An uncountable countable set
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From: Lester Zick on 27 Oct 2006 18:37 On 27 Oct 2006 11:01:57 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> Ah, Brian, ever the amanuensis. > >Zick, ever the nuisance. Ah, Moe, truth is often a nuisance. ~v~~
From: Tony Orlow on 27 Oct 2006 18:40 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(t) = 1 if A_n <= t < R_n, >>>>> 0 if t < A_n or t >= R_n. >>>>> >>>>> Let V(t) = sum_n B_n(t). >>>>> -------------------------- >>>>> >>>>> Just looking at these definitions of sequences and functions from R (the >>>>> real numbers) to R, and assuming that the sum is defined as it would be >>>>> in a Freshman Calculus class, are you saying that V(0) is not equal to >>>>> 0? >>>> On the surface, you math appears correct, but that doesn't mend the >>>> obvious contradiction in having an event occur in a time continuum >>>> without occupying at least one moment. It doesn't explain how a >>>> divergent sum converges to 0. Basically, what you prove, if V(0)=0, is >>>> that all finite naturals are removed by noon. I never disagreed with >>>> that. However, to actually reach noon requires infinite naturals. Sure, >>>> if V is defined as the sum of all finite balls, V(0)=0. But, I've >>>> already said that, several times, haven't I? Isn't that an answer to >>>> your question? >>> I think it is an answer. Just to be sure, please confirm that you agree >>> that, with the definitions above, V(0) = 0. Is that correct? >> Sure, all finite balls are gone at noon. > > Please note that there are no balls or time in the above mathematics > problem. However, I'll take your "Sure" as agreement that V(0) = 0. > Okay. > Let me ask you a question about this mathematics problem. Please answer > without using the words "balls", "vase", "time", or "noon" (since these > words do not occur in the problem). I'll try. > > First some discussion: For each n, B_n(0) = 0 and B_n is continuous at > zero. What??? How do you conclude that anything besides time is continuous at 0, where yo have an ordinal discontinuity???? Please explain. In fact, for a given n, there is an e < 0 such that B_n(t) = 0 for > e < t <= 0. There is no e<0 such that e<t and B_n(t)=0. That's simply false. In other words, B_n is not changing near zero. Infinitely more quickly but not. That's logical. And wrong. Now, V is the > sum of the B_n. As t approaches zero from the left, V(t) grows without > bound. In fact, given any large number M, there is an e < 0 such that > for e < t < 0, V(t) > M. We also have that V(0) = 0 (as you agreed). > > Now the question: How do you explain the fact that V(t) goes from being > very large for t a little less than zero to being zero when t equals > zero even though none of the B_n are changing near zero? > I'll consider answering that when you correct the errors above. Sorry.
From: Lester Zick on 27 Oct 2006 18:44 On 27 Oct 2006 11:38:10 -0700, imaginatorium(a)despammed.com wrote: > >David Marcus 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. > >Gosh, you obviously haven't seen Lester when he's in full swing. (Have >_you_ searched sci.math for "Zick transcendental"?) No but obviously you have, Brian. ~v~~
From: Lester Zick on 27 Oct 2006 18:45 On Fri, 27 Oct 2006 16:23:44 -0400, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >imaginatorium(a)despammed.com wrote: >> >> David Marcus 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. >> >> Gosh, you obviously haven't seen Lester when he's in full swing. (Have >> _you_ searched sci.math for "Zick transcendental"?) > >I did briefly, but there are so many posts, I didn't read them all. You can read? ~v~~
From: Virgil on 27 Oct 2006 18:49
In article <454227d7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > > > What are you talking about? I defined two sets. There are no > > balls or vases. There are simply the two sets > > > > IN = { n | -1/(2^floor(n/10)) < 0 } > > OUT = { n | -1/(2^n) < 0 } > > > > For each n e N, IN(n)=10*OUT(n). For each n in N, {x in IN: x <= n} >= {x in OUT: x <= n} > > >>> So apparently you do not think such an n exists, yet you > >>> think there are elements in IN that are not in OUT. > >>> Those are contradictory positions. > > > >> No, it is the only conclusion consistent with the notion that a proper > >> subset is alway smaller than the superset. How is N a PROPER subset of N? > > But OUT is not a proper subset of IN, unless you believe that > > there exists an n such that > > -1/(2^floor(n/10)) < 0 > > but > > -1/(2^n) >= 0 > > > > If you claim that OUT is a proper subset of IN, you must > > be able to identify an element that is in IN that is not in OUT. > > > > Stephen > > For each n e N, IN(n)=10*OUT(n). For each n in N, {x in IN: x <= n} >= {x in OUT: x <= n} So that IN must be a subset of OUT. > > You appear to think that the times are all that matters, but the times > included in IN each apply to 10 elements, whereas the times in OUT each > apply to only one. The only relevant question is "According to the rules set up in the problem, is each ball inserted before noon also removed 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 find any balls in the vase after every ball has been removed? |