An uncountable countable set
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From: cbrown on 18 Oct 2006 03:30 Tony Orlow wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > >> cbrown(a)cbrownsystems.com wrote: > >>> Tony Orlow wrote: > >>>> cbrown(a)cbrownsystems.com wrote: > >>>>> 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> > > > > <snipitty-snip> > > > >>> Do you accept the above statements, or do you still claim that there is > >>> /no/ valid proof that ball 15 is not in the vase at t=0? > >>> > >> 15 is a specific finite number for which we can state its times of entry > >> and exit. > > > > Agreed, > > > >> At its time of exit, balls 16 through 150 reside in the vase. > > > > Agreed. > > > >> For every finite n in N, upon its removal, 9n balls remain. > > > > "upon its removal" = "at the time of ball n's removal"; Agreed. > > > >> For every n > >> e N, there is a finite nonzero number of balls in the vase. > > > > "For every n e N, there is a finite non-zero number of balls in the > > vase at t = -1/n". Agreed. > > > >> Every > >> iteration in the sequence is indexed with an n in N. > > > > "Balls are only added or removed at a time t = -1/n for some natual n." > > Agreed. > > > >> Therefore, nowhere > >> in the sequence... > > > > ..., i.e, at no time t such that t = -1/n for some natural n, ... > > > >> is there anything other than a finite nonzero number of > >> balls in the vase. > > > > Agreed. > > > >> Now, where, specifically, in the fallacy in that argument? > >> > > > > Well, what do you state is the conclusion of this argument? > > You have agreed with everything so far. At every point before noon balls > remain. To be precise, the assertions above all imply that at every time t = -1/n, where n is a natural number, there are balls in the vase. But that *alone* does not even include every time t before noon; let alone every time t. For example, notice that nowhere above do you or I /explicitly/ assert: "at t=-2/3, the number of balls in the vase is a positive finite number". We assert something specific about t = -2/4, and something specific about t = -1/3, but nowhere do we directly state somthing about t = -2/3. On the other hand, given the problem statement, I think we would both /agree/ that there "should be" an obvious (perhaps even unique) well-defined answer to the question : "what is the number of balls in the vase at time t = -1/pi?" Assuming in the remaining statements that one agrees with the previous statement, this leads us to the question: what are the unstated assumptons that allow to agree that this must be the case? I attempted to describe those assumptions in my previoius post. Did you read those assumptions? If so, do you agree with those assumptions? > You claim nothing changes at noon. Where, exactly, above do I claim that "nothing changes at noon"? > Is there something between > noon and "before noon", when those balls disappeared? If not, then they > must still be in there. In the statements I made to which you are referring, I am responding to the argument you gave in my previous post; which I naturally assumed (as it directly followed) was a response to my question: > >>> Do you accept the above statements, or do you still claim that there is > >>> /no/ valid proof that ball 15 is not in the vase at t=0? At this point in the post, you had not stated your "nothing changes, so how can something change?" argument; so it is not surprising that I did not, at this point in the post, refute it, or even comment in any way upon it. I refuted your identical statement, in the section you snipped. Did you read it? > > > > > If the conclusion of this argument is "we cannot therefore state that > > ball 15 is not in the vase at t=0", I really don't see how you have > > addressed the issue. You agree that ball 15 is removed, and not put > > back in the vase at any time before or at noon; and I think you would > > agree that if if a ball is not put in the vase, it cannot be in the > > vase. Therefore ball 15 is not in the vase at noon; and nothing you > > said above challenges the logic of this conclusion. > > Of course not. Ball 15 is gone. > You say here, "of course not", but you previously stated that