Cantor Confusion
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From: briggs on 12 Oct 2006 08:09 In article <1160591832.201554.78960(a)c28g2000cwb.googlegroups.com>, cbrown(a)cbrownsystems.com writes: > briggs(a)encompasserve.org wrote: >> In article <1160546562.540946.205860(a)e3g2000cwe.googlegroups.com>, cbrown(a)cbrownsystems.com writes: >> > Dik T. Winter wrote: >> >> In article <virgil-372F10.17374709102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: >> >> > In article <J6w6LC.9rL(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> >> >> > wrote: >> >> > > In article <virgil-9E9CC6.02103209102006(a)comcast.dca.giganews.com> Virgil >> >> > > <virgil(a)comcast.net> writes: >> >> > > > > Dik T. Winter wrote: >> >> > > ... >> >> > > > > > The balls in vase problem suffers because the problem is not >> >> > > > > > well-defined. Most people in the discussion assume some implicit >> >> > > > > > definitions, well that does not work as other people assume other >> >> > > > > > definitions. How do you *define* the number of balls at noon? >> >> > > ... >> >> > > > How about the following model: >> >> > > >> >> > > And you also start with definitions, or a model. I did *not* state that >> >> > > it was difficult to define, or to make a model. But without such a >> >> > > definition or model we are in limbo. I think other (consistent) definitions >> >> > > or models are possible, giving a different outcome. >> >> > >> >> > Can you suggest one? One that does not ignore the numbering on the balls >> >> > as some others have tried to do. >> >> >> >> That does not matter, nor is that the problem. You gave a model where you >> >> find 0 as answer. I only state that I think there are also models where >> >> that is not the answer. Why is a limit of the number of balls over time >> >> not an answer? >> >> >> >> Let's give a simpler problem. At step 1 you add ball 1. At step n you >> >> remove ball n-1 and add ball n (simultaneously, I presume). >> > >> > When you say "at step n", do you have some particular time t associated >> > with that step? >> >> That's somewhat irrelevant. What matters is not what numeric time t is >> associated with each step. What matters is the [partial] ordering on >> the steps. Associating a numeric time t with each step is a way to >> ensure a total ordering. But that's more than we need. > > What time ensures in this problem is that the notion "after all steps > have completed, the state of the vase is..." can be well-defined from > the problem statement. Delusions of physicality. If you want to define the state of the vase after all steps have completed all that is neccessary is to define the state of the vase after all steps have completed. Waving the magic wand, parameterizing by t and appealing to an intuitive notion of "if all steps of a process are defined it follows that the outcome is uniquely defined" is a poor substitute. That notion turns out to be false. >> In particular, arranging matters so that all the step times come before a >> particular finite time is irrelevant -- it's a trick designed to fool >> our intuitions into delusions of physicality and all the implicit >> assumptions that come with physicality. >> > > On the contrary, the red herring here is to assume that the problem is > of the form: let V = {V_n} be a sequence of subsets of N; what subset > of N which corresponds to the lim n->oo V_n? _The_ red herring? I don't think there is just one. > Now, there is an obvious notion of limit we can apply here (pointwise > convergence); but as Dik asserted, in that case we need to /define/ > that notion, independent of the given formulation of the problem. > > This opens the door to complaints such as "noon never arrives" (which > is essentially the claim that there is no such thing defined in the > problem as the limit of the sequence V); or that some other form of > convergence should be used. > > For example, suppose the steps occured at t=1, t=2, etc. We still have > the total ordering you describe below, but now one can argue that "when > all steps are completed, the vase is empty" is a meaningless statement, > because there is no such time "when" this state is achieved. My point is that the problem is equally meaningful or meaningless whether you augment your t scale by contemplating t=oo or whether you scale so that t=12 occurs after all relevant events. If someone is willing to contemplate the one and not the other then you're not dealing with the mathematical part of the problem. You're dealing with delusions of physicality. The way out of that morass is not to scale things so that our intuitions are satisfied. It is to define things clearly enough that our implicit intuititive assumptions need not be silently invoked. > What the element of time in the problem statement provides us with is > the same definition of convergence you outline below. It allows us to > conclude that if t > -1/n, then ball n is not in the vase at time t; > and this holds for /all/ t > -1/n (not just for those t such that t = > -1/m for some natural number m). Note that I did not define a notion of convergence. I proposed a framework within which such a notion could be defined without the use of a real-valued parameter called "time", thus demonstrating by construction that the notion of "time" is not essential to the analysis of this problem.
