An uncountable countable set
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From: Randy Poe on 28 Oct 2006 15:54 Lester Zick wrote: > On Fri, 27 Oct 2006 14:23:58 -0400, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > > >Lester Zick wrote: > >> On Fri, 27 Oct 2006 16:30:04 +0000 (UTC), stephen(a)nomail.com wrote: > >> >A very simple example is that there exists a smallest positive > >> >non-zero integer, but there does not exist a smallest positive > >> >non-zero real. > >> > >> So non zero integers are not real? > > > >That's a pretty impressive leap of illogic. > > "Smallest integer" versus "no smallest real"? Seems pretty clear cut. You must be joking. I can't believe even you can be this dense. Is 1 the smallest positive non-zero integer? Yes. Is it the smallest positive non-zero real? No. 1/10 is smaller. Ah well, then is 1/10 the smallest positive non-zero real? No, 1/100 is smaller. Is that the smallest? No, 1/1000 is smaller. Does that second sequence have an end? Can I eventually find a smallest positive non-zero real? How about the first? Is there something smaller than 1 which is a positive non-zero integer? - Randy
From: Virgil on 28 Oct 2006 15:56 In article <45435a9c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> Tony Orlow wrote: > >>>> Eat me. Do you maintain that the two theories are compatible with each > >>>> other? Is there, and also not, a smallest infinity. > >>> They're not in conflict, becuase 'smallest infinite' means something > >>> DIFFERENT in the different contexts. How many times will I say that > >>> while you STILL refuse to hear it? > >> So, either smallest has two meanings, or infinite has tow meanings, or > >> both. Would you like to elucidate the matter by enumerating the various > >> definitions of "small" and "infinite"? A table might be nice... > > > > As many have said, "infinite" has many meanings. I'm afraid it isn't > > practical to produce a table. > > > > How about a list? ;) As lista are merely function having N as domain, and functions can often be given by formulae, it is easy to produce some lists. For example, f:N -> R: n |-> n, is a simple list.
From: cbrown on 28 Oct 2006 15:57 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > >>> Tony Orlow wrote: > >>>> cbrown(a)cbrownsystems.com wrote: <snip> > >>>>> So at least one of the assumptions (1)..(8) and (T1) must be discarded > >>>>> if we are to resolve this. What do you suggest? Which of (1)..(8) do > >>>>> you want discard to maintain (T1)? > >>>>> > >>>>> Cheers - Chas > >>>>> > >>>> This is a very good question, Chas. Thanks. I'll have to think about it, > >>>> and I'm rather tired right now, but at first glance it seems like it > >>>> could be a sound analysis. I've cut and pasted for perusal when I'm > >>>> sharper tomorrow. > >>>> > >>> Here's some of my thoughts: > >>> > >>> When you say "noon doesn't occur"; I think "he doesn't accept (1): by a > >>> time t, we mean a real number t" > >> That doesn't mean t has to be able to assume ALL real numbers. The times > >> in [-1,0) are all real numbers. > > > > And I would say that assuming that by a time t, we mean a real number > > in [-1,0) is a different assumption than (1). > > > > There is no contradiction between them, is there.... Sure there is. (1) states that we can speak of a time 0. Your version states we cannot speak of a time 0. Those are contradictory statements. > > >>> When you say "if we always add more balls than we remove, the number of > >>> balls in the vase at time 0 is not 0", I think "he doesn't accept (8): > >>> if the numbers of balls in the vase is not 0, then there is a ball in > >>> the vase." > >> No, I accept that. There is no time after t=-1 where there is no ball in > >> the vase. > >> > > > > I.e., there is a ball in the vase. But then by the argument I > > previously gave, there is then a ball in the vase which is not in the > > vase. Your reference to "unspecified" balls in the vase at noon I > > interpret to be a way of saying that (8) should instead state something > > like "if the number of balls in the vase is not 0, then there may be no > > /specific/ ball in the vase (because there is instead an /unspecific/ > > ball in the vase)". > > > > You claim no balls are added at noon, because nothing can be, but then, > nothing can be removed at noon, either. Yes it follows from (1)..(8).. > Either it grows or stays un-zero. > No, that is not a conclusion from (1)..(8); although it follows from different assumptions such as (T1). > >>> When you say "an infinite number of balls are removed at time 0", I > >>> think "he does not agree with (6) if balls are removed at some time t, > >>> they are removed in accordance with the problem statement: i.e. there > >>> exists some natural number n s.t. n = -1/t and (some other stuff)". > >> I didn't say that exactly. If 0 occurs, then all finite balls are gone, > >> but infinite balls have been inserted such that 1/n=0 for those balls. > >> So, at noon the vase is not empty, even if it occurs in the problem, > >> which it doesn't. > >> > > > > If infinite balls are inserted at some time t = -1/n = 0, then by (5) > > each of them are inserted at time t; and at that time exactly 10 balls > > are inserted. 10 is not infinite. > > > > It i larger than 1, and at an infinite rate of insertion, yes, an > infinite number of balls are added at t=0. > Therefore, you don't accept (5); so you assert that it is not the case if a ball is placed in the vase at time t, it must be placed in the vase in accordance with the rules of the problem. Alternatively, you may be claiming that (2) is false; balls can be in the vase at time t without being placed in the vase at some time earlier than or equal to t. > >>> All these assertions follow a simgle theme: "If I require that my > >>> statemnents be /logically/ consistent, does the given problem make > >>> sense; and if so, what is a reasonable resonse?". > >>> > >>> Cheers - Chas > >>> > >> That there is a contradiction in your conclusion if you assume that all > >> events must occur at some time... > > > > The "occurence" of these events (ball insertions and removals at > > particular times) is described by (1), (5), (6), and (7). > > > >> ... and that becoming empty is the result of > >> events that happen in the vase. > > > > There is no "becoming" empty described in (1)..