Cantor Confusion
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From: David Marcus on 9 Oct 2006 02:05 Dik T. Winter wrote: > In article <virgil-148331.15522708102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > ... > > 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. > > Hm. I humbly submit that the probability for a particular rational number > in the range [0,1) the probability to get it when doing a random choice is 0. > Nevertheless, the sum of all the probabilities is 1. The sum of countably > many 0's is not always 0. I don't follow. Usually, probability measures are countably additive. As for what Virgil wrote, presumably he meant for the sum in his definition of B(t) to be the usual infinite sum from calculus/analysis (i.e., the usual epsilon-delta definition). In which case, sum_{i=1}^infty 0 = 0. -- David Marcus
From: cbrown on 9 Oct 2006 03:03 Dik T. Winter wrote: > In article <1160302222.613036.300930(a)h48g2000cwc.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > > 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. > I agree. > 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. > I agree again! > 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? I disagree. This is no more difficult than asking "how do you *define* the number of balls at pi/10 seconds before midnight?" To ask "how many balls are in vase A at time t?" is equivalent to asking "what is the cardinality of the set of balls in vase A at time t?" As the question is phrased, we can define a function f of the (continuous) time t yielding a well-defined set of naturals: f(t) = {n in N : 1/(10*n) < -t < 1/n}. This is consistent with, e.g., f(-9/10) = f(-1); and to my mind, "n in f(t)" is the only sensible interpretation to the question "Is ball n in vase A at time t?" We /then/ define a function numBalls(t) on these sets, equivalent to the cardinality of f(t). And so we can ask "what is the number of balls at time t?", and always get a well-defined (and finite) value. It is fallacious to base one's argument on the assumption that the problem is /equivalent/ to defining a function g on N ("the number of balls at step n"), satisfying g(n)*9 = g(n+1), and that therefore either (a) t = 0 implies n = oo, and that therefore at time t, f(n) = f(oo) = oo, or (b) since n is always finite in the problem, "noon never occurs". This approach misses the point. /Time/ is the /independent/ variable here, not n. The question isn't "Based on the step B in the process which is equivalent to noon, how many balls are there at step B?", which would indeed be a limit problem; the question is, "Given that we can define the function f(t), what is |f(0)|?" > You can not use limits, because the > limit does not exist when you use standard mathematics. And in fact this is not a problem of limits. Cheers - Chas
From: Albrecht on 9 Oct 2006 03:40 Virgil schrieb: > 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. You are unable to understand? You have to explain why you use ZF and not man-in-the-moon-mouse or any other set of axioms (not for me, I know, but perhaps for yourself?). That's the point. You decide this with the criteria how many and how intersting the possible derivations are? How do you know that ZF is the best in this concern? Best regards Albrecht S. Storz
From: Virgil on 9 Oct 2006 04:10 In article <1160377400.288823.275240(a)c28g2000cwb.googlegroups.com>, cbrown(a)cbrownsystems.com wrote: > Dik T. Winter wrote: > > In article <1160302222.613036.300930(a)h48g2000cwc.googlegroups.com> > > Han.deBruijn(a)DTO.TUDelft.NL writes: > > > > 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. > > > > I agree. > > > 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. > > > > I agree again! > > > 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? > > I disagree. > > This is no more difficult than asking "how do you *define* the number > of balls at pi/10 seconds before midnight?" > > To ask "how many balls are in vase A at time t?" is equivalent to > asking "what is the cardinality of the set of balls in vase A at time > t?" > > As the question is phrased, we can define a function f of the > (continuous) time t yielding a well-defined set of naturals: f(t) = {n > in N : 1/(10*n) < -t < 1/n}. This is consistent with, e.g., f(-9/10) = > f(-1); and to my mind, "n in f(t)" is the only sensible interpretation > to the question "Is ball n in vase A at time t?" > > We /then/ define a function numBalls(t) on these sets, equivalent to > the cardinality of f(t). And so we can ask "what is the number of balls > at time t?", and always get a well-defined (and finite) value. > > It is fallacious to base one's argument on the assumption that the > problem is /equivalent/ to defining a function g on N ("the number of > balls at step n"), satisfying g(n)*9 = g(n+1), and that therefore > either (a) t = 0 implies n = oo, and that therefore at time t, f(n) = > f(oo) = oo, or (b) since n is always finite in the problem, "noon never > occurs". > > This approach misses the point. /Time/ is the /independent/ variable > here, not n. > > The question isn't "Based on the step B in the process which is > equivalent to noon, how many balls are there at step B?", which would > indeed be a limit problem; the question is, "Given that we can define > the function f(t), what is |f(0)|?" > > > You can not use limits, because the > > limit does not exist when you use standard mathematics. > > And in fact this is not a problem of limits. > > Cheers - Chas How about the following model: 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 transitions (changes state from before to after that time). 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: Virgil on 9 Oct 2006 04:17
In article <1160379651.668583.120910(a)m7g2000cwm.googlegroups.com>, "Albrecht" <albstorz(a)gmx.de> wrote: > Virgil schrieb: > > > 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. > > > You are unable to understand? You have to explain why you use ZF and > not man-in-the-moon-mouse or any other set of axioms (not for me, I > know, but perhaps for yourself?). That's the point. Some sets of axioms are sufficient to produce much, if not all, of analysis. Such sets are interesting. Other sets of axioms are sufficient to produce the entire theory of groups, another interesting set. A.S.S.'s set is nowhere near as interesting. > You decide this with the criteria how many and how intersting the > possible derivations are? How do you know that ZF is the best in this > concern? I don't. NBG seems quite as good in many ways. And if A.S.S. has a set that he considers better than either, let him by all means present it. But as yet, I have seen none I consider better than both ZF and NBG are for set theory. |