From: William Hughes on 12 Oct 2006 08:18 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > It is not > > > contradictory to say that in a finite set of numbers there need not be > > > a largest. > > > > There is no such thing as a number with arbitrary size > > (i.e. a number that has the property that its size is > > arbitrary). > > The number I will write down here has that property, before I write it > down. > > 7 > > Now it has no longer this property. > > > The prime number to be discovered next has the property to be unknown. > - Until it is discovered. You are confusing "unknown" with "arbitrary". They do not mean the same thing. > > > > Take a finite set B. > > > > By definition, B must have a finite > > number of elements. This > > number does not have arbitrary size. > > > > Each element has a size This size is not > > arbitrary. > > > > So we have a finite number (which does not > > have arbitrary size) > > of elements, each of which does not > > have arbitrary size. > > > > Thus B has a largest element. > > What is the largest number of the set of numbers mentioned in the New > York Times in 2007? "unknown" but not "arbitrary". > You see, your proof is rubbish. B will have a largest element. And the > set of all numbers ever used in the universe in eternity also will have > a largest element. But it has not yet. Therefore it is unknown. However, it is not arbitrary. - William Hughes
From: Dik T. Winter on 12 Oct 2006 10:17 In article <1160637643.391973.280990(a)i3g2000cwc.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: > Dik T. Winter schrieb: > > In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: > > > David Marcus schrieb: > > ... > > > > I don't follow. How do you know that the procedure that you gave > > > > actually "defines/constructs" a natural number d? It seems that you keep > > > > adding more and more digits to the number that you are constructing. > > > > > > What is the difference to the diagonal argument by Cantor? > > > > That a (to the right after a decimal point) infinite string of decimal > > digits defines a real number, but that a (to the left) infinite string > > of decimal digits does not define a natural number. > > > I claim that the diagonal number of Russell isn't infinite (to the > left) since it is finite different from any natural number by > construction. In what way is it finite different? You claim that the difference with 1 is finite? If so, what is the difference? > In respect to the diagonal number of Cantor, you have to proof if the > sentence "any infinite string of decimal digits (to the right after a > decimal point) defines a real number" is a correct sentence. Or it is > just an arbitrary axiom? The initial finite segments form a Cauchy sequence in the rational numbers, and so by theorems has a limit in the completion of the rational numbers, which are the real numbers. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 12 Oct 2006 10:25 In article <1160640069.503756.100380(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1160551520.221069.224390(a)m73g2000cwd.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: > > > David Marcus schrieb: > > ... > > > > I don't follow. How do you know that the procedure that you gave > > > > actually "defines/constructs" a natural number d? It seems that you keep > > > > adding more and more digits to the number that you are constructing. > > > > > > What is the difference to the diagonal argument by Cantor? > > > > That a (to the right after a decimal point) infinite string of decimal > > digits defines a real number, but that a (to the left) infinite string > > of decimal digits does not define a natural number. > > And why is this so? Because an infinite string of digits is not at all > defined. Only by the factors 10^(-n) this is veiled. Cauchy sequences, completion of the rationals, etc. Pray read a bit about that all (and also about how the reals are defined by e.g. Dedekind). > But this has > been forgotten by Cantor whose diagonal proof attaches the same weight > to every digit. When comparing for equality each digit has equal weight. Otherwise there would be no trichonomy. (Or do you claim that some numbers are more equal to a given number than others? Or that some numbers are more unequal to a given number than others?) > Or it could also be applied to > the left of the decimal point, constructing an infinite natural number > and showing that every list of natural numbers is incomplete. It can be done if you use some metric where the sequence 1, 11, 111, ... converges. And indeed, it can be done in the n-adics, where such sequences indeed do converge, and the result is 1/(1-n). And *indeed* you can show (with the diagonal proof) that the n-adics are *not* countable for any n. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 12 Oct 2006 10:27
In article <1160646886.830639.308620(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > If every digit position is well defined, then 0.111... is covered "up > to every position" by the list numbers, which are simply the natural > indizes. I claim that covering "up to every" implies covering "every". Yes, you claim. Without proof. You state it is true for each finite sequence, so it is also true for the infinite sequence. That conclusion is simply wrong. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |