Cantor Confusion
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From: mueckenh on 19 Oct 2006 12:24 William Hughes schrieb: > There are only a limited number of messages that it is > possible to communicate. One of these messages describes a > larger number than any of the other messages (description includes > description of any compression method used). Correct. There will be a largest number ever communicated. But that does not mean that this number is fixed as yet (unless you would favour strict determinism.) > > > > Of course, but necessarily it also changes the maximum sizes of > > elements. As long as the sizes all are finite, the cardinality is > > finite too. > > Nope. It is possible to have a set with infinite cardinality > composed of element all of which have finite size > (consider the real numbers in [0,1]). Yes, but it is impossible to have such an infinite set of numbers with finite differences. > > >You assume that only the one is changed, the other is not. > > But you seem not to be aware that in natural numbers size and > > cardinality are strictly the same > > I take it you mean that for a set of natural numbers of the form > {1,2,3,...,n}, the cardinality is equal to the size of the maximum > element. > This is only true if the set has a maximum element It is true for a set of finite numbers. An infinite number of finite numbers would imply an infinite difference but that would imply an infinite number. (Given actual infinity.) > > > You state something about the set {2,4,6,...} > > > > That are finite numbers. > > That is also an infinite set. No. > > > > > Piffle. Numbers are not infinite, sets are infinite. Sets are numbers , number are sets. What is Piffle? > > > > > The brain is contstrained by physical laws. The concepts produced by > > > > > the brain are not contrained by physical laws. > > > > > > > > The puppet hangs on the string. The feet of the puppet hang on the > > > > puppet, but not on the string? They do not fall down when the string is > > > > cut? > > > > > > The beauty of the puppet cannot exist without the puppet. If the > > > string > > > is cut the beauty of the puppet does not fall down. > > > > If the puppet falls down and if it is of porcelain, the beauty is gone. > > > > So? The beauty may be gone, but the beauty does not fall down. > The beauty of the puppet depends on the puppet, but the beauty of the > puppet is not a physcial entity. I wonder how beauty would look like without physics. Regards, WM
From: mueckenh on 19 Oct 2006 12:28 Virgil schrieb: > In article <1161159181.827351.41670(a)e3g2000cwe.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Wrong. The connection between finite paths and partial sums of edges > > > > leads to > > > > (1-(1/2)^n+1)/(1 - 1/2) edges per path. > > > Which, as written, is negative for all positive naturals n. > > Indeed? In your world? Then I can guess why your vase > > sometimes empty. > And (1-(1/2)^n+1)/(1 - 1/2) <= 4 for all n in N, > and lim_{n -->oo} (1-(1/2)^n+1)/(1 - 1/2) = 4. I knew this calculation when seeing your reply. But I have not yet understood why you consider 4 as being negative. > And if edges are to be split, so can paths be split, and into as many > pieces. Please split the paths. Then you will get more pieces. You will get a larger set which consists of what? Of all edges. Regards, WM
From: mueckenh on 19 Oct 2006 12:30 William Hughes schrieb: > > A constructible number is a number which can be constructed. Definition > > obtained from Fraenkel, Abraham A., Levy, Azriel: "Abstract Set > > Theory" (1976), p. 54: "Why, then, the restriction to the digits 1 and > > 2 in our proof? Just to kill the prejudice, found in some treatments of > > the proof, as if the method were purely existential, i.e. as if the > > proof, while showing that there exist decimals belonging to C but not > > to C0, did not allow to construct such decimals." > > > > Definition (by me): A number which can be constructed like pi, sqrt(2) > > or the diagonal of a list is that what I call constructible. If you > > dislike that name, you may call these numbers oomflyties. Anyhow that > > set is countable. > > Nope. By the definitions you use, that set is not countable. > Every set of constructions is countable due to the finite alphabet of any language. > > And that set cannt be listed. > > And here is your problem. Uncountable means unlistable. Not my problem. Countably infinite means unlistable too. Regards, WM
From: georgie on 19 Oct 2006 12:31 David Marcus wrote: > georgie wrote: > > Virgil wrote: > > > In article <1160675643.344464.88130(a)e3g2000cwe.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > With the diagonal proof you cannot show anything for infinite sets. > > > > > > Maybe "Mueckenh" can't but may others can. > > > > Only a very very small group of self-proclaimed experts better known > > as the mathematics community think they can. But they do so > > with circular arguments as this thread shows. > > > > The only explanation to the OP from the math community so far: > > > > #2 is not self-referential because #1 says ANY. > > #1 is correct in saying ANY because #2 holds. > > The OP said that a definition was invalid because it was self- > referential. However, there is no rule against self-reference in modern > mathematics, so the OP's objection is not valid. So the set of sets containing themselves is ok by you. > Almost a century ago, Russell and Whitehead attempted to develop such > rules as a way of avoiding the paradoxes, but their approach was too > cumbersome. So, ZFC avoids the paradoxes in a different way. The > diagonal argument follows the rules of ZFC. If you want more details on > what the rules are, there are quite a few good books on the subject. This is a bunch of BS. There are no references to ZFC in Cantor's proof.
From: mueckenh on 19 Oct 2006 12:32
MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > MoeBlee wrote: > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > You haven't yet noticed it? Each digit of the infinitely many digits of > > > > > the diagonal number has the same weight or importance for the proof. In > > > > > mathematics, the weight of the digits of reals is 10^(-n). Infinite > > > > > sequences of digits with equal weight are undefined and devoid of > > > > > meaning. > > > > > > > > The proof doesn't contradict the fact that the members of the sequence > > > > are divided by greater and greater powers of ten. That fact is > > > > mentioned in the previous proof showing the correspondence between the > > > > sequences and real numbers. We prove that every sequence corresponds to > > > > a real number where the real number is the limit of the sum of the > > > > sequence made by taking greater and greater powers of ten in the > > > > denominators, and that every real number corresponds to such a > > > > sequence. THEN we proceed to the diagonal argument. > > > > > > > P.S. Again, if you disagree with the proof, then please just say what > > > axiom or rule of inference you reject. In the meantime, again, there is > > > no rational basis whatsoever for disputing that the argument does > > > indicate a proof from the axioms per the rules of inference. > > > > Neglecting the powers 10^(-n) converts an infinite sequence which can > > possibly yield a meaningful result into an impossible sequence, which > > cannot be treated at all. But I believe your intuition will hinder you > > accept that. > > You just went right past what I wrote. You just completely ignored what > I wrote, which was in direct response to you. We do NOT neglect the > exponentiation. You do. Otherwise there was no reason to explicitly exclude the case 1 = 0.999. In real numbers we have this equation due to exponentiation. With Cantor's diagonal we have not. Regards, WM |