Cantor Confusion
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From: David Marcus on 8 Oct 2006 12:43 mueckenh(a)rz.fh-augsburg.de wrote: > > What does the sentence "...the contents of the vase grows in infinity" > > mean at all?? > > The contents does never decrease. > > > It sounds just like the christian fundie sentence "I love > > you in Christ": what does this mean, in the holy name of Woody > > Allen?!?!? And axioms can yeild whatever they want: that's what they're > > axioms for! > > If a set of axioms yields the theorems A and nonA, then this set is > useless. The axioms of ZFC yield the theorems "the vase is empty at > noon" and "the vase is not empty at noon". Please state the vase problem in words (vase, balls, enter, leave, noon) and also state your translation of the problem into the language of Mathematics (sets, functions, numbers). -- David Marcus
From: Alan Morgan on 8 Oct 2006 16:36 In article <1160296113.211935.299880(a)c28g2000cwb.googlegroups.com>, <mueckenh(a)rz.fh-augsburg.de> wrote: > >Virgil schrieb: > >> > We know, not by >> > intuition, but by logic, that the vase at any time contains more balls >> > than have escaped. >> >> Absolutely false. > >Bold words, but unfortunately simply wrong. The contents of the vase >increases any time by 9 balls. > >> Before any balls are put in the vase, it is empty, and >> after al balls have been removed from the vase it is equally empty. >> It is only between these time that there are any balls in the vase at >> all. >> >We have a contradiction n ZFC >1) Every ball will have left the vase at noon. >2) At noon there are more balls in the vase than at any time before. That would be a contradiction. Fortunatly, you can not conclude #2. You *can* conclude that at every point before noon there are more balls in the vase than at any time before, but that's not the same thing. Before noon != noon. Consider the even simpler case of putting balls in the vase and never removing them. Put in one ball at one minute before noon, another ball at 1/2 minute to noon, another at 1/4 of a minute to noon, etc. At every point in time before noon there are a finite number of balls in the vase, but just after noon...... Alan -- Defendit numerus
From: Virgil on 8 Oct 2006 17:09 In article <1160295281.279569.143920(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > If a set of axioms yields the theorems A and nonA, then this set is > useless. The axioms of ZFC yield the theorems "the vase is empty at > noon" and "the vase is not empty at noon". Not so. One has to add "Mueckenh"'s, or at least other assumptions, to ZFC to get a nonempty vase at noon. The only ZFC analysis coincides with: 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 There is no ZFC-compatible argument for having a non-empty vase at or after noon that does not require assumptions beyond those of ZFC. And it is with those additional assumptions that "Mueckenh" and others make, that the conclusions of ZFC conflict.
From: Virgil on 8 Oct 2006 17:10 In article <1160295415.826549.50480(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Tonico schrieb: > > > Not even Brouwer and his rather > > weird intuitionist movement could, imfho., have used such nonsense to > > defend their ideas... > > Oh, it seems I am in good company, in your eyes. > > Regards, WM I read that as excluding you even from that company.
From: Virgil on 8 Oct 2006 17:15
In article <1160295497.967885.243740(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > 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. > > But you cannot derive that the vase is not empty at noon from the > observation that its contents cannot decrease? As there is nothing in the gedankenexperiment that specifically prohibits a decrease, except at times strictly before noon, I see no reason to come to any such conclusion about what the status is at or after noon. And in fact: 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, or limit 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. It is, of course, undefined at every transition time. 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 |