Cantor Confusion
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From: Virgil on 8 Oct 2006 17:37 In article <1160302387.877639.179920(a)c28g2000cwb.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > mueckenh(a)rz.fh-augsburg.de schreef: > > > > But you cannot derive that the vase is not empty at noon from the > > observation that its contents cannot decrease? > > A picture says more than a thousand words. Isn't it? > > http://hdebruijn.soo.dto.tudelft.nl/jaar2006/ballen.jpg > > Han de Bruijn A misrepresentative picture misrepresents like a thousand words.
From: Virgil on 8 Oct 2006 17:42 In article <1160308871.194701.44520(a)c28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 0) There is a bijection between the set of balls entering the vase and > |N. When? > 1) There is a bijection between the set of escaped balls and |N. When? > 2) There is a bijection between (the cardinal numbers of the sets of > balls remaining in the vase after an escape)/9 and |N. This does not occur ever.
From: Virgil on 8 Oct 2006 17:50 In article <1160319212.308670.87550(a)h48g2000cwc.googlegroups.com>, "Albrecht" <albstorz(a)gmx.de> wrote: > Virgil wrote: > > In article <1160223586.604282.269450(a)h48g2000cwc.googlegroups.com>, > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > > > Tonico wrotef: > > > > > > > Maths, just like the other sciences, isn't grounded on dogma, and > > > > people forwarding REASONABLE, well-based objections, opinions or ideas > > > > on whatever are always welcome. > > > > > > _There_ is your problem! Ask Virgil, and the other mathematicians here: > > > > > > MATHEMATICS IS _NOT_ A SCIENCE > > > > > > And therefore there is NO guarantee that it "isn't grounded on dogma". > > > > > > HdB's version of mathematics certainly seems to be grounded in dogma. > > > > My mathematics is merely grounded on determining what can be derived > > from a given set of axioms. > > 1. The moon exists > 2. The moon consits of green cheese > 3. The man in the moon is a mouse > > Now do maths. Else explain why you prefer a special set of axioms. > > Best regards > Albrecht S. Storz Does A.S.S suggest that enough can be derived from his set of axioms, without any additional assumptions, to be of any interest? Does that man-in-the-moon-mouse exist or not? Does he/it, if extant, survive by eating the green cheese? If so, what does he/it excrete, and for how long has he been doing it? A.S.S. needs a more interesting axiom set to interest me.
From: Virgil on 8 Oct 2006 17:52 In article <MPG.1f92eeb49307dc6a98969d(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > David Marcus schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > You got it several times already. According to the axiom of infinity > > > > the set of all natural numbers does exist. And with it the following > > > > statements are true: > > > > > > > > 1) Before noon every ball comes out of the vase. At noon the vase is > > > > empty. > > > > 2) Before and at noon there are more balls in the vase than have come > > > > out. > > > > > > How do you translate the words of the problem into mathematics? > > > > 0) There is a bijection between the set of balls entering the vase and > > |N. > > 1) There is a bijection between the set of escaped balls and |N. > > 2) There is a bijection between (the cardinal numbers of the sets of > > balls remaining in the vase after an escape)/9 and |N. > > > > Instead of "balls", use "elements of X where X is a variable". > > Sorry, perhaps I wasn't clear. That's not what I meant. Please state the > problem using mathematics. Mathematics doesn't include "balls", "vases", > "enter", "escape". Mathematics uses sets, functions, numbers. Actually, > please state the problem in words, then state how you translate it into > (or model it using) mathematics. How is this: Let A_n(t) be equal to 0 at all times, t, when the nth ball is out of the vase, 1 at all times, t, when the nth ball is in the vase, and undefined at all times, t, when the nth ball is in transition. Note that noon is not a time of transition for any ball, though it is a cluster point of such times. let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase at any non-transition time t. B(t) is clearly defined and finite at every non-transition point, as being, essentially, a finite sum at every such non-transition point. Further, A_n(noon) = 0 for every n, so B(noon) = 0. Similarly when t > noon, every A_n(t) = 0, so B(t) = 0
From: Dik T. Winter on 8 Oct 2006 21:18
In article <1160302222.613036.300930(a)h48g2000cwc.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > Dik T. Winter wrote: > > > I would say that all forms of mathematics are grounded on axioms (or dogmas > > as you prefer to say). But contrary to dogmas, axioms can be negated to > > get another form of mathematics. Dogmas are absolute truths, axioms are > > only absolute truths within some realm of discourse. > > If it is so simple, where then come these heated debates (about the > Balls in a Vase at noon) come from? Because some of the people in the discussion only use intuition, and not provable concept, but they keep stating that there intuition tells them that there are contradictions, without giving proof of any of this. If you are, say, discussing in a context where the axiom of infinity is assumed, you can not get a contradiction when during your proof you, at one stage or another, use the contradiction of that axiom. 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? You can not use limits, because the limit does not exist when you use standard mathematics. So using standard definitions there is no answer. More precise, given the sequence of sets: {1, ..., 10) {2, ..., 20} {3, ..., 30} etc., is there a limit? Well, no, there is no defined limit unless you define what a limit of sets looks like. I have never seen a definition that tells me how the limit of a sequence is defined. The limit of the size of the sets also gives no answer, because that limit does not exist. Strange enough, when somebody goes on to define things, *you* question his definitions, rather than the result. > And why then are some axiom systems > so much more dominant than others? That is easy. Some axiom systems give results easier than others. For instance, the axiom of infinity asserts that the set of natural numbers does exist. This means a simple definition of limits, also this means that the definitions of the reals in their various ways work, differentiation and integration get properly defined, and in the end, even things like eigenvalues of matrices are properly defined. To build the same without the axiom of infinity may be possible, but to me it does not look as being exactly easy. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |