Obections to Cantor's Theory (Wikipedia article)
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From: Robert Kolker on 25 Jul 2005 15:15 Han de Bruijn wrote: > > > That's true. But mathematical axioms start to behave _as if_ they were > physical laws, as soon as they become being _applied_ to i.e. physics. The mathematical axioms remain so. It is the mapping of an abstract mathematical system into a system of physical measurements and operations that produces empirical content. The mathematics, per se, has no empirical content whatsoever. Here is the bottom line. Mathematics has no empirical content whatsoever. None. Zip. Nada. Zero. Bupkis. K'ducchis. Got it? > >> The axioms of set theory (ZFC) make a really, really interesting game. > > > Set theory doesn't deserve such a predominant place in mathematics. > After the discovery of Russell's paradox et all, everybody should have > become most reluctant. Without set theory there would be no rigorous theory of real or complex variables. Without set theory there would be no point set topology, a general theory of spaces based on the abstract notion of nearness. Set theory has made modern mathematics what it is today. Bob Kolker
From: David Kastrup on 25 Jul 2005 15:15 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > What is sad is that you say you have heard the same arguments > repeatedly, and still don't seem to understand them. If there is a > constant finite difference of one between any two consecutive > natural numbers, and there are an infinite number of them, then > indeed the overall range of values becomes infinite. Sure does. But the difference between any given two fixed numbers remains finite, even though there is no finite overall range. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Robert Kolker on 25 Jul 2005 15:17 Han de Bruijn wrote: > Affirmative. After I discovered (in 1973) that Set Theory was just a > burden on analysis (Lie groups), I've managed to do analysis without > set theory for another 30 years. Your ignorance and incompetence has no empirical or philosophical import. And I still cannot imagine what its > additional value is, Your problem. Why don't you learn something? Bob Kolker
From: Virgil on 25 Jul 2005 15:20 In article <MPG.1d4eb8002ff83793989f69(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > So, I can play your stupid game, and say that alpha is > > > the largest finite, but alpha+1=alpha. Tada! I am as senseless as you! > > > Isn't > > > that great? Let me know when you want to get off the "largest finite" > > > kick. I > > > won't be responding to it any more. > > > > It is not me who keeps requiring a largest finite, but TO, by his > > delusional insistence that there must be non-finite naturals within the > > Peano system. > > > While it may seem that counting ala Peano can never produce any infinite > numbers, the insistence on the set being infinite requires the existence of > infinite values in the set. Perhaps you should call your set "boundless" or > "bigfinite" or something, but it's not "infinite" if you only count a finite > number of times and only have finite values. TO must be having a different definition of an infinite set than the rest of us. The mapping n -> n+1 on N clearly injects N into a proper subset of itself without ever mapping any finite member of N onto any infinite member of N. So the finite members of N are enough. > You know, the Peano axioms can > easily be inverted to produce infinite whole numbers that count down, with > exact symmetry compared to the finite end of the number circle. If one uses > such a set of axioms, does that mean that finite whole numbers cannot exist, > because that set of axioms doesn't seem to allow us to count down that far? > This insistence on determining the dividing point between finite and infinite > is clearly a waste of time, and your repetitions of this non-point don't make > it any more important. You can't count through the divide in any finite > number > of steps. So what? Since there is no step from finite naturals to infinite naturals, and all stepping is done one step at a time, from some n to n+1, there is no getting from finite naturals to anything else by one-stepping. That doesn't mean that one side exists and the other > doesn't. Infinite set sizes ARE infinite whole numbers. I can't > understand why this isn't clear to everyone. Because everyone else reserves the phrase "whole numbers" for "finite" numbers. Whatever TO wants to call them, it should not be some word or phrase that already has another meaning. The common term is 'cardinals'.
