Obections to Cantor's Theory (Wikipedia article)
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From: david petry on 19 Jul 2005 17:14 >The truth of the matter is that the article you wrote constitutes >*original research* on your part, despite your attempt to ascribe your >views to the "anti-Cantorians", which is, as quasi pointed out, not a >well-defined group. Thus, it is not acceptable for inclusion into >Wikipedia. I'm objectively describing, as best I can, a debate which has appeared in these newsgroups. That's not against any rules I know about. It's not "original research". You are free to edit the article any way you want, and I will be free to do likewise.
From: Robert Low on 19 Jul 2005 17:16 Tony Orlow (aeo6) wrote: > Stephen J. Herschkorn said: >>To those who insist there is a smallest positive real number? > 000...000.000...001 And how many 0's are there after that decimal point? N? log_2(N)? N-1? N+1? (Whatever the hell any of those answers mean...)
From: David Kastrup on 19 Jul 2005 17:15 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > David Kastrup said: >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >> >> > David Kastrup said: >> >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >> >> >> >> > Alec McKenzie said: >> >> >> "Stephen J. Herschkorn" <sjherschko(a)netscape.net> wrote: >> >> >> >> >> >> > Can anti-Cantorians identify correctly a flaw in the proof >> >> >> > that there exists no enumeration of the subsets of the >> >> >> > natural numbers? >> >> >> >> >> >> In my view the answer to that question a definite "No, they >> >> >> can't". >> >> >> >> >> >> However, the fact that no flaw has yet been correctly >> >> >> identified does not lead to a certainty that such a flaw >> >> >> cannot exist. Yet that is just what pro-Cantorians appear to >> >> >> be asserting, with no justification that I can see. >> >> >> >> >> > Even though every subset of the natural numbers can be >> >> > represented by a binary number where the first bit denotes >> >> > membership of the first element, the second bit denotes >> >> > membership of the second element, etc? >> >> >> >> Well, what number will then represent the set of numbers dividable by >> >> three? >> > 100100...100100100100 >> >> > Of course, you will argue that this infinite value is not a >> > natural number, since all naturals are finite, but that is >> > clearly incorrect, as it is impossible to have an infinite set of >> > values all differing by a constant finite amount from their >> > neighbors, and not have an overall infinite difference between >> > some pair of them, indicating that at least one of them is >> > infinite. >> >> You have not shown such a thing, and of course it would be >> inconsistent with the Peano axioms defining the naturals. > > That is simply not true. Sulking won't help. > There is nothing in Peano's axioms that states explicitly that all > natural numbers are finite. It is an immediate consequence. > The fifth axiom, defining inductive proof, is used to prove this > theorem, but it is a misapplication of the method. An axiom is not a "misapplication". > I offered, and you saw, a deductive proof that proves that the > largest natural in a set must be at least as large as the set > size. But the set of naturals does not have a largest element. So you can prove any property you want about it, like it having webbed feet and dancing swing polka with a vengeance. Doesn't make a difference, since no such beast exists anyway. > So, which inductive proof do you believe? You cannot add 1 an > infinite number of times to your maximal element, There is no maximal element, and the Peano axioms don't define "infinite number of times" or similar processes. > without it achieving an infinite value. I provided two other proofs > that an infinite set of naturals must include infinite values, which > were dismissed, but never refuted, or any flaw pointed out. You are delusional. > I will have my web pages published before too long, so I am not > getting into a mosh pit with you again right now. Just be aware that > anti-Cantorians are sick of being called crackpots, and the day will > soon come when the crankiest Cantorians will eat their words, and > this rot will be extricated from mathematics. Oh good grief. Successor in interest to JSH, are we? >> >> Let's take the number representing the set of numbers dividable >> >> by three. Is this number dividable by three? >> > >> > Why does it have to be? >> >> It does not have to be. But if it is a natural number, it either >> is dividable by three, or it isn't. You claim that it is a natural >> number. So what is it? Is it dividable by three, or isn't it? >> >> It must be one, mustn't it? > > The number is 100100...00100100. It's certainly even, and a multiple > of 4. Is it divisible by 3? That can only be determined in a binary > system with a finite number of digits, as far as I can > tell. Oh, certainly not. Natural numbers are defined by the Peano axioms. Now let us define the set of all numbers with a well-defined remainder from division by 3: a) 0 has a well-defined remainder of 0 b) If n has a well-defined remainder of 0,1,2 respectively, S(n) has a well-defined remainder of 1,2,0 respectively. c) different numbers n with well-defined remainder have different successors S(n) with well-defined remainder d) 0 is not the successor of any natural with well-defined remainder e) if a set contains 0, and for each of its elements x contains S(x), then this set contains all numbers with well-defined remainders. Shiver me timbers, looks just like we have the Peano axioms here. So all natural numbers have a well-defined remainder from division by 3. Looks like 100100...00100100 is not a natural number. If it is, just point out which of the above laws is wrong. > Infinite whole numbers aren't always as convenient as finite ones, > but they still must exist for the set to be infinite, Says you. > and they still can be used to represent infinite subsets of the > naturals. Unfortunately not subsets in- or excluding themselves at will. > If you divide this number by 3 (11) you find it is divisible or not, > depending on whether you have an odd or even number of 100's in your > infinite string. Of course, this question is not really answerable, > so I don't have an answer for you. What do you think? What is > aleph_0 mod 3? Oh, I never claimed that aleph_0 was a member of the natural numbers, so I don't need to make claims about aleph_0 mod 3. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Chan-Ho Suh on 19 Jul 2005 17:18 In article <42DD6183.5030705(a)netscape.net>, Stephen J. Herschkorn <sjherschko(a)netscape.net> wrote: > Should a reputable encyclopedia contain an entry devoted entirely to > people who think the earth is flat? > An entry only for those who think that sun revolves the earth? > An entry devoted specifically to those who think that man never landed > on the moon? > To those who insist there is a smallest positive real number? Hmmm...I don't know if this makes the Wikipedia more or less reputable, but yes, it does contain entries for almost all that you ask. http://en.wikipedia.org/wiki/Flat_earth http://en.wikipedia.org/wiki/Geocentric_model probably more along the lines of what you're thinking is: http://en.wikipedia.org/wiki/Modern_geocentrism http://en.wikipedia.org/wiki/Apollo_moon_landing_hoax_accusations
From: Alan Morgan on 19 Jul 2005 17:19
In article <MPG.1d473242aff1e303989f30(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >That is simply not true. There is nothing in Peano's axioms that states >explicitly that all natural numbers are finite. The fifth axiom, defining >inductive proof, is used to prove this theorem, but it is a misapplication of >the method. I offered, and you saw, a deductive proof that proves that the >largest natural in a set must be at least as large as the set size. No, you didn't. You proved this for finite sets only. You claimed, without proof, that this result applied to infinite sets. Alan -- Defendit numerus |