An uncountable countable set
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From: cbrown on 15 Oct 2006 18:11 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> > > > >>>>>>> Putting aside the question of /how/ (limit? sum of binary functions?) > >>>>>>> one determines the /number/ of balls in the vase at time t for a > >>>>>>> moment... > >>>>>>> > >>>>>>> Do you then agree that there is some explicit relationship described in > >>>>>>> the problem between what time it is, and whether any particular > >>>>>>> labelled ball, for example the ball labelled 15, is in the vase at that > >>>>>>> time? > > > >>>>>> For any finite time before noon, when iterations of the problem are > >>>>>> temporally distinguishable, yes, but at noon, no. > >>>>>> > > > >>>>> I don't understand why you think this would be the case. > >>>>> > >>>>> Why do you think the relationship holds for t < 0? > >>>>> > >>>>> Why you do think it does not hold for t >= 0? > >>>>> > >>>>> Cheers - Chas > >>>>> > > > >>>> Because for t>=0, n>=oo. > > > >>> Actually, for t>=0, there is /no/ natural number n such that t = -1/n. > >>> Similarly, for t = -1/pi, there is no natural number n such that t = > >>> -1/n. > > > >> Yeah, no idding. Who said oo was a natural number? > >> > > > > You just implied it; when you claimed that at t>=0, n>=oo; where I > > presume that by "n", you refer to the statement in the problem: > > > > "At time t = -1/n, where n is a natural number, we add balls labelled > > (10*(n-1)+1) through 10*n inclusive, and remove the ball labelled n". > > > > That statement obviously does not refer to removals or additions of > > balls at time t = 0, becuase there is no natural number n such that > > -1/n = 0. > > > > Do you agree with this conclusion? > > > > Of course. That's what I was saying. Was it? When you said: > >>>> Because for t>=0, n>=oo. you seemed to be saying that we could use the rule: > > "At time t = -1/n, where n is a natural number, we add balls labelled > > (10*(n-1)+1) through 10*n inclusive, and remove the ball labelled n". and the fact that "for t>=0, n>=oo" to prove that we cannot conclude that ball 15 is not in the vase at t=0. Since "n>=oo" is never true, I don't see how your logic applies. In fact, from the problem statement, it follows logically that ball 15 is not in the vase at t=0. Now, I will grant this (alone) does not imply that you cannot /also/ prove that ball 15 /is/ in the vase at t=0. If you can /also/ prove that, then you have proven that the problem is contradictory - i.e., it contains assumptions that allow us to prove that something is both true and false. But even if you /did/ have such a proof, that would not change the fact that we can /also/ prove that ball 15 is not in the vase at t=0. 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? > Your statement concerning n does > not cover noon, because noon=f(oo), and oo is outside your range. You've lost me. What is f? What does it mean to say "noon = f(oo)"? How does this disprove the assertion that ball 15 is not in the vase at t=0? > So, > you really don't have any claim with regard to what happens at noon. Its > beyond your purview. On the contrary, in a mathematical sense, a thing "happens" (i.e., can be concluded from the problem statement) if, and /only/ if, it can be logically deduced from assertions in the problem. Suppose we modify the original problem by appending "and, at each time t = -1/n, where n is a natural number, we do not add a solid gold statuette of Richard Nixon kissing Henry Kissinger to the vase". Do you conclude that therefore "it is beyond our purview" to state "there is not a solid gold statuette of Richard Nixon kissing Henry Kissinger in the vase at noon"? Assuming you see the absurdity of the above, why do you then claim "because there is not natural number n such that -1/n >= 0, therefore it is our beyond our purview to claim that ball 15 is not in the vase at t>=0"? > > >>> But what do either of those statements have to do with whether or not > >>> ball 15 is in the vase at t=0? > >> Nothing to do with ball 15. That has a specific time of removal. Every > >> specific ball does. > > > > This contradicts what you said above regarding whether we can determine > > if a particular ball is in the vase at some time t; you wrote: > > > >>>>>> For any finite time before noon, when iterations of the problem are > >>>>>> temporally distinguishable, yes, but at noon, no. > > How does that contradict that? Well, you claimed above that your statements have /nothing to do/ with whether ball 15 is in the vase at t=0. You previously claimed that your statements /do/ have something to do with whether the ball 15 is in the vase at t=0. That seems contradictory to me. > Any specific finitely indexed ball has a > specific finite time before noon at which it is inserted and another at > which it is removed. And since it is never replaced in the vase at any other time after its removal, we can then conclude: "there is no ball in the vase at noon which is labelled with a natural number". Go on... > At any of those times, there are a growing number > of balls as t approaches noon. We are in agreement so far... > The set-theoretic claim is that, even > though nothing happens AT noon, nevertheless BY noon the vase is empty, > even though BEFORE noon there are potentially infinitely many balls in > the vase. Ahem. When does this occur, if at all? There is a misstatement in your assertion (the number of balls in the vase is always finite at any t<0); but I think your main objection is in fact linguistic and not mathematical. Very generally, suppose I say "For all t<0, something happens to X. For all t >= 0 nothing happens to X". Now let f be the function defined by: f(t) = 1 if something happens to X at time t, and f(t) = 0 if nothing happens to X at time t. Now I claim that something happens at time t = 0; to whit, f(t) stops being 1 and begins to be 0. To me, your argument seems to be "how can that be possible? Nothing is explicitly described as happening at t>=0, so how can something happen at t=0?" In the original problem, the only exp
