Cantor Confusion
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From: Dik T. Winter on 6 Oct 2006 19:33 In article <1160145108.340148.16790(a)k70g2000cwa.googlegroups.com> "Albrecht" <albstorz(a)gmx.de> writes: .... > That's really untrue. I had read several books and papers of academics > (who do not post in this or the german math newsgroups) in which they > formulate (very cautious) criticism about ZF, axiomatic set theory or > especially the axiom of infinity. Mathematicians? But pray, name a single book where that is done. > I understand, why they are cautious. > They fear the defamation they must be aware of if they would be very > concrete in criticism. I have experienced this defamation (not for me > but other persons). You completely misunderstand. It is when people announce that the axiom of infinity is self-contradictionary, without giving proper proof, that they are severely criticised. If you look at all the proofs given in this newsgroup upto now, you will find that in all cases a conclusion is drawn that depends on the falsity of that axiom. That is not the way to prove a contradiction. Also when somebody writes a bold statement that the axiom of infinity leads to nonsense, that is just opinion, and nothing more than that. Everybody is free to develop mathematics without the axiom of infinity, nobody will criticise somebody for doing that (unless it is done using belittling statements about that axiom). But until such a form of mathematics has been reasonable properly defined, you will not find many followers. So, if you come here, and simply state: "the axiom of infinity is false, because no infinite sets do exist", that is just a bald statement of opinion. And if you expect that every mathematician will immediately see that you are justified, you are deluded. Mathematicians will not take away one of the founding blocks of much of mathematics when there is no adequate replacement. Or perhaps even not when there is adequate replacement. Because, within mathematics, it is just a question of opinion. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 6 Oct 2006 20:40 mueckenh(a)rz.fh-augsburg.de wrote: > Hi, Dik, > > I would like to publish our result to the mathematicians of this group > in order to show what they really are believing if they believe in set > theory. > > > There is an infinite sequence S of units, denoted by S = III... > > This sequence is covered up to any position n (included) by the finite > sequences > I > II > III > ... What do you mean by "cover"? > But it is impossible to cover every position of S. > > So: S is covered up to every position, but it is not possible to cover > every position. > > A further discussion is not useful, but the statement should be known. -- David Marcus
From: William Hughes on 6 Oct 2006 22:34 Albrecht wrote: > William Hughes schrieb: > > > Albrecht wrote: > > > William Hughes schrieb: > > > > > > > Albrecht wrote: > > > > > > > > <...> > > > > > > > > > I don't controvert the axiomatic methode anymore. But I claim that it > > > > > isn't the only and the important one in math. In teaching and in the > > > > > mind of the people the axiomatic method appears to be the only right > > > > > way to do math. That's not correct. > > > > > The nondenumerable infinity of the reals is not the only one truth. > > > > > Nobody is wrong who claims only one kind of infinity, the one we only > > > > > can know: the endless infinity. > > > > > > > > > > > > > > > > > The problem is not that someone who believes > > > > in your intuitive "endless infinity" (intuitive because it cannot > > > > be put on a mathematical footing) is wrong. > > > > > > Oh yes, it is the problem. I came to these subject by reading a bunch > > > of popular books about math. When I read the diagonal argument the > > > third or fourth time I started to wonder. The textes were of differnt > > > quality but all of them had a special sort of feeling. And all stated, > > > that this proof is so elementary, easy and absolute right that nobody > > > had anything to reflect or critizise about it. > > > But I found in shortest time a lot of questions about the issue. > > > Later I read professional works about set theorie and I found a similar > > > feeling in the textes about the diagonal argument. And then I started > > > to learn about the role of ZF in the teaching on universities and I had > > > a lot of disputes about the matter in newsgroups. > > > > Don't you find it interesting that of all the places you looked, > > the only place where anyone disagreed with the diagonal > > argument was the newsgroups? > > That's really untrue. I had read several books and papers of academics > (who do not post in this or the german math newsgroups) in which they > formulate (very cautious) criticism about ZF, axiomatic set theory or > especially the axiom of infinity. Did any of them disagree with the diagonal argument? This is the problem. Not someone saying, "your arguments lead to conclusions I find counterintuitive, so the axioms you chose need to be modified", but someone saying "your arguments lead to conclusions I find counterintuitive, so there must be something wrong with your arguments" or "your arguments lead to conclusions I find counterintuitive, so your axioms must be inconsistent". - William Hughes
