Obections to Cantor's Theory (Wikipedia article)
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From: Dik T. Winter on 27 Jul 2005 10:05 In article <MPG.1d5025572ee38731989f8e(a)newsstand.cit.cornell.edu> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > Dik T. Winter said: .... > > > > Back on your horse again. Tell me about the binary numbers > > > > (extended to the left with 0's) where the leftmost 1 is in a > > > > finite position. Are all those numbers finite? Are there > > > > only finitely many of them? > > > > > > > yes and yes > > > > I think you should apply for the reward for solving Collatz' problem. > > > I just looked that up. Does this have anything at all to do with that > problem? Your assertion at least makes Collatz' problem decidable. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Robert Kolker on 27 Jul 2005 10:52 Han de Bruijn wrote: > MoeBlee wrote: > >> Meanwhile, I'm still fascinated by your inconsistent theory, posted at >> your web site, which is: >> >> Z, without axiom of infinity, bu with your axiom: Ax x = {x}. >> >> Would you say what we are to gain from this inconsistent theory? > > > No. But first we repeat the mantra: > > A little bit of Physics would be NO Idleness in Mathematics Many parts of mathematics are inspired or motivated physical phenomena. For example calculus (the theory of real variables) was invented by Newton specifically to quantify motion of material bodies. So physics is a potent motivator for some branches of mathematics and has been so for 400 years. However, physical applicability is not a requirement for creating a mathematical theory. Analytic number theory is not motivated by physical considerations althought it turns out it has some physical applications. In any case, physical applicablility is not the sina qua non for the validity or soundness of a mathematical theory. Internal consistency is. > > It is physically correct that every member of a set is at the same time > a _part_ of the set, Sets are not physical. They are abstract. Search the world over and you will not find a set. In the real world trees exist but forests do not. Forests exist in our heads. Bob Kolker
From: Robert Kolker on 27 Jul 2005 10:55 Han de Bruijn wrote: > > So to speak. The more accurate formulation is that a set in mathematics > should be an idealization from something that is called a set in common > speech, say physics in a broad sense. That explains the psychological basis for the abstract concept set. But it does not change the fact that sets are not physical entities. Like numbers they are abstract. Search the world over and you will not find the integer one. It has no physical existence. Bob Kolker Take that and you will find that Take the proposition that your grandma has balls it it will be correct that she is your grandfather. Assuming something is correct does not make it correct. Mathematics has its uses in understanding the physical world, but mathematical entities do not exist outside of our skulls. Bob Kolker
From: Robert Kolker on 27 Jul 2005 10:57 Han de Bruijn wrote: > > > You're right. But you don't even try to understand what I mean. Do you? If what you mean or intend is balderdash and poppycock why should he try to understand it? Your problem is that your thinking is sloppy, inaccurate and incoherent. Bob Kolker
From: Tony Orlow on 27 Jul 2005 11:16
Robert Low said: > Tony Orlow (aeo6) wrote: > > > Barb, you're not saying anything new. I have heard it all before. I am not > > drawing my conclusions in any such confused way, > > Differently confused, then. It's hard to tell. > > > If I > > thought the way you are suggesting, then wouldn't I also be claiming that the > > reals in [0,1) must have infinite values too? > > So, consider the rationals in [0,1). Each of them is (by definition) > of the form p/q, where p is a natural number, q is a natural number > other than 0, and p and q have no common factors. > > Are there rationals in [0,1) where p and/or q have to be infinite? > Interesting question. In order to have an infinite number of them, yes, you would need either an infinite number of numerators or an infinite number of denominators, both of which are whole numbers. So, you would require infinite values in either numerator or denominator, in order to achieve an infinite set. Now, of course, you would counter that, in order for the value to be in the interval [0,1) and have either an infinite numerator or denominator, it must have both, which is true, and that a ratio between two infinite values cannot be between 0 and 1, but I really have no problem declaring that N/(N+1) is in that range, even though N is infinite. I mean, it is trivial to prove inductively that for all n in N, 0 <= n/(n+1) < 1, and for me that includes infinite n. -- Smiles, Tony |