An uncountable countable set
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From: stephen on 30 Oct 2006 12:25 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> Tony Orlow <tony(a)lightlink.com> wrote: >>> imaginatorium(a)despammed.com wrote: >>>> Tony Orlow wrote: >> >> <snip> >> >>>>> The formulaic relationship is lost in that statement. When you state the >>>>> relationship given any n, then the answer is obvious. >>>> Do "state the relationship given any n"... I mean, what is it, exactly? >>>> >> >>> Uh, here it is again. in(n)=10n. out(n)=n. contains(n)=in(n)-out(n)=9n. >>> lim(n->oo: contains(n))=oo. Basta cosi? >> >> >> What is in(n)? The sets I and everyone but you are talking about are >> IN = { n | -1/2^(floor(n/10)) < 0 } >> OUT = { n | -1/2^n < 0 } >> Noone has ever mentioned or defined in(n) >> >> What is the definition of in(n)? Is is a set? >> >> Stephen >> > out(n) is the number of balls removed upon completion of iteration n, > and is equal to n. > in(n) is the number of balls inserted upon completion of iteration n, > and is equal to 10n. But there are no balls or iterations in the problem I posed. So why do you keep talking about balls and iterations? Are you really that incapable of participating in a discussion? > contains(n) is the number of balls in the vase upon completion of > iteration n, and is equal to in(n)-out(n)=9n. > n(t) is the number of iterations completed at time t, equal to floor(-1/t). > contains(t) is the number of balls in the vase at time t, and is equal > to contains(n(t))=contains(floor(-1/t))=9*floor(-1/t). > Lim(t->-0: 9*floor(-1/t)))=oo. The sum diverges in the limit. > See how that all fits together? Its almost like physics, eh? > Tony The balls and vase problem has absolutley nothing to do with physics. Try it out. Stephen
From: Lester Zick on 30 Oct 2006 13:11 On Sun, 29 Oct 2006 20:45:57 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On Sun, 29 Oct 2006 15:11:48 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >> >Lester Zick wrote: >> >> According to MoeBlee mathematical definitions require a "domain of >> >> discourse" variable such as IN(x) and OUT(x). >> > >> >I think you've used this joke enough already. Why don't you come up with >> >a new one? >> >> Mainly because everyone seems to want to ignore the point Moe raised >> regarding mathematical definitions. It would seem either Moe is right >> or Stephen (I think) drew an improper mathematical definition or Moe >> is not right. > >Or, you misunderstood what Moe said. I would think that would be the >heavy favorite. Gee how hard can it be to understand "card(x)=least ordinal(x) with equinumerous(x)" where x represents the domain of discourse? I mean I can appreciate it might be difficult to understand the validity of such particular definitions but I don't agree the form itself which Stephen didn't choose to employ is especially difficult to comprehend. Quite possibly you just misunderstand what Moe said and prefer not to consider the possibility that one of the two is wrong. >> The last time around it was Stephen who was telling me >> that dr is velocity. Who did what is unimportant. I've seen both types >> of mathematical definition and I don't know as either is absolutely >> correct to the exclusion of others. But it does seem curious in such a >> didactic domain of discourse as mathematical definition one should be >> unable to tell which if either is which. You bust my chops over every >> nickel and dime mathematical issue passing my lips with accusations of >> trolling for no better reason than you decline to pay close attention >> to any point I raise and demand I come up with new jokes. Maybe you >> should come up with a few new jokes besides the same old modern math. > >I never said you were trolling. I asked someone else whether they >thought you were trolling. Oh yes, of course. I thought the question was rhetorical. Clearly it was just a candid request for information. My mistake. > In truth, I don't think you are trolling. Of course you don't (wink, wink, nod, nod). Problem is I'm paranoid about particular definitions such as "crackpot(x)=disagree(u)" where the domain of discourse can't be demonstrated true to begin with and just because I'm paranoid doesn't mean "they" aren't out there. >So, I guess you are going to stick with this joke for a while. Only as long as you insist on sticking with your little joke. ~v~~
From: Lester Zick on 30 Oct 2006 13:16 On Sun, 29 Oct 2006 15:10:18 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On 28 Oct 2006 12:54:51 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> >> wrote: >> >> > >> >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. >> >> Oh I dunno. I can be pretty dense. Just not as dense as you, Randy, >> but that's nothing new. >> >> >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? >> >> See the problem here, Randy, is that you're explaining an issue I >> didn't raise then pretending you're addressing the issue I raised. I >> don't doubt there is no smallest real except in the case of integers. >> But that is not what was said originally. What was said is that there >> is a least integer but no least real. Now these strike me as mutually >> exclusive predicates. But then who am I to analyze mathematical >> predicates in logical terms especially when there are self righteous >> neomathematikers around who prefer to specialize in name calling >> rather than keep their arguments straight in reply to simple queries. > >I'll probably regret asking, but what the heck. Are you saying that the >following two statements are contradictory? > >1. There is a smallest positive integer. >2. There is no smallest positive real. No. In response to these two propositions I'm simply asking whether you consider integers real? Or you might try reading what I originally asked which you pronounced illogical without apparently bothering to read what I wrote. ~v~~
