An uncountable countable set
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From: Tony Orlow on 24 Oct 2006 12:55 Virgil wrote: < endless reiterations of the following > > > The only question is "According to the rules set up in the problem, is > each ball which is inserted into the vase before noon also removed from > the vase before noon?" > > An affirmative answer confirms that the vase is empty at noon. > A negative answer violates the conditions of the problem. > > Which answer does TO choose? God, are you a broken record, or what? Let's take this very slowly. Ready? Each ball inserted before noon is removed before noon, but at each time before noon when a ball is removed, 10 balls have been added, and 9/10 of the balls inserted remain. Therefore, at no time before noon is the vase empty. Agreed? Events including insertions and removals only occur at times t of the form t=-1/n, where n e N. Where noon means t=0, there is no t such that -1/n=0. Therefore, no insertions or removals can occur at noon. Agreed? Balls can only leave the vase by removal, each of which must occur at some t=-1/n. The vase can only become empty if balls leave. Therefore the vase cannot become empty at noon. Agreed? It is not empty, and it does not become empty, then it is still not empty. Agreed? When you bring t=0 into the experiment, if anything DOES occur at that moment, then the index n of any ball removed at that point must satisfy t=-1/n=0, which means that n must be infinite. So, if noon comes, you will have balls, but not finitely numbered balls. In this experiment, however, t=0 is excluded by the fact that n e N, so noon is implicitly impossible to begin with. Have a nice lunch. Tony
From: Tony Orlow on 24 Oct 2006 13:11 David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> Either something happens an noon, or it doesn't. Where do you stand on >>>>>>>> the matter? >>>>>>> What does "something happens" mean, please? I really don't know what you >>>>>>> mean. >>>>>> ??? Do you live in the universe, or in a static picture? When "something >>>>>> happens" o an object, some property or condition of it "changes". That >>>>>> occurs within some time period, which includes at least one moment. >>>>>> There is no moment in this problem where the vase is emptying, >>>>>> therefore, that never "occurs". If you are going to insist that time is >>>>>> a crucial element of this problem, then you should at least be familiar >>>>>> with the fact that it's a continuum, and that events occurs within >>>>>> intervals of that continuum. >>>>> Thanks. That explains what "something happens" means. Now, please >>>>> explain what "emptying" means. >>>> "Empty" means not having balls. To become empty means there is a change >>>> of state in the vase ("something happens" to the vase), from having >>>> balls to not having balls. >>>> >>>> Now, when does this moment, or interval, occur? >>> A reasonable question. Before I answer it, let me ask you a question. >>> Suppose I make the following definitions: >>> >>> For n = 1,2,..., define >>> >>> A_n = -1/floor((n+9)/10), >>> R_n = -1/n. >>> >>> For n = 1,2,..., define a function B_n by >>> >>> B_n(t) = 1 if A_n < t < R_n, >>> 0 if t < A_n or t > R_n, >>> undefined if t = A_n or t = R_n. >>> >>> Let V(t) = sum{n=1}^infty B_n(t). >>> >>> Then V(-1) = 1 and V(0) = 0. If we consider V to be a function of time, >>> at what time does it become zero? >> Just answer the question, and stop beating around the bush. > > To recap, you wrote, "When 'something happens' to an object, some > property or condition of it 'changes'. That occurs within some time > period, which includes at least one moment." You also wrote, "To become > empty means there is a change of state in the vase ('something happens' > to the vase), from having balls to not having balls." > > From the definition of V, if t equals A_n or R_n, then V(t) is not > defined. From your definition, but I would say that B_n(t) is 1 at A_n and 0 at R_n, and let V(t) be defined at every point. There are always a specific number of balls, if additions and removals occur instantaneously. > For other 1 <= t < 0, V(t) is positive. Yes, there are balls in the vase (a growing number) at every time before noon. And, V(0) = 0. I notice you summed to oo: "Let V(t) = sum{n=1}^infty B_n(t)." Can you sum to oo when the set is limited to N? When n e N and t=-1/n, does t=0 exist? This is the kind of fallacy you folks try to accuse me of. t=0 ^ t=-1/n ^ n e N = FALSE. > You asked when does the vase change from having balls to having no > balls? Since, V(t) is positive (or undefined) for 1 <= t < 0 and V is > zero at time zero, it would seem that according to *your* definition of > "become empty", the vase becomes empty at noon. Correct, as much as I could expect. If the vase is empty at noon, but not before, then it must have become empty at that point in time. > > The only reason I say "seem" is that I don't know whether your > definition allows V to be undefined at the times of addition and > removal. We could either agree that at the addition and removal > instants, the ball is not in the vase (thus changing the definition of > V) or we could agree that "becomes empty" requires V to be defined at > all times (in which case the vase never becomes empty). Just like we are saying that the vase becomes empty at t=0 since it is not empty at t<0, at the moment of insertion or removal we should consider the event completed from that moment forward. That's consistency, and it removes any undefined states of the vase at any time t<0. Regardless, V(0) > = 0, so there are no balls in the vase at noon. > No, t=0 is proscribed by n e N and t=-1/n. Contradiction. Sorry. You can't have it both ways. Either noon is in the experiment and something occurs, which involves infinitely numbered balls, or you stick with the original constraint that n e N and so t=0 is not allowed. Your choice.
From: Lester Zick on 24 Oct 2006 13:11 On 23 Oct 2006 10:38:09 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> >I don't insist that conversations must be about set theory. However, >> >when various incorrect things are said about set theory, then I may see >> >fit to comment, as well as I may see fit to mention set theoretic >> >approaches to certain mathematical subjects. In the present case, a >> >poster mentioned certain things about set theory, then I responded, >> >then he asked me questions, then I gave answers, then you commented on >> >my answers. That hardly presents me as insisting that the scope of this >> >newsgroup be confined to set theory. >> >> Technically set "theory" is only a method not a theory since it can't >> be proven true or false. > >By a 'theory' I mean a set of sentences closed under entailment. That's >what *I* mean when I use the word in such contexts as this one. I deny >that there are other senses of the word; I'm just telling you exactly >what I mean in this context. And by "theory" I mean a series of predicates which are either true or false in combination. I have no idea what you mean by "closed under entailment", Moe, but if a series of of predicates can't be proven true or false they can't be a theory and the best we can say for them is they're an arbitrary method of analysis. Now there are many methods of analysis possible and standard set theory is just one of many. >> Moe, you get all bent >> out of shape and claim they're spouting nonsense on the internet for >> no better reason than they don't follow the party line on the subject. > >No, I have no problem with someone rejecting the axioms of any >particular theory, thus to reject the theory. What I post against is >people saying things that are incorrect about certain theories. And I don't see any theories which can be demonstrated true or false. > Saying >that one rejects the axioms or even that one rejects classical first >order logic is fine with me (though, I'm interested in what >alternatives one offers). You've already rejected the alternative I offer although I don't know why. > But that is vastly different from saying >untrue things about existing theories. Well see the problem here, Moe, is that you and standard set analysis have no demonstrable basis for truth. So "untrue" things cannot be said of a standard analytical method for sets which has no method of demonstrating "truth" except in reference to its own assumptions of truth the first among which would seem to be that what it assumes is true is true. In other words the only standard of truth in standard set analysis would seem to be the lack of inconsistency with its own assumptions of truth. Which means that standard set analysts would insist on being able to use the words "truth" and "true" as if they meant nothing more or less than "no lack of inconsistency" with standard set analytical techniques. So by not saying things which are "untrue" of standard set analysis I would be pretty much restricted to saying only those things which standard set analysts accepts as being not inconsistent with its own analytical techniques. In other words it's not just a matter of accepting or rejecting isolated assumptions of standard set analysis in their own terms but of rejecting their analytical techniques. And when someone like me comes along and suggests the analytical techniques of standard set analysis are flawed you get all bent out of shape because you insist not that I accept individual assumptions but because I reject the techniques which standard set analysis insists are somehow "true" and if I don't I'm therefore saying things about standard set theory which are somehow "untrue". ~v~~
From: Lester Zick on 24 Oct 2006 13:11 On 23 Oct 2006 11:44:09 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >MoeBlee wrote: >> I deny >> that there are other senses of the word; I'm just telling you exactly >> what I mean in this context. > >Of course I meant I do NOT deny that there are other senses. Check. ~v~~
From: Tony Orlow on 24 Oct 2006 13:15
David Marcus wrote: > Tony Orlow wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> stephen(a)nomail.com wrote: >>>>> Also, supposing for the sake of argument that there are "infinitely >>>>> number balls", if a ball is added at time -1/(2^floor(n/10)), and removed >>>>> at time -1/(2^n)), then the balls added at time t=0, are those >>>>> where -1/(2^floor(n/10)) = 0. But if -1/(2^floor(n/10)) = 0 >>>>> then -1/(2^n) = 0 (making some reasonable assumptions about how arithmetic >>>>> on these infinite numbers works), so those balls are also removed at noon and >>>>> never spend any time in the vase. >>>> Yes, the insertion/removal schedule instantly becomes infinitely fast in >>>> a truly uncountable way. The only way to get a handle on it is to >>>> explicitly state the level of infinity the iterations are allowed to >>>> achieve at noon. When the iterations are restricted to finite values, >>>> noon is never reached, but approached as a limit. >>> Suppose we only do an insertion or removal at t = 1/n for n a natural >>> number. What do you mean by "noon is never reached"? >> 1/n>0 > > Sorry, I meant t = -1/n. So, I assume your answer is that -1/n < 0. > > But, I don't follow. Translating "-1/n < 0" back into words, I get "all > insertions and removals are before noon". However, I asked you what > "noon is never reached" means. Are you saying that "noon is never > reached" means that "all insertions and removals are before noon"? > Yes, David. What else happens in this experiment besides insertions and removals of naturals at finite times before noon? If the infinite sequence of events is actually allowed to continue until t=0, then you are talking about events not indexed with natural numbers, so you're not talking about the same experiment. If noon is not allowed, and all times in the experiment are finitely before noon, well, at none of those times does the vase empty, as we all agree. This is why I am asking when this occurs. It can't, given the constraints of the problem. |