From: Tony Orlow on 25 Jul 2005 15:27
Chris Menzel said: > On Wed, 20 Jul 2005 16:16:44 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > > Chris Menzel said: > >> On Wed, 20 Jul 2005 09:08:02 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > >> > ... > >> >> To say that a list of the reals is aleph_1 in length assumes the > >> >> continuum hypothesis; is that what you intend? (And if so, why?) > >> >> Moreover, granted, aleph_1 is omega_1 in pure set theory, but one should > >> >> use ordinal numbers rather than cardinals when talking about such things > >> >> as the length of a list, as there are many lists of different ordinal > >> >> lengths that are the same size. > >> > >> > I don't think it assumes any such thing. > >> > >> If the assumption you are referring to is that there is a list of > >> reals of length omega_1 (= aleph_1 in pure set theory), then all you > >> are doing by denying that this assumption requires the continuum > >> hypothesis is showing your abject ignorance of the subject. > > The continuum hypothesis states, if I'm not mistaken, that there are > > no infinities between aleph_0 and aleph_1. > > Sorry, dude, no, you are quite mistaken. aleph_1 is *by definition* the > next cardinal after aleph_0. The continuum hypothesis is that aleph_1 > is the size of the set of real numbers. Interesting. MathWorld says: The proposal originally made by Georg Cantor that there is no infinite set with a cardinal number between that of the "small" infinite set of integers aleph_0 and the "large" infinite set of real numbers c (the "continuum"). Symbolically, the continuum hypothesis is that aleph_1=c. Apparently it says both, dude. > > > So, what does this have to do with assuming that a complete list of > > the reals has aleph_1 members? > > Get it now? > > > The continuum hypothesis is mularkey as far as I can see. > > Yes, and we've seen how far you can see. No, you really haven't looked. > > > There is a whole spectrum of infinities between those alephs. > > Really, you are making such a fool of yourself by talking through your > hat instead of just learning some simple, basic set theory. Do yourself > a favor. I don't need a favor. > > >> > In fact, it is my position that there is an infinite spectrum of > >> > infinities between that of the naturals and that of the reals. > > Guess what? That means you reject the continuum hypothesis! Isn't that > exciting? Don't you just want to go out and learn all about it rather > than just spouting vague, uninformed, and often silly nonsense? Why do I want to spend my time studying something that I disagree with? Why don't you go study Mormonism? Wouldn't that be exciting? > > >> Well, then your assumption that there is a list of reals of length > >> omega_1 is curious indeed. > > > > If you are curious, why don't you try asking a question, instead of > > making declarations? What, exactly, do you find curious? > > Hope that's been explained now that I've explained a bit of the > mathematics you didn't understand to you. Not really, and that doesn't answer the question. > > >> > I have not immersed myself deeply in set theory, no. But, I can see > >> > clearly that some of the conclusions are wrong, and in arguing > >> > this, I have stumbled upon a few of the obvious flaws in the logic. > >> > >> David Kastrup has noted the problems with this reasoning. Many > >> thinkers through the years have thought they clearly knew one thing > >> or another, only to be shown wrong by deeper advances in the relevant > >> discipline. As I noted earlier, actually learning the material you > >> criticize as an intelligent but uneducated amateur will help you to > >> see where your conceptual errors lie. > > > > Well, it might help if you identified some conceptual errors on my > > part, instead of sending me off to read books. Obviously, you haven't > > indentified any major flaws in my logic, or you would be pointing them > > out specifically, wouldn't you? > > Oh but I have -- just pointed out one conceptual error above re CH. Not really. besides, that has nothing to do with the arguments I put forth. Try again. > Another recently identified is your belief that if there is no upper > bound on the length of the strings in a set, then the set must contain a > string of infinite length. That was not my statement, but someone else's paraphrase of their understanding of my position. Pay attention. I have repeatedly stated that N is infinite IFF N contains infinite n. > Now, unfortunately, you are confusing your > inability to see your errors when they are clearly pointed out to you > with the idea that no one has found any. That is why I am, quite > sincerely, pointing you to texts containing real mathematics, in the > hope that some sincere study will enable you to see the errors you are > making. If I read as well as you do, I won't learn anything anyway. Try responding to MY statements, if you want to claim that you are doing so. > > >> > Hopefully I will find time soon to decipher the specific axioms of > >> > ZF and see whether the roots of the problem lie there, or with > >> > subsequent assumptions. > >> > >> Or with your own thinking -- an important part of being a scholar is > >> always to bear in mind that one *might* be wrong. > > > > Yes, I have been corrected in the past, and that's okay. I haven't > > been corrected correctly on this, and refuse to concede simply on the > > basis of peer pressure. The flaws I see are glaring in my eyes, and > > the refutations of them all have holes I can stick my hand through. > > Well, look. There are a lot of people who have a lot of years of > advanced study behind them who are claiming that you are making some > mistakes, and who, moreover, are going to some lengths to try to explain > them to you. Actually, not. Generally my claims are refuted by being mischaracterized and made to look stupid. That is not a valid mathematical refutation, or an honest communication. > Moreover, it is more than evident, as you yourself admit, > that you in fact are quite ignorant of the fields in which the things > you criticize are studied; you don't understand some of their most > elementary concepts. Now, just probabilistically, what are the odds > that *all* of those folks (not to mention several generations of great > mathematicians) are rather hopelessly confused about the foundations of > their disciplines, and that you -- uneducated in those disciplines -- > are right? Now, I grant you, it is not logically impossible -- perhaps > God is playing a trick on everyone but you. But ponder the odds. Oh, I have. What are the odds of being alive to begin with. What are the odds of miracles? Perhaps they happen every day, and perhaps every sun has some life around it. What were the chances of every scholar in Europe being wrong about the Sun revolving around the Earth? Truth is not decided democratically. > > >> > Now, if I came up with a theory that "proved" that 1=2, would you > >> > believe it, at first glance? > >> > >> Of course not at first glance. But I would not dismiss it out of > >> hand, either, given what we know from Gýdel about the unprovability > >> of consistency. I would be greatly skeptical, of course, if the > >> proof came from someone who obviously doesn't know much about > >> arithmetic or set theory. > > > > Basically what Godel proved was that no system can prove itself > > correct from within, so that proving any system correct becomes an > > infinite regress into the systems from which it is derived or the > > context in which it exists. Godel did some good work there. It's too > > bad he got involved with the continuum hypothesis and ended up with > > similar mental issues as Cantor. > > Ah, yes, working on CH causes mental illness. Perhaps. It's certainly not a fruitful endeavor, obviously. > > Metacomment: What could *possibly* drive someone who probably *knows* > he doesn't understand Gýdel's theorem, and probably *knows* he doesn't > have any idea what Gýdel proved about CH, to make pronouncements like > this on a public forum? Conviction? > > Chris Menzel > > -- Smiles, Tony |