Obections to Cantor's Theory (Wikipedia article)
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From: MoeBlee on 25 Jul 2005 15:58 >From a post by Tony Orlow (aeo6): > What is sad is that there are a number of informed people here who are willing to help you, but your stubborn unwillingness to even inform yourself about the subject prevents you from benefiting from the knowledge of these people. What is sad is that people who have a tremendous amount of knowledge and wisdom to share with you, if you'd only let them, spend so much energy, fruitlessly trying to raise you from your self-imposed ignorance. If you would just read a book on the subject, then you'd appreciate your fortune at having someone like Chris Menzel to answer a question now and then and even to disagree with you when there is some rational basis for disagreement. MoeBlee
From: Virgil on 25 Jul 2005 16:02 In article <MPG.1d4ed3b45f5665d2989f6d(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > The proof regarding strings is so simple, you really can't complain. N=S^L > precisely describes the relationship between the number of strings N, the > symbol set size S, and the length of strings L. I am not assuming anything > except for this fact, and for the fact that S^L can only be infinite with > either infinite S or L. That would indicate that one character can only produce one number, which is one error. But also, if either the S or L may be unboundedly large, as they both are, so may S^L be unboundedly large. Every non-empty set of naturals has a smallest member. TO claims that the set of string lengths has an upper bound, so the set of naturals exceeding that bound must be non-empty. Can TO prove that there is a natural number greater that any possbile string length? If not, TO fails again. > Do you argue with either of these facts? If not, then > can you see any way to have your infinite set of strings without strings of > infinite length, besides having an infinite alphabet? By having a set of strings of unbounded lengths. > I don't see where you pointed out any specific flaw, except to rant about > your > largest finite number again. If no string in the set can be infinitely long, > then how, without infinite L, can S^L EVER be infinite? It can't. Willful ignorance, such as TO exhibits, does not cinvince anyone. > > You say it is obvious nonsense, and that I have quantifier dyslexia, but that > is bullshit and you know it. If the naturals satisfy the Peano properties, then any non-empty subset of the naturals will have a smallest (first) member. If the set of infinite naturals is not empty, it must have a smallest member. Subtract one and that must be the largest finite natural. Unless TO can produce evidence either of a largest finite natural or a smallest infinite natural, he argues in violation of the Peano properties. So that whatever TO's "numbers" are, they are not the Peano naturals.
From: Robert Kolker on 25 Jul 2005 16:05 Han de Bruijn wrote: > > The mathematics or the physicists, whoever does it. The empirical content of any mathematically formulated theory of physics lies in the mapping of the mathematatical theory into the space of measurables and physical operations. Mathematics, as such, has no empirical content whatsoever. Bob Kolker
From: Virgil on 25 Jul 2005 16:11 In article <MPG.1d4ed8dc13bc1ee989f6f(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Now, what do you not understand about N=S^L. The number of binary > strings of length L is 2^L, so you cannot have an infinite set of > binary strings unless L is allowed to be infinite, in which case the > binary value is infinite. There can only be a finite number of finite > strings with a finite alphabet. What is the largest number of different characters that one can use in any one position in a string? What is the largest number of characters that one can string together into a string? TO assumes some finite bound on both the number of different characters in the character set and a limit on the number of characters in a character string, neither of which bounds exist. herefore there is no finite bound on the number of strings, nor on the number of naturals which they can represent.
From: Virgil on 25 Jul 2005 16:13
In article <MPG.1d4eea3fa2b7da2b989f70(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Robert Low said: > > Now, until you give a definition of 'bigger', and explain why my > > first paragraph above is rendered irrelevant, we'll all be very > > grateful. Or at the very least, very surprised. > > > Your definition of "bigger" works fine for finite sets. When it comes > to infinite sets, however, it gives a lot of false positives in the > form of equivalences. Sizes of such sets need to be built upon the > properties of their members, so the method differs a little for whole > numbers vs strings, although all of these sets are related in ways. > An infinite set of numbers, for instance, can be compared to another > infinite set of numbers by looking at the functions which describe > them; if one is defined by a function that is always larger than the > function describing the other, then it is a smaller set. Sets of > strings are measured using N=S^L, so we could say, for instance, that > all decimal numbers which include only 1's and 0's would be > 2^(log10(N)) in size. A more precise notion of size for infinite sets > requires a slightly more complex method than bijection, as far as I > can see. And the silliness of his argument shows that TO cannot see very far at all. |