Cantor Confusion
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From: David Marcus on 29 Oct 2006 12:49 mueckenh(a)rz.fh-augsburg.de wrote: > stephen(a)nomail.com schrieb: > > > Can you describe a continuous version of the problem where each > > "unit" of water has a well defined exit time? A key part of > > the original problem is that the time at which each ball is > > removed is defined and reached. This is crucial to the problem. It is > > not just a matter of rates. If you added balls 1-10, then 2-20, > > 3-30, ... but you removed balls 2,4,6,8, ... then the vase is > > not empty at noon, even though the rates of insertions and removals > > are the same as in the original problem. So you cannot just > > say the rate is 10 in and 1 out and base an answer on that. > > The answer for any time t *before noon* is independent of the chosen > enumeration of the balls. Doesn't that fact make you think a bit > deeper? In other words, we have two (or more) situations (depending on how we decide when the balls are removed). For these different situations, the number of balls in the vase before noon are the same. Are you saying that this implies that the number of balls in the vase at noon are the same for the different situations? Suppose we define two functions by f(x) = 1 if x < 0, 0 if x >= 0, g(x) = 1 if x <= 0, 0 if x > 0. Then for x < 0, f(x) = g(x). Are you saying that this implies that f(0) = g(0)? -- David Marcus
From: David Marcus on 29 Oct 2006 12:58 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > David Marcus wrote: > > > Not sure if he ever said precisely "within Z set theory", but he > > > certainly said things very similar. Below are a few messages that I > > > found. There are probably others. > > > > > > In the first, he says that "standard mathematics contains a > > > contradiction". > > as ar as standard mathematics is derived from set theory an comes to > the conclusion of an empty vase at noon or an uncountable set of reals. > Classical mathematics is free of such contradictions. > > > >In the next two, he states there are "internal > > > contradictions of set theory". In the next, I say that he says that > > > "standard mathematics contains a contradiction", and he does not > > > dispute this. > > > > Thanks. And at least a couple of times I said that I was reading his > > argument to see whether it does sustain his claim about set theory, as > > I mentioned specifically Z set theory. > > My proof of the binary tree covers all possible theories. And it should > not cost you too much time to see that it is true. Unfortunately, I don't know what "covers all possible theories" means. (You seem to use the word "cover" a lot.) Are you saying that your proof works *within* all possible theories, i.e., all possible theories contain your proof? For example, does your proof work within ZFC (i.e., are the axioms and rules of inference of ZFC all that you need for your proof)? You just said: "The tree is that mathematics which deserves this name. It is outside of your model, independent of ZFC". Either you have a proof that can be given in ZFC or you don't. Which is it? This shouldn't be a difficult question to answer. -- David Marcus
From: David Marcus on 29 Oct 2006 13:03 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > And once again WM deliberately confuses, "all positions are > > > > finite", with "there are a finite number of positions". > > > > > > There is no confusing! Every finite position belongs to a finite > > > segment of positions (indexes). If you don't believe that and assert > > > the contrary, then try to find a finite position which dos not belong > > > to a finite segment (of indexes). > > > > > > It is simply purest nonsense, to believe that "all positions are > > > finite" if "there are an infinite number of positions". > > > > Before any of us wastes more time on this, please pick one: > > > > 1. You are making a statement that you say is provable within some > > standard mathematical system, e.g., ZFC. > > > > 2. You are making a statement that is true within your own system. > > My above statement is true within any system which is free of self > contradictions. That's nice, but it isn't what I asked. I'll try again with a more specific question: 1. Is your statement provable within ZFC, i.e., using the axioms and rules of inference of ZFC? -- David Marcus
From: David Marcus on 29 Oct 2006 13:05 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > Virgil wrote: > > > In article <1162067841.239007.74250(a)e64g2000cwd.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > Please name the mathematicians that agree that your argument is correct. > > > > > (Han doesn't count, since he says he is a physicist.) > > > > > > > > Every mathematician whose mind has not yet suffered the drill of set > > > > theory you probably were exposed to knows that my argument is correct. > > > > > > "Mueckenheim" cannot speak for all those people. > > > > The set of mathematicians who don't know any set theory is probably > > pretty small. So, Mueckenheim's statement might be true (vacuously). > > Most know a bit set theory, but in their study they do not take curses > in set theory. In Germany it is not obligatory. I doubt that is > different in other countries. 90 % will not be able to tell the ZFC > axioms when asked. Most of them will not know that there is no > definable well order of the real numbers. Many will not even know what > a well order is. They all can be very good mathematicians, and in > general they are. Considering that I wrote "set theory", not "axiomatic set theory", I fail to see the relevance of your comment. However, feel free to name any of these "very good mathematicians" who agree that your argument is correct. -- David Marcus
From: Virgil on 29 Oct 2006 12:32
In article <1162135300.256879.37750(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > No, you must tell me what that limit is, until now you have not provided > > any definition of a limit of sets so I can verify whether the above is > > true or false. Until now you have defined the limit of exactly *one* > > sequence of sets, but that definition is not sufficiently general to > > determine whether the above statement is true or false. > > The definition is sufficiently special (= non-general) to determine > that lim [n-->oo] {1,2,3,...,n} = N is correct. Only to "WMueckenheim", but not to anyone else. > > > But this is again different, you state here that N is the set of natural > > numbers, and you define the limit of a particular sequence of sets as > > being that set N. But it does *not* cover arbitrary sequences of sets. > > What, according to that definition, is: > > lim {n -> oo} {-1, 0, 1, ..., n} > > the definition does not make that clear. So the definition is not about > > the "operator" but about a particular sequence of sets. > > The operator "lim [n-->oo]" defines N. Not unless N is already defined. Your definition is circular in that correct undersatnding of its meaning depends on one already knowing its meaning. > > > In the first > > > case, we have 1/n < epsilon for every positive epsilon and we may > > > *define or put* > > > lim [n-->oo] 1/n = 0. > > > In the second case we have without further ado > > > lim [n-->oo] 1/n = 0. > > > That is the difference between potential and actual infinity. > > > > Well, in mathematics the first form is valid, the second is not valid. > > If actual infinity is assumed to exist, then the second case is valid. Not in any actual mathematics, only in that garbage "WM" mistakes for it. |