From: mueckenh on 7 Oct 2006 04:59 Tonico schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Tonico schrieb: > >> > ************************************************************************* > > > Whoever talked bout "interesting" or "important"? I just pointed out > > > that the vase-balls question was an ill-posed one since it mixed maths > > > and real life in an improper way, imfho. > > > > You should know that Cantor designed his theory in order to describe > > real life, physics, chemistry, economy, ... > > ******************************************************************** > I really didn't know that, and I very heavily suspect this isn't true, > though I can't be 100% sure. Cantor to Hilbert, 20.9.1912, concerning a publication (I am too lazy to translate it, but you may recognize the words derived from latin): "Der dritte Theil bringt die Anwendungen der Mengenlehre auf die Naturwissenschaften: Physik, Chemie, Mineralogie, Botanik, Zoologie, Anthropologie, Biologie, Physiologie, Medizin etc. Ist also das, was die Engländer "Natural philosophy" nennen. Dazu kommen aber auch Anwendungen auf die sogenannten "Geisteswissenschaften", die meines Erachtens als Naturwissenschaften aufzufassen sind, denn auch der "Geist" gehört mit zur Natur." (saying: the application of set theory to various branches of science) > What does foundations of maths in general, and trying to describe stuff > about infinity from a mathematical point of view have in common with > describing "real life", whatever that is?? What does math in general > HAS TO HAVE in common with real life? In my opinion, nothing. Math. started with applications to real life like the shepherds counting their cattle and the Harpedonapts in Egypt. Therefore its results could be tested experimentally. The change to a game which uses arbitrary axioms and fixed laws of logic (why not arbitrary logic) leading to meaningless results started later. It fully developed based on Cantor's set theory. This is an irony of history because Cantor was the last to find such games interesting. He disliked axioms and wrote: Whenever I see axiom put ahead a paper I know that the contents is wrong. (He called them Hypothesen): to Veronese, 17.11.1890 "Hypothesen" welche gegen diese Grundwahrheiten verstoßen, sind ebenso falsch und widersprechend, wie etwa der Satz 2 + 2 = 5 oder ein viereckiger Kreis. Es genügt für mich, derartige Hypothesen an die Spitze irgend einer Untersuchung gestellt zu sehen, um von vorn herein zu wissen, daß diese Untersuchung falsch sein muss. >Now if > somebody ever finds some application to "real life" of some part of > maths, as the cas has been MANY times and in some rather important > occasions, good. This is the reason I can't think of Cantor developing > his stuff because he wanted to describe whatever from "real life"... Cantor to Mittag-Leffler, 22.9.1884 "Mit diesen Ideen einer genaueren Ergründung des Wesens alles Organischen beschäftige ich mich schon seit 14 Jahren, sie bilden die eigentliche Veranlassung, weshalb ich das mühsame und wenig Dank verheissende Geschäft der Untersuchung von Punctmengen unternommen und in diesem Zeitraum keinen Augenblick aus den Augen verloren habe." (saying: Cantor had developed his set theory of point sets in order to discover the essence of all organic matter.) > I wonder what do you call "actual infinity" to, and I'm afraid that > this attitude of yours begins to ressemble a lot what sometimes little > kids do: they cry, kick and make a fuss out of something just because > life, or some part of it, happens not to be like they've decided it > SHOULD be. Cantor distinguished furiously between potential and actual infinity. Cantor, collected works, p. 374: "Trotz wesentlicher Verschiedenheit der Begriffe des potentialen und aktualen Unendlichen, indem ersteres eine veränderliche endliche, über alle Grenzen hinaus wachsende Größe, letztere ein in sich festes, konstantes, jedoch jenseits aller endlichen Größen liegendes Quantum bedeutet, tritt doch leider nur zu oft der Fall ein, daß das eine mit dem andern verwechselt wird." Who does not understand this distinction cannot understand Cantor and set theory. Deplorably most of modern mathematics does not make this distinction. > Omega is NOT a set at all: it is the ordinal of the natural > numbers set N with the usual ordering. Period. omega is a fixed quantum! omega is also a cardinal and as every cardinal is a set omega is a set. (omega as a cardinal is used in modern set theory but already Cantor in his later years did use it this way. See G. KOWALEWSKI: Bestand und Wandel, Oldenbourg, München (1950) 202.) > ??!!!?!?! Did cantor EVER talk of a "whole number" "Whole number" is the direct translation of the German word "ganze Zahl". It is derived from the fact that numbers ae sets and no set has broken elements >(I supose you must > mean an integer, positive one...and this already points towards the > heavy doubt I had: you are not a mathematician, which of course doesn't > automatically rule you out as a debater in these matters, but it does > make some of your arguments highly suspicious and even dismissable from > a mathematical point of view. Why? Because you simply doesn't know > enough maths, apparently)?? I have been teaching analysis and algebra for 15 years now. > No, he didn't...and again: omega is NOT any > number at all, but an ordinal. No need for you to waste your better > efforts in vain: omega has NEVER been a number. Read Cantor's works. You have to learn a lot. One of many examples: "erste ganze Zahl der ersten oder zweiten Zahlenklasse" means first whole number of the first or second number class (the second number class is the set of countable transfinite numbers). > This is maths: neitehr infinity nor differentiable functions, matrices > or infinite (!!) abelian groups require or demand belief from someone. > This is a wonderful, enjoyable and intellectual game called maths. > Wanna play? Yes, after you will have absorbed its rules. > About the "And we know that never all day of his life are described > because always 50,000 remain to be written."[sic]: it'd be interesting > if you could be more precise and point out who "we" in your sentence > is. We is all those who do not accept or do not know the actual (i.e. completed) infinity. So you may belong to "we" because of lacking
From: Han.deBruijn on 7 Oct 2006 08:19
Tonico wrotef: > Maths, just like the other sciences, isn't grounded on dogma, and > people forwarding REASONABLE, well-based objections, opinions or ideas > on whatever are always welcome. _There_ is your problem! Ask Virgil, and the other mathematicians here: MATHEMATICS IS _NOT_ A SCIENCE And therefore there is NO guarantee that it "isn't grounded on dogma". Han de Bruijn |