han de Bruijn wrote: >>> stephen(a)nomail.com wrote: > >>>> how many balls are in the vase at noon? >>>> >>>> What does your "mathematics" say the answer to this >>>> question is, in the "limit" as n approaches infinity? >>> My mathematics says that it is an ill-posed question. And it doesn't >>> give an answer to ill-posed questions. >>> >>> Han de Bruijn >>> >> Actually, that question is not ill-posed, and has a clear answer. The >> vase will be empty, if there is any limit on the number of balls, and >> balls can be removed before more balls are added, but it is not the >> original problem, which states clearly that ten balls are inserted, >> before each one that is removed. That's the salient property of the >> gedanken. Any other scheme, such as labeling the balls and applying >> transfinitology, violates this basic sequential property, and so is a ruse. > > One can pose any gedanken one likes. > > If TO does not like to be able to tell one ball from another, he does > not have to play the game, but he should not ever try to pull that in > games of pool or billiards. If distinguishing balls gives a less exact answer, and a nonsensical one to boot, then that attention can be judged to be ill spent, and not contributing to a solution at all. It is clear that sum(x=1->oo: 9) diverges, is infinite, not 0. It's ridiculous to think otherwise. |