From: David R Tribble on 30 Oct 2006 13:18 Tony Orlow wrote: David R Tribble wrote: >> You misunderstand. Your H-riffics are simply finite-length paths >> (a.k.a. the nodes) of a binary tree. Your definition precludes >> infinite-length paths as H-riffic numbers. > Tony Orlow wrote: >> What part of my definition says that? For the positives: >> >> 1 e H >> x e H -> 2^x e H >> x e H -> 2^-x e H > Tony Orlow wrote: >> Exactly. If you list these H-riffic numbers as a binary tree, each one >> is a node in the tree along a finite-length path. > David R Tribble wrote: >> If you extend your definition and allow the H-riffics to include the >> infinite paths of your binary tree as well as the finite-length paths >> (nodes), then, yes, the H2-riffics are uncountable and probably >> cover the reals. >> But then the H2-riffics [H-riffics?], like the reals, are not well-ordered. >> You're back to square one. > Tony Orlow wrote: > Yes, you're right, and I admitted that in "Well Ordering the Reals". One > can linearly order the reals in this way, but eliminating any "countably > infinite descending sequences" appears to be a matter of infinite > regression, and I don't see how to prove that it's a *countably* > infinite regression. If it were, then perhaps it could be considered > well ordered, but well ordering an uncountably infinite set requires > predecessor discontinuities as one finds in the limit ordinals. Really, > despite the Axiom of Choice, I don't see that it's possible to > explicitly state a well ordering on any uncountable set, besides the > concoction of ordinals. So, I agree, on that point. > > See? I can admit failure, especially when I failed in an experiment that > didn't guarantee success. :) David R Tribble wrote: >> (If you don't see this, consider the H2-riffic, which we'll call p, as >> the infinite path in your binary tree, where each successive fork taken >> is left (2^x) if the next digit in the binary fraction for pi is 0, and >> the right fork (2^-x) if the next digit is 1. Then we ask, what is the >> successor to p in the well-ordering? There is no way to know, just >> as there is no way to name the successor of pi in the usual ordering >> of the reals.) > Tony Orlow wrote: > Oh. You're using "H2-riffics" as the uncountable set. Okay. I get it. > > However, you're wrong here. Sorry. There are two successors to every > H-riffic h, 2^h and 2^-h. So, successors to pi are 2^pi and 1/2^pi. > Those exist, don't they, as reals? I am sure they (those points on the > real line) can be calculated (specified) to any arbitrary accuracy. No? No, there are no successors in your tree for that number p, because there is no last L/R branch in the path defining that H2-number. So there can be no different or next branching for the "last" node for either successor of p, because there is no last node in the path. I'm talking over your head here, because you obviously do not follow what I'm saying.
From: Lester Zick on 30 Oct 2006 13:24
On 30 Oct 2006 08:34:21 -0800, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: > >Lester Zick wrote: >> On 28 Oct 2006 12:54:51 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> >> wrote: >> >> > >> >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. >> >> Oh I dunno. I can be pretty dense. Just not as dense as you, Randy, >> but that's nothing new. >> >> >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? >> >> See the problem here, Randy, is that you're explaining an issue I >> didn't raise then pretending you're addressing the issue I raised. > >I'm providing explicit descriptions of why, as the original >quote said, there is a least positive non-zero integer, >but not a least positive non-zero real. Who cares? That has nothing to do with the question I asked. This isn't rocket science, Randy. You really need to learn to construe plain language more accurately. >> I >> don't doubt there is no smallest real except in the case of integers. >> But that is not what was said originally. What was said is that there >> is a least integer but no least real. > >Since the original text is above, we can actually see whether >that is what was said. >"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." > >Are you equating "least integer" with "smallest positive >non-zero integer" and "least real" with "smallest >positive non-zero real"? It certainly seemed to me that I was just asking a question regarding implications of the point made. And I reinterpreted that question just above for clarification. >> Now these strike me as mutually >> exclusive predicates. > >"There is a least integer" and "there is a least real" >are both false. They are? Perhaps you should take that up with theologians then. > But neither of those was what was said. >You have provided the original quote above. Reread it. I have many times. I still ask the same question. You don't answer it. ~v~~ |