An uncountable countable set
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From: Virgil on 1 Oct 2006 16:16 In article <452014e5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <451df1cb(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> Since there is a specific time prior to noon at which any given numbered > >>> ball is removed, one must conclude that they have all been removed by > >>> noon. > >> Numbers aside > > > > The whole point is that according to the definition of the problem, one > > is not allowed to put number aside. > > There are many riddles and joke which throw in extraneous information to > distract one from the obvious answer. This is not one of them. The result is specifically dependent on the numbering of the inserted and balls removed, so any alteration presents a different problem. > > >>> Suppose the instead of being put in in batches of 10, they are all put > >>> in when the first one is put in, but removed according to the original > >>> scheduled. In this case it is clear that every ball is removed. > >> In that case addition is all performed first, then all subtraction. > >> Sure, that can work. But the measure of the addition is divorced from > >> that of the subtraction. If one measures the whole process in some kind > >> of common time frame, given the numbers for the additions and > >> subtractions, one can get a "rate of filling", which isn't going to > >> change in this case and cause the thing to suddenly empty. > > > > I beg to differ! There are in this game infinitely many balls in the > > vase at every instant after the balls are put in andbefore noon, and > > none left at noon. > > The something occurs at noon which requires explanation. Roll the tape > backwards from noon, one iteration at a time, and you'll see what I mean. How does one roll back a "last" iteration when there isn't one? > > >>> So that TO is claiming that putting the balls in earlier, but taking > >>> them out the same way, leaves fewer balls than the original way. > >> At least I have an explanation. You still haven't explained why I can't > >> relabel the balls afterwards and make them disappear from the vase. :) > > > > Yes I have. > > No, you haven't. Assume the labels AREN'T fixed. Can I do that now? Only by changing the components, which may very well change the result. If a chemical experiment called for an alkali and you used an acid instead, would you expect the same result? > > > > > If TO wants to play a different game, fine, but he cannot then pretend > > that it is not a different game. > > Assume it is. What then? How far does Virgilogic stretch? Not at all. It is merely not my logic but mathematical logic which saya that changing the rules changes the ruels. If one is given a certain set of rules which produce a certain outcome then any alteration of those rules will produce an outcome irrelevant to the consequences from the original set of rules.
From: Virgil on 1 Oct 2006 16:19 In article <45201554(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <451df41c(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <451dd1f2(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>> > >>>>>> Are you saying that aleph_0 naturals only require ln(aleph_0+1)/ln(2) > >>>>>> bit positions? > >>>>> Not at all. I am talking about indvidual natural numbers as members of > >>>>> N, ,not N itself, which is not a member of N. > >>> > >>>> And also for every set of contiguous naturals starting at 0 EXCEPT for > >>>> N. Why EXCEPT for N? > >>>> > >>> For the same reason that a paper sack holding oranges is not an orange. > >>> > >> A set is a sack? It is nothing besides the elements it includes. > > > > 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.
From: Virgil on 1 Oct 2006 16:21 In article <45201597(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <451df438(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <451dd293(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>>>> In article <451d83c4(a)news2.lightlink.com>, > >>> > >>>>> So what balls remain in the vase at noon, oh waffler extraordinary? > >>>> For n balls inserted, balls n/10+1 through n remain at the end of any > >>>> iteration n. You specify the number of iterations, I'll give you the > >>>> sum. What was it? Aleph_0? > >>> All iterations executed before noon. > >> And that would be how many? > > > > All of them. > > How many in quantitative terms? All the people in the world doesn't > describe the global population as a quantity. "Quantitative terms" is TO's bag. If he want that he will have to figure it out for himself. "Al of them" is enough for me.
From: Tony Orlow on 1 Oct 2006 16:22 David R Tribble wrote: > Tony Orlow wrote: >>> For the sake of this argument, we can talk about infinite reals, of >>> which infinite whole numbers are a subset. > > David R Tribble wrote: >>> What are these "infinite reals" and "infinite whole numbers" that you >>> speak of so much? >>> >>> If you've got a set containing the finite naturals and the "infinite >>> naturals", how do you define it? N is the set containing 0 and all >>> of its successors, so what is your set? > > Tony Orlow wrote: >> The very same, with no restriction of finiteness. Any T-riffic number >> has successor. :) > > Well, 0 is finite, and the successor of 0 is finite, and the successor > of any finite in N is just another finite in N. Therefore N must > contain only finite naturals. No, the successor to an infinite, after an infinite number of successions from 0, is infinite. > > It's sporting of you to drop the requirement that all the naturals in N > have to be finite, but since all of them are, it's meaningless to say > "with no restriction of finiteness". That's kind of like saying N > contains all naturals "with no restriction of non-integer values". > I can say that, but it does not change the fact that all the members > of N are integers. Is the successor to ...11110000 not equal to ...11110001? > > So I ask again, where are those infinite naturals and reals you keep > talking about? It's obvious they are not in N. > Not it's not.
From: Tony Orlow on 1 Oct 2006 16:25
mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > >> In article <1159648683.273921.24350(a)e3g2000cwe.googlegroups.com>, >> mueckenh(a)rz.fh-augsburg.de wrote: >> >>> Dik T. Winter schrieb: >>> >>>> In article <1159438112.240001.268540(a)m7g2000cwm.googlegroups.com> >>>> mueckenh(a)rz.fh-augsburg.de writes: >>>> > >>>> > Dik T. Winter schrieb: >>>> > >>>> > > > The successor function *is* counting (+1). >>>> > > >>>> > > Wrong. >>>> > >>>> > After a while you will have run out of the predefined successor, >>>> > unavoidably. >>>> >>>> succ(x) = {x}. >>> That is nothing else but a veiled form of +1. (This form of addition >>> of 1 is due to Zermelo's, a little bit different from that of von >>> Neumann's.) >> "Mueckenh" has it backwards, "+1" is merely a veiled form of "another", >> which predates counting by millennia. And both Zermelo's and von >> Nuemann's successor echo "another" faithfully without any requirement >> for "+1". > > Another one. Yes, that is +1. > > Regards, WM > I dunno Wolfgang. I think you're going out on a limb to say that one is the same as '1', and that "+" means "in addition to what's already there". ;) Tony |