it is "not in my purview" to claim /anything at all/ about the state of affairs at noon; because "noon does not occur". Therefore, we can conclude that you now retract these statements, is that correct? > > > > If your conclusion is "therefore, at t=0, there must be a finite > > nonzero number of balls in the vase", then the fallacy is called non > > sequituur - it doesn't logically follow. > > There must be a nonzero number, unless soemthing occured between "before > noon" and noon. Is there something between those two? > In light of my observations, why won't you directly address the validity of your argument as you /actually stated it/ in your previous post, instead of haring off on some new and unrelated argument? I addressed the question you ask above in the previous post. It does not provide a well-defined property "change happened to (something) at some time t". Do you need me to amplify on my response? "Noon" is a time. Is t=-1/2 "before noon"? How about t=-2/3? Are not the times t = -1/2 and -2/3 "indistinguishable" as regards their both being "before noon"? "Before noon" is /not/ a time, it is a /set/ of times. But then, how do you define /the/ state of the vase at /the/ set of times "indistinguishable" from the time "before noon"? Surely there is, at least, "room for disagreement" regarding these questions? But I digress... > > > > * Because t=0 is /not/ a time such that t = -1/n for some natural n; > > Is there something between x=0 and x<0? > Jeez. Earth to Tony! Earth to Tony! Sometimes it seems like you are both interested in, and capable of, a mathematical conversation (which is a part of why I bother to converse with you at all, as opposed to illucid cases like Ross, or out-and-out trolls such as Lester). But sometimes it seems that you simply won't /let/ yourself adopt the convemtions that make a conversation a /mathematical/ conversation, because you don't actually give enough of a damn about that sort of thing to expend the effort to learn what it is, /gi
From: Ross A. Finlayson on 18 Oct 2006 03:37 imaginatorium(a)despammed.com wrote: .... > > Hmm. So your "those balls" must have been introduced into the vase in > this mysterious zone "between before noon and noon". But, see, in > mathematics, it's quite clear there is no such zone - here's a proof. > > Let B = { t : t is a time, and t is before noon } // the set of all > times before noon > Let N = {noon} // the singleton set of noon > > I suggest that if there is a time _between_ before noon and noon, it > must be a member of the following set: > > Let Z = { t : t is a time, t is after b for all b in B, t is before n > for all n in N } > > Do you agree? > > Would you like to prove that Z is the empty set, just as a little > exercise? > > Brian Chandler > http://imaginatorium.org Would you care to try to prove the opposite, just as a little exercise? How about as a little exorcise? You have a lot of undefined terms there, Brian. ZF is inconsistent. So: well-order the reals. How about if z encompasses b and n in the vague fugue? The point here is that there are counterexamples to standard real analysis, and, that means there is more to real analysis than the "standard". Ross
From: Virgil on 18 Oct 2006 03:46 In article <9010a$4535d38b$82a1e228$21861(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > David Marcus wrote: > > > Han de Bruijn wrote: > > > >>No matter how much textbooks about set theory I read, it all remains > >>abacedabra for me. > > > > I just looked in two books on set theory on my shelf and both explicitly > > state that the set in the axiom of infinity need not be the natural > > numbers. The books are "Naive Set Theory" by Paul R. Halmos and "Set > > Theory, An Introduction to Independence Proofs" by Kenneth Kunen. > > Hmm, I have that book by Halmos myself ... > > But why are the finite ordinals not equivalent with the naturals (I mean > in mainstream mathematics)? > > > Which math textbooks have you read? Have you worked the problems in the > > ones you've read? Taken any math courses at a university? > > I'm a (theoretical) physicist by education. > > Han de Bruijn And one who measures everything by how well it fits into his personal vision of physics.