From: Han de Bruijn on 16 Oct 2006 03:45 Virgil wrote: > In article <1160935613.121858.178420(a)m7g2000cwm.googlegroups.com>, > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >> I find the idea absurd that natural numbers can be built >>by putting curly braces around the empty set. > > It appears as if much of useful mathematics is ultimately based on what > HdB finds absurd. The natural numbers can be defined without employing set theory. Han de Bruijn
From: imaginatorium on 16 Oct 2006 11:07 Han de Bruijn wrote: > Virgil wrote: > > > In article <1160935613.121858.178420(a)m7g2000cwm.googlegroups.com>, > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > >> I find the idea absurd that natural numbers can be built > >>by putting curly braces around the empty set. > > > > It appears as if much of useful mathematics is ultimately based on what > > HdB finds absurd. > > The natural numbers can be defined without employing set theory. I should think they could be. Though I fancy the natural numbers will never be properly defined by anyone who is incapable of understanding the set theoretic definition. Brian Chandler http://imaginatorium.org
From: Ross A. Finlayson on 16 Oct 2006 11:22 Virgil wrote: > In article <4530434f(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > David Marcus wrote: > > > Han de Bruijn wrote: > > >> Virgil wrote: > > >> > > >>> In article <9020$452f46c4$82a1e228$31963(a)news2.tudelft.nl>, > > >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > >>> > > >>>> Virgil wrote about the Balls in a Vase problem: > > >>>> > > >>>>> Everything takes place before noon, so that by noon, it is all over and > > >>>>> done with. > > >>>> Noon is never reached, because your concept of time is a fake. > > >>> > > >>> No one expects the experiment to take place anywhere except in the > > >>> imagination, so that everything about it, including its time, is > > >>> imaginary, but logic continues to hold even there, at least for > > >>> mathematicians. And logic says that a ball removed from a vase is not > > >>> later in the vase. > > >> Since your logic and the logic of others give contradictory results for > > >> the same problem, logic alone is unreliable. > > > > > > Are you saying that Mathematics gives contradictory results for a > > > problem? If so, please state the problem. > > > > > > > The problem, among others, is the vase. If you haven't gotten a clue > > about the problem yet, well, get on the bus. > > Along with TO, HdB, "Mueckenh", Ross, and others of that ilk. > > I am surprised that JSH has not joined in. There is no universe in ZF, ZF is inconsistent. Model theory posits the existence of a maximal ordinal in, for example, ZF, where there exists no maximal ordinal. That the maximal ordinal does and does not exist is self-contradictory, in ZF and similar regular/well-founded theories. The generic extension of N, that bijects to R, contains no elements not in N. Sets are defined by their elements. For no finite differential, as algebraically manipulable as it is, does analysis work. The differential is infinitesimal. There are only real numbers between zero and one. There are also everywhere real numbers between zero and one. If infinitesimals exist between zero and one, they're reals. The least positive real is called iota, and according to "Counterexamples in Real Analysis" such a thing exists, arithmetic on those things involves book-keeping of related rates and an implicit universe. There are a variety of developments of "non-standard" real numbers, and the hyperintegers are the standard integers and the hyperreals are the standard reals, because the transfer principle is what's of actual importance in extending various, but not all, results true on the finite to the infinite. A variety of modern mathematical methods divide by zero. A (regular) set, even an infinite set, is demonstrably larger than a proper superset. ZF has no numbers in it. In general finite von Neumann ordinals are considered to be mechanistically the natural numbers, but that is a definition. Cauchy/Dedekind is insufficient to describe the real numbers. N E N, in terms of the naturals number similarly to how U E U, the universe would contain itself, besides that the universe as a set is an element of itself. Where N is considered in a similar vein, because it's infinite and Spinoza's continuum, N < N besides N = N. The integers are superarchimedean ring. Less than absolute zero is hotter than any finite temperature. The closer the subatomic particles are examined, the smaller they appear to be, as reflected in CODATA. Similarly, the more that is known about the physical universe, the larger it appears to be, it's infinite. There's only one theory with no non-logical axioms, the null axiom theory. If it's not the null axiom theory, Goedel says it's not a theory of everything. I'm among many who see utility in bijecting the naturals to the unit interval of reals, in reasonable ways, preserving true analytical results. In Parrradise you're naked and ignorant. If you perceive otherwise, is it not Hell? There are NO known uses of transfinite cardinals in physics. Be a post-Cantorian, but it might be wise not to tell people that, because they might not understand. Consider the ball and vase as a related rate problem. The vase is a bucket with a hole in it, and it overflows. Find something in the actual real world, not the fantasy "here's a finite vase that does not cover the entire galaxy that can contain any finite number of balls", which could never exist. Find an actual particle/wave system to which to apply Zeno, and then maybe you'll learn about HUP, and how it doesn't always apply, photons, time, and some of the notions about the continuum of time. Ross
From: MoeBlee on 16 Oct 2006 14:32
Tony Orlow wrote: > Your take on logic is very, shall we say, provincial. "Your logic is provincial." Surely, that is a stunning rejoinder in any mathematical discussion. MoeBlee |