Cantor Confusion
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From: David Marcus on 30 Oct 2006 16:45 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > 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)? > > If you think that you can arbitrarily define the value at t = 0, how > can you be sure that in the vase problem this value is "empty"? Before we discuss the vase problem, how about answering my question, i.e., are you saying that the above implies that f(0) = g(0)? -- David Marcus
From: William Hughes on 30 Oct 2006 16:47 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > Let our alphebet be {0,1,2}. Let our diagonal construction be > > 0->1, 1->2, 2->0. Define a finite sequence as one that has only 0's > > after a certain point. The set A only has sequences that have only 0's > > after a certain point. > > > > A begins > > > > 000... > > 1000... > > 2000... > > 11000... > > 12000... > > > > The diagonal is an unending string of ones. The set A does not contain > > the diagonal. > > > > > > > > > Thus there is no contradiction. You need a diagaonal before > > > > you can get a contradiction, therefore you need set B. > > > > > > Why should we not construct the diagonal of these sequeces (words) of > > > A? > > > > . > > We can do this but the diagonal is not a finite sequence, so it is > > not a member of A. > > If the list consists of finite sequences, then the diaogonal is a > finite sequence too. Because it cannot be broader than the list > Correct. The diagonal is exactly as long as the list is broad. The list is broader than any line. So the diagonal is longer than any line. > All entries of the list have a finite number of letters. Correct. And given any integer, we can take a set of lines such that the number of letters in this set is greater than the integer. > An infinite > sequence is larger than any finite sequence. The diagonal of a list > cannot have more letters than the lines. > Correct. The number of letters in the lines is greater than any integer. So the greatest number of letters the diagonal can have is greater than any integer. So the diagonal has infinite length (call this potentially infinite length if you get your kicks by saying potentially). - William Hughes
From: David Marcus on 30 Oct 2006 16:48 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > 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.) > > Here it means, the result of my proof is valid for all possible > theories which allow to define the infinite binary tree. Sorry. I still don't know what you mean. ZFC is enough to define an infinite binary tree. Are you saying that your proof can be done in ZFC or are you not saying this? If not, then what are you saying? > > 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. > > But it is not an interesting question. Perhaps not interesting to you, but please answer it anyway. -- David Marcus
From: Virgil on 30 Oct 2006 16:50 In article <c5a1b$4545ba52$82a1e228$12545(a)news1.tudelft.nl>, Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > Are you still doing physics, water, and a finite number of molecules? > > Let us know when you switch to mathematics. > > I'm DOING mathematics. Mathematics is NOT independent of Physics. That dependency is a cross only physicists bear. For non-physicists, mathematics is quite independent of physics.
From: Virgil on 30 Oct 2006 16:52
In article <1162216166.774865.153700(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > If you think that you can arbitrarily define the value at t = 0, how > can you be sure that in the vase problem this value is "empty"? Every ball inserted before noon is removed before noon. |