From: imaginatorium on 18 Oct 2006 04:18 Ross A. Finlayson wrote: > imaginatorium(a)despammed.com wrote: > ... > > > > Hmm. So your "those balls" must have been introduced into the vase in > > this mysterious zone "between before noon and noon". But, see, in > > mathematics, it's quite clear there is no such zone - here's a proof. > > > > Let B = { t : t is a time, and t is before noon } // the set of all > > times before noon > > Let N = {noon} // the singleton set of noon > > > > I suggest that if there is a time _between_ before noon and noon, it > > must be a member of the following set: > > > > Let Z = { t : t is a time, t is after b for all b in B, t is before n > > for all n in N } > > > > Do you agree? > > > > Would you like to prove that Z is the empty set, just as a little > > exercise? > > > > Brian Chandler > > http://imaginatorium.org > > Would you care to try to prove the opposite, just as a little exercise? > How about as a little exorcise? > > You have a lot of undefined terms there, Brian. > > ZF is inconsistent. So: well-order the reals. > > How about if z encompasses b and n in the vague fugue? > > The point here is that there are counterexamples to standard real > analysis, and, that means there is more to real analysis than the > "standard". You've said most, if not all, of the above before. The question here is whether any of the "counterexamples to standard real analysis" that you speak of can be expressed in mathematical language, or even in any language, in a way that can be understood - or even parsed - by anyone other than yourself. You said a day or so ago that you thought you were not liked - I think that's a misunderstanding. I can't see any reason not to like you - you do not resort to abuse, let alone to threats to contact employers etc. The reason most people ignore you is, I'm sure, that you have never really said anything that makes enough sense not to ignore. If you think this is wrong, or unfair, please explain to me one of your counterexamples to standard real analysis. Brian Chandler http://imaginatorium.org
From: Ross A. Finlayson on 18 Oct 2006 12:34
imaginatorium(a)despammed.com wrote: > Ross A. Finlayson wrote: > > imaginatorium(a)despammed.com wrote: > > ... > > > > > > Hmm. So your "those balls" must have been introduced into the vase in > > > this mysterious zone "between before noon and noon". But, see, in > > > mathematics, it's quite clear there is no such zone - here's a proof. > > > > > > Let B = { t : t is a time, and t is before noon } // the set of all > > > times before noon > > > Let N = {noon} // the singleton set of noon > > > > > > I suggest that if there is a time _between_ before noon and noon, it > > > must be a member of the following set: > > > > > > Let Z = { t : t is a time, t is after b for all b in B, t is before n > > > for all n in N } > > > > > > Do you agree? > > > > > > Would you like to prove that Z is the empty set, just as a little > > > exercise? > > > > > > Brian Chandler > > > http://imaginatorium.org > > > > Would you care to try to prove the opposite, just as a little exercise? > > How about as a little exorcise? > > > > You have a lot of undefined terms there, Brian. > > > > ZF is inconsistent. So: well-order the reals. > > > > How about if z encompasses b and n in the vague fugue? > > > > The point here is that there are counterexamples to standard real > > analysis, and, that means there is more to real analysis than the > > "standard". > > You've said most, if not all, of the above before. The question here is > whether any of the "counterexamples to standard real analysis" that you > speak of can be expressed in mathematical language, or even in any > language, in a way that can be understood - or even parsed - by anyone > other than yourself. You said a day or so ago that you thought you were > not liked - I think that's a misunderstanding. I can't see any reason > not to like you - you do not resort to abuse, let alone to threats to > contact employers etc. The reason most people ignore you is, I'm sure, > that you have never really said anything that makes enough sense not to > ignore. > > If you think this is wrong, or unfair, please explain to me one of your > counterexamples to standard real > analysis. > > Brian Chandler > http://imaginatorium.org Hi Brian, I actually recommend that you read the book "Counterexamples in Real Analysis." That contains scores of what are called "counterexamples in standard real analysis." The vase overflows. No HUP means straight line motion. When I say "iota" or "natural/unit equivalency function", please keep in mind that's post-Cantorian, AND post-Goedelian, and I explain why that is so for having introduced those terms into the discussion. Many applied analytical domains have terms explicitly equivalent to those, for example if you see or hear "infinitesimal", it often means iota, the least positive real, or a differential, with much the same meaning. Sum the areas under a curve that is not a line segment, i.e. integrate, Riemann or Lebesgue, the differential only works when it's positive, non-zero, and less than any ratio of finites. That's a counterexample, Brian. I have my own "nonstandard" real numbers, thank you very much. Did you know model theory posits the existence of a maximal ordinal of which there is none in ZF? Yes, I've said most, if not all, of the above before. Ross |