An uncountable countable set
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From: Tony Orlow on 29 Sep 2006 16:57 Randy Poe wrote: > Tony Orlow wrote: >> Virgil wrote: >>> In article <451bac34(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>>>> If the vase is empty at noon, but not before, how can that not be the >>>>>> moment that it becomes empty? >>>>> Saying that it is empty is quite different from saying anything about a >>>>> "last ball". andy does not deny that the vase becomes empty, he just >>>>> does not say anything about any "last ball out". >>>> Does that answer the question of **when** this occurs? Of course not. >>> It does answer the question of "whether" it occurs. "When" is of lesser >>> importance. >> So, you have no answer. > > If something doesn't occur, the question "when does it occur" > does not have an answer. "[R]andy does not deny that the vase becomes empty". That sounds like it occurs. > > If I ask you what date you took a trip to Mars last year, > would you have an answer? Does the vase become empty? > >> And so it's not important. I see. > > I think you would consider a trip to Mars very important, > but nevertheless you would not be able to tell me when this > trip occurred. > > - Randy > I hear it's kind of boring up there anyway. Buncha rocks and dust.
From: Tony Orlow on 29 Sep 2006 17:03 Virgil wrote: > In article <451d5d29$1(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <1159438112.240001.268540(a)m7g2000cwm.googlegroups.com>, >>> mueckenh(a)rz.fh-augsburg.de wrote: >>> >>>> Dik T. Winter schrieb: >>>> >>>>> > The successor function *is* counting (+1). >>>>> >>>>> Wrong. >>>> After a while you will have run out of the predefined successor, >>>> unavoidably. >>> If that were ever to happen, one would have discovered a largest >>> possible number. But it does not ever happen, because for every set x >>> there is a set UNION(x,{x}) which is its successor. >>> >> I believe Wolfgang is saying that, once you run out of the starting >> known successive symbols of your language, your alphabet, you then have >> to employ an actual number system, using those elements recursively. >> Since alphabets are generally finite, you can never represent "infinite" >> quantities, in terms of string length. > > One does not need to, as every finite natural is representable by a > finite string. But not every finite real, such as pi. That's why WM says pi doesn't really exist as a number. Personally, I think it exists as a point on the real line. >> As you know, I like to represent specific infinite quantities using >> finite strings, but of course it's only a countable set of infinite >> numbers, since the infinite sequences are defined using repeating >> patterns, making them rational fractions of declared infinities. They're >> T-riffic! :) > > More like Horrific! You used to call them T-errible. You should reserved that for the H-riffics, but then, they already know how evil they are. ;) All the finite strings are already in use for finite > naturals, so how does TO propose to re-use them for infinite numbers? > Print them in a different color? Won't work in NGs. We've been through the T-riffics. That's just an e-fart.
From: Tony Orlow on 29 Sep 2006 17:07 Virgil wrote: > In article <451d5e15(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Randy Poe wrote: >>> Tony Orlow wrote: >>>> Han de Bruijn wrote: >>>>> Virgil wrote: >>>>> >>>>>> In article <d12a9$451b74ad$82a1e228$6053(a)news1.tudelft.nl>, >>>>>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>>>>> >>>>>>> Randy Poe wrote, about the Balls in a Vase problem: >>>>>>> >>>>>>>> It definitely empties, since every ball you put in is >>>>>>>> later taken out. >>>>>>> And _that_ individual calls himself a physicist? >>>>>> Does Han claim that there is any ball put in that is not taken out? >>>>> Nonsense question. Noon doesn't exist in this problem. >>>>> >>>>> Han de Bruijn >>>>> >>>> That's the question I am trying to pin down. If noon exists, that's when >>>> the vase supposedly empties, >>> Why does the existence of noon imply there is a time >>> which is the last time before noon? >>> >>> It doesn't. >>> >>> - Randy >>> >> I never said it did. When did I say that? I will offer this simple >> logical argument. If the vase ever became empty, it would be because one >> ball was removed, as per the gedanken, but 10 balls would have been >> inserted immediately beforehand. The vase would therefore have had to >> contain -9 balls, which I'm afraid is simply impossible. Don't you? It's >> a ridiculous set-theoretical result. > So is TO's conclusion that there ought to be infinitely many naturally > numbered balls in the vase for which he can not find the number of any > one of them. If you stuck aleph_0 consecutive balls in there, you have aleph_0/10+1 through aleph_0 left in the vase. > > Let us consider a slightly modified experiment in which as each ball is > removed from the vase, it is put into an initially empty urn. That was my idea a year ago. > > Now at or after noon, ball 1 is in the urn (and not in the vase). > Furthermore for every n-marked ball in the urn, ball n+1 is also in the > urn. > > Thus, by induction, EVERY naturally numbered ball is in the urn (and > not in the vase). > > So which balls are still in the vase and not in the urn, TO? What's amazing about that situation is that you are adding 10 balls to the vase for every one you remove and put in the urn, but the urn ends up with all the balls and vase becomes empty. When's your birthday? I'll get you a label maker.
From: Randy Poe on 29 Sep 2006 17:09 Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <451bac34(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>>>> If the vase is empty at noon, but not before, how can that not be the > >>>>>> moment that it becomes empty? > >>>>> Saying that it is empty is quite different from saying anything about a > >>>>> "last ball". andy does not deny that the vase becomes empty, he just > >>>>> does not say anything about any "last ball out". > >>>> Does that answer the question of **when** this occurs? Of course not. > >>> It does answer the question of "whether" it occurs. "When" is of lesser > >>> importance. > >> So, you have no answer. > > > > If something doesn't occur, the question "when does it occur" > > does not have an answer. > > "[R]andy does not deny that the vase becomes empty". That sounds like it > occurs. > > > > > If I ask you what date you took a trip to Mars last year, > > would you have an answer? > > Does the vase become empty? Virgil and I differ on terminology here. As I have already said, you are trying to pin down an identifiable pair of contiguous moments where the vase is non-empty in one, and empty in the next. As I have already said, a verb like "emptying" conveys to me the existence of a PAIR of moments with that property, of an identifiable "change moment". I would not use the word "become" for the same reason. So I will continue to say what I have said. The vase is empty at noon, because before noon every ball put in was taken out. There is no moment when the vase "becomes empty". The first time when the vase IS empty is noon. - Randy
From: Tony Orlow on 29 Sep 2006 17:12
Virgil wrote: > In article <451d6037(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <451bac34(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>>>> If the vase is empty at noon, but not before, how can that not be the >>>>>> moment that it becomes empty? >>>>> Saying that it is empty is quite different from saying anything about a >>>>> "last ball". andy does not deny that the vase becomes empty, he just >>>>> does not say anything about any "last ball out". >>>> Does that answer the question of **when** this occurs? Of course not. >>> It does answer the question of "whether" it occurs. "When" is of lesser >>> importance. >> So, you have no answer. And so it's not important. I see. > > When have all the balls, whether removed or not, all been inserted? Not until noon. > That is the time at which all have also been removed, as no ball can be > removed before its insertion nor after all have been inserted. Repeat that to yourself 10 times, and you will see God. > > Except for the first 10 balls, each insertion follow a removal and with > no exceptions each removal follows an insertion. Which is why you have to have -9 balls at some point, so you can add 10, remove 1, and have an empty vase. Can you have -9 balls in your vase? Is it like a ball bank account that can be overdrawn? Are you sure it's not Cantor's ashes in there? |