Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 26 Jul 2005 11:35 Peter Webb said: > >> >>> 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. > >> > >> > > I never claimed you would be cast from the ranks of mathematics, but that > > you > > will see the errors that you are currently ignoring, and that the rot of > > Cantorian cardinality will be removed from mainstream thought and replaced > > with > > ideas that don't lead to absurdity like Banach-Tarski. I do see the > > ramifications of this nonsense in many areas. Until you can demonstrate > > that > > the theory is really correct, > > > Correct? What does that mean? Really, I have no idea what the word means in > this context. Inconsistent perhaps? If so, how? In line with reality, or at least other areas of math. Corroborated. Justified. Externally consistent. > > > >I am well within my rights to disagree with your > > axioms and conclusions, > > > How do you disagree with an axiom? You question its basic justification, examine it inductively, compare its results with those of other axiom systems. When you discover inconsistencies, you analyze your onstruction to see which axioms are involved, and where the inconsistency came from. > > How about Group Theory, for example? Do you disagree that there is element e > such that g*e=g for every g? What does it mean that you disagree with an > axiom? Is it that you don't think that there is a model which satisfies the > axiom? If so, are you contending that N doesn't model PA? Could you explain > exactly which axiom(s) you "disagree with", what "disagree with" means, and > why you "disagree with" them? Various axioms have their various issues. The most pertinent to this discussion right now, it seems, is Peano's 5th. I don't disagree with the axiom or with the concept of inductive/recursive proof, but in order to eb careful that what we are doing is correct, we need to keep in mind the original justifications for axioms when applying them. If you are applying a method such as inductive proof, with an inherent infinite loop, you cannot maintain finiteness through an infinity of iterations, each involving finite increase in value. Other axioms only apply to a particular situation, and that needs to be recognized, and minimized, as much as possible. For instance, if you look at Hilbert's axioms of incidence, 1.7 states that if two planes intersct at one point they intersect at at least one other point. WHile this is true in three dimensions, it is not generally true. In four or more dimensions, two planes can intersect at exactly one point. There is a more general statement that can be made about intersections in spaces of any number of dimensions, which subsumes this axiom. Similarly, some of the other axioms can be combined into one, when generalized for objects of different dimensionalities. > > We can then move on to conclusions, where I don't understand what "disagree > with" them means either. It means they disagree with reality as I understand it. In my world, math IS about reality, and if it's mathematically true, then it is probably manifested somewhere in reality. > > > > and if that right is challenged, I will continue to > > defend it and challenge your theory. It takes two to tango, and if you end > > up > > with people vowing vengeance, well hell, you probably deserve it. Then > > again, > > maybe JSH is mentally unstable, but then so was Cantor, and so was Godel. > > -- > > You might think that infinite ordinals and cardinals are "absurd" and > "nonsense". You are in good company historically. Lots of people thought > that zero was absurd and nonsense, as were negative numbers (show me -3 > cows), imaginary numbers, irrational numbers, positional notation ... of > course, 100 years after each of these absurd and nonsensical concepts were > introduced, it was only the cranks that continued to rail against them. You entirely miss my point. I am not against the concept of infinity, or the concept of different infinities. Quite the contrary. I perceive an entire spectrum of infinities between your alephs, and a way to order them precisely, rather than just tossing them into one of a small number of bags. > > Its like somebody saying 100 years after Copernicus that the geocentric > model leads to absurdities and obvious nonsense like the earth is spherical, > when it is obviously flat. This would be likely to lead to the same type of > contempt from astronomers as you are getting from the mathematicians here. Your response is like someone listening to Copernicus and thinking he is saying the world is a disk. You don't seem to get my position at all. > > > > > > -- Smiles, Tony
From: Daryl McCullough on 26 Jul 2005 11:16 Tony Orlow writes: >Barb, you're not saying anything new. I have heard it all before. I am not >drawing my conclusions in any such confused way Uh, yes you are. Why don't you write down what you consider to be valid axioms for working with infinite sets, and then try to write a formal proof of your claim If a set S of strings is infinite, then S contains some infinite strings. A proof consists of a sequence of statements such that every statement is *either* an axiom, or follows from previous statements by logical deduction. -- Daryl McCullough Ithaca, NY
From: Tony Orlow on 26 Jul 2005 11:48 Dik T. Winter said: > In article <MPG.1d48369d8fca2174989f42(a)newsstand.cit.cornell.edu> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > Dik T. Winter said: > > > In article <MPG.1d4722e516a9e4df989f2b(a)newsstand.cit.cornell.edu> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > ... > > > > The only reason to reject this bijection is > > > > if one clings to the idea that all natural numbers are finite, which is > > > > impossible. > > > > > > 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? -- Smiles, Tony
From: David Kastrup on 26 Jul 2005 11:50 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > Various axioms have their various issues. The most pertinent to this > discussion right now, it seems, is Peano's 5th. I don't disagree > with the axiom or with the concept of inductive/recursive proof, There is no such thing as "recursive proof" in this context. > but in order to eb careful that what we are doing is correct, we > need to keep in mind the original justifications for axioms when > applying them. Wrong. An axiom needs to stand on its, absolutely. If it requires additional considerations, it was ill-chosen. Fortunately, this does not appear to be the case with the 5th Peano axiom. > If you are applying a method such as inductive proof, with an > inherent infinite loop, you cannot maintain finiteness through an > infinity of iterations, each involving finite increase in value. Completely irrelevant chitchat to the 5th Peano axiom. It is not bothered about "increase" in value, it is not bothered about "maintaining finiteness", it is not bothered about "iterations" or an "infinity" of them. It works without you having to keep an eye on all that folderol. That's what makes it a good choice. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Daryl McCullough on 26 Jul 2005 11:38
Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >It seems like axioms are given a status that makes >them unquestionable and almost incomprehensible, beyond the application of >them. They are not incomprehensible to the people who work with them. There purpose is to *clarify* what is going on, and they serve that purporse well. Axioms allow for precise communication between mathematical workers, and they allow for objective criteria for when a proof is valid or not. In contrast, what you consider to be a proof seems to be purely subjective. >For instance, everyone's dismissal of the infinity inherent in the >recursive nature of inductive proof is a sign that the axiom is not really >understood, but accepted without question. Nothing in mathematics is excepted without question. Not by mathematicians, anyway. Yes, it is certainly the case that *if* you can prove by induction "forall x, Phi(x)", *then* you can write a corresponding recursive function that given a number n, produces a proof of Phi(n). Nobody disputes that. What people are disputing is your bizarre belief that proving "forall x, Phi(x)" by induction means that you have proved Phi(0), Phi(1), Phi(2), ... It means that you *can* prove all those infinitely many statements, not that you *have*. >For me, mathematics without meaning is unsatisfying, and >symbolic manipulation without understanding is boring. That's true for all mathematicians. The difference with you is that you don't want to do all the work necessary to understand real mathematics. -- Daryl McCullough Ithaca, NY |