(8). There is only > > "being" empty; which is described by (1), (2), (3), and (4), and (8). > > > > Put it in there, in whatever form you think might be correct. Okay. ($) If for some times t1, t2 with t1 < t2, it is the case that for all t with t1 <= t < t2, the number of balls at time t is not 0, and at time t2 the number of balls in the vase is 0; then the number of balls becomes 0 at t2. We can also add my previously stated "balls must be removed for the vase to become empty": (*) If for some times t1, t2 with t1 < t2, it is the case that the number of balls in the vase at time t1 is not 0, and the number of balls at time t2 is 0; then there is a time t with t1 <= t <= t2 such that a ball is removed at time t. > I assure > you, it will make you question your conclusions. It doesn't; because ($) and (*) together do not imply (T1). On the other hand, neither do they contradict (T1) in the absence of (5), (6), and (7). > Then, there is a > question of logical validity there. We have to ascertain to root of the > problem. Or, we don't. We can just stay dumb. > > >> It cannot become empty until noon, when > >> nothing happens to cause it. > > > > And that is your premise that I call (T1). It is incompatible with > > (1)..(8); so either we must reject something in (1)..(8), or we must > > reject (T1) in favor of my previously described (*). > > > > Cheers - Chas > > > > What exactly is the contradiction, specifically? Can you formulate that? It follows from my previous argument that (1)..(8) alone implies that the vase is empty at noon. By (5), (6), (7), and (3), there is a ball in the vase for all time t < 0; therefore it follows from ($) that the vase becomes empty at noon. By (6) no balls are removed at noon; therefore, we conclude that the vase becomes empty at noon, and no balls are removed at noon. That directly contradicts (T1), without relying on the assumption that (T1) is either true or false to start with (although I will note that it does /not/ contradict (*)). Since all statements (1)..(8), ($), (*), and (T1) are assumptions no
From: Randy Poe on 28 Oct 2006 15:58 Tony Orlow wrote: > Randy Poe wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <4542201a(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> cbrown(a)cbrownsystems.com wrote: > >>>>> When you say "noon doesn't occur"; I think "he doesn't accept (1): by a > >>>>> time t, we mean a real number t" > >>>> That doesn't mean t has to be able to assume ALL real numbers. The times > >>>> in [-1,0) are all real numbers. > >>> By what mechanism does TO propose to stop time? > >> By the mechanism of unfinishablility. > > > > But that's why I asked you a question about variables labelling > > times yesterday, when noon clearly occurred. > > > The experiment occurred in [-1,0). Talk of time outside that range is > irrelevant. Times before that are imaginary, and times after that are > infinite. Only finite times change anything, so if something changes, > it's at a finite, negative time. > > > > > I can define a list of times t_n = noon yesterday - 1/n seconds, > > for all n=1, 2, 3, ... > > Are there balls in the vase for t<-1? No. What balls? What vase? I'm naming times. They're just numbers. > > Clearly this list of times has no end. But didn't noon happen? > > Nothing happened at noon to empty the vase, \ What vase? Why are you obsessed with vases? Do you deny me the ability to create a set of variables t_n, n = 1, 2, ...? Why do vases have to come into it? - Randy
From: Virgil on 28 Oct 2006 16:03
In article <454360a5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> David Marcus wrote: > >>> Tony Orlow wrote: > >>>> David Marcus wrote: > >>>>> Tony Orlow wrote: > >>>>>> David Marcus wrote: > >>>>>>> You are mentioning balls and time and a vase. But, what I'm asking is > >>>>>>> completely separate from that. I'm just asking about a math problem. > >>>>>>> Please just consider the following mathematical definitions and > >>>>>>> completely ignore that they may or may not be > >>>>>>> relevant/related/similar > >>>>>>> to the vase and balls problem: > >>>>>>> > >>>>>>> -------------------------- > >>>>>>> For n = 1,2,..., let > >>>>>>> > >>>>>>> A_n = -1/floor((n+9)/10), > >>>>>>> R_n = -1/n. > >>>>>>> > >>>>>>> For n = 1,2,..., define a function B_n: R -> R by > >>>>>>> > >>>>>>> B_n(x) = 1 if A_n <= x < R_n, > >>>>>>> 0 if x < A_n or x >= R_n. > >>>>>>> > >>>>>>> Let V(x) = sum_n B_n(x). > >>>>>>> -------------------------- > > I thought we agreed above to not use the word "time" in discussing this > > mathematics problem? > > If that's what you want, then why don't you remove 't' from all of your > equations? I have taken the liberty of replacing the 't' with 'x' in those equations. It does not change the conclusions. > > > > As for your question, let's look at B_2 (the argument is similar for the > > other B_n). > > > > B_2(x) = 1 if A_2 <= x < R_2, > > 0 if x < A_2 or x >= R_2. > > > > Now, A_2 = -1 and R_2 = -1/2. So, > > > > B_2(x) = 1 if -1 <= x< -1/2, > > 0 if x < -1 or x >= -1/2. > > > > In particular, B_2(x) = 0 for x >= -1/2. So, the value of B_2 at zero is > > zero and the limit as we approach zero is zero. So, B_2 is continuous at > > zero. > > > > Oh. For each ball, nothing is happening at 0 and B_n(0)=0. That's for > each finite ball that one can specify. As there are no other balls, what is your point? The only relevant 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 directly violates the conditions of the problem. How does TO answer? |