An uncountable countable set
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From: Randy Poe on 29 Sep 2006 14:06 Tony Orlow 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 was responding to Han, who said that "If noon exists, that's when the vase empties". Noon exists. But in order for the vase to transition from not-empty to empty, there would have to be a last non-empty moment. That would be the last time before noon. > I will offer this simple > logical argument. If the vase ever became empty, it would be because one > ball was removed, Hence my continued statement that the vase does not "become empty". It is non-empty at certain times and empty at others. There is no transitional moment. Noon is the first moment at which the vase is empty. But noon is not the transitional moment. There's no time just before noon where the transition happened. - Randy
From: Tony Orlow on 29 Sep 2006 14:16 Virgil wrote: > In article <451bafc9(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> 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, since it doesn't do before then. If the >> limit doesn't "actually occur", then vase never empties (not that it >> would anyway). > > If the vase were not empty after noon, someone ought to be able to say > which balls were in it. Since no one will say, I will continue to > maintain that after noon the vase is empty. Since it is impossible to say how many iterations are performed by noon, it's impossible to number the balls. If balls 1 through n are inserted, balls n/10+1 through n will remain.
From: Tony Orlow on 29 Sep 2006 14:17 Virgil wrote: > In article <451be86a(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >>> What is the number of the ball which, when removed, >>> makes the vase empty? >>> >> There is no such number, since for each ball n removed, balls n+1 >> through balls n*10 remain. I have maintained throughout that, despite >> your "labeling schemes", 9/10 of the balls remain, if you add 10 and >> remove 1 repeatedly. It is precisely like adding 10 gallons and removing >> 1 per minute. The ocean will never empty. Think measure. > > So which balls are left after noon, TO? If n balls are inserted, balls n/10+1 through n remain.
From: Tony Orlow on 29 Sep 2006 14:29 stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> stephen(a)nomail.com wrote: >>> Tony Orlow <tony(a)lightlink.com> wrote: >>>> stephen(a)nomail.com wrote: >>>>> Randy Poe <poespam-trap(a)yahoo.com> wrote: >>>>> >>>>> <snip> >>>>> >>>>>> What is the number of the ball which, when removed, >>>>>> makes the vase empty? >>>>>> I know the kind of nonsense you will spout in answer to >>>>>> those questions, but the answers within our axiom system >>>>>> are: (1) there is no t<noon which is the moment just >>>>>> before noon. For any t<noon, there is t < t' < noon. >>>>>> (2) There is no such ball. >>>>>> Here are the Tony gobbledgook answers: >>>>>> (1) noon - 1/oo >>>>>> (2) Ball number omega >>>>>> In TO-matics, one can confidently give an answer like >>>>>> number 2 despite the fact that one can also agree >>>>>> that no ball numbered omega is ever put into the >>>>>> vase. >>>>> In TO-matics, it is also possible to end up with >>>>> an empty vase by simply adding balls. According to TO-matics >>>>> >>>>> ..1111111111 = 1 + 1 + 1 + 1 + ... >>>>> >>>>> and >>>>> ..1111111111 + 1 = 0 >>>>> >>>>> So if you just keep on adding balls one at a time, >>>>> at some point, the number of balls becomes zero. >>>>> You have to add just the right number of balls. It is not >>>>> clear what that number is, but it is clear that it >>>>> exists in TO-matics. >>>>> >>>>>> But in mathematics and logic, we don't get to >>>>>> keep a set of self-contradictory assumptions around, >>>>>> only using the ones we want as needed. >>>>>> - Randy >>>>> Where's the fun in that? :) >>>>> >>>>> Stephen >>>> You drew that from my suggestion of the number circle, and that ...11111 >>>> could be considered equal to -1. Since then, I looked it up. I'm not the >>>> first to think that. It's one of two perspectives on the number line. >>>> It's either really straight, or circular with infinite radius, making it >>>> infinitesimally straight. The latter describes the finite universe, and >>>> the former, the limit. But, you knew that, and are just trying to have fun. >>>> Tony >>> I am just pointing out that according to your mathematics >>> that if you keep adding balls to the vase, you can end up >>> with an empty vase. The fact that other people may have >>> considered a number circle does not change the fact that the >>> number circle implies that if you keep on adding balls, eventually >>> you will have zero balls. > >> That's a bastardization of the concept. There are two ways to look at >> the number circle, and you are combining them in mutually contradictory >> ways. > > How is it a bastardization of the concept? You claim that > 1+1+1+1+ ... = ..11111111 I wouldn't put it that way. ...1111 can be interpreted as the largest binary natural, if you claim all bit positions are finite. > and that > ..11111111 + 1 = 0 ....11111 can be interpreted indeed as -1, as is done every millions of times per microsecond all over the world in computers. Those are two different interpretations of -1. They aren't really compatible, as far as I can see, although Ross used to like to point out that temperatures below 0 Kelvin were somehow hotter than any positive temperature. There could be a connection. > > Why does that not apply to balls in the vase? Each ball > is a 1. If a add balls, I add 1's. Do I not eventually get 0? > If it does not apply to balls, what does it apply to, > and how do you determine when it applies? > They are two different interpretations of the string. Do you honestly think that I think sum(x=1->oo: 1)=0? For god's sake, how long have I been saying it's infinite, and that's why any infinite set of naturals would have to include infinite naturals? Oy. Like I said, you're combining concepts that are mutually contradictory, from two different number systems. And, people complaint hat I try to use normal operations normally on infinite numbers.... >>> So why is it okay to end up with zero balls, when you never remove >>> any at all, but it is not okay to end up with zero balls when >>> each ball is clearly removed at a definite time? > >> Because the model of the number circle where all strings are positive is >> incompatible with the number circle where any string with a >> left-unending string of 1's (in binary) is negative. Duh. > > That seems to be a non-answer. The most I can glean from > that is that the number circle is not relevant to the balls > in the vase problem. Is the number circle relevant to anything? And > how does one determine when it is relevant? If it is not > relevant to anything, why did you bring it up in > the first place? There are an number of topics going on here besides the crazy vase. It's relevant to number systems, and came up initially with my suggestion that ...1111 represents the largest finite if all bit positions are finite. I only mentioned the alternative interpretation of ...1111 being -1 as a sidebar. It's an *alternative* interpretation of the *string*. > > Like most of the other cranks, your "system" is only > usable by yourself, as the only way to know when > one of your "rules" applies is by asking you. Not if you read more carefully. > > So apparently sometimes 1+1+1+1+ ... = ...1111111 > and sometimes ...1111111+1 = 0 but at other times > they equal something different and there appears > to be know way to know which is which. > > Stephen Sum(x=1->n: 1) = n. "...1111" can mean a number of things. Tony
From: Virgil on 29 Sep 2006 14:31
In article <c87e0$451cc5b4$82a1e228$4275(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > MoeBlee wrote: > > > Han de Bruijn wrote: > > > >>It's a priorities issue. Do axioms have to dictate what constructivism > >>should be like? Should constructivism be tailored to the objectives of > >>axiomatics? I think not. > > > > Fine, but if you don't give a formal system, then your mathematical > > arguments are not subject to the objectivity of evaluation that > > arguments backed up by formal systems are subject to. > > Exactly! Constructivism is not Formalism. > > Han de Bruijn Do constructivists have any statements which they accept as true without proof? If not how do they prove anything from nothing? If so, then aren't those things they accept equivalent to axioms. |