Cantor Confusion
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From: William Hughes on 12 Nov 2006 14:28 mueckenh(a)rz.fh-augsburg.de wrote: > > "Goes on forever" is a property of the set of natural numbers, Yes. > as well > of the elements n as of initial the segments 1,2,3,...,n. Yes Let the set of "the elements n as of initial the segments 1,2,3,...,n" be K. Then K is simply the set of natural numbers. Do you thing the sets are different? >It does not > make you believe that there are infinite natural numbers n. Why does it > make you believe that there is an infinite initial segment 1,2,3... We do not need an infinite natural number ot have an infinite initial segment 1,2,3 ... As you stated "Goes on forever" is a property of the natural numbers. The set {1,2,3,...} is just the natural numbers. So this set must go on forever. - William Hughes ? > > Regards, WM
From: Virgil on 12 Nov 2006 14:29 In article <1163352469.769509.39380(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Cardinal numbers like one, tweo, ... and ordinal numbers like first, > second, ... are closely connected. So every iinitial segment of natural > numbers is counted by a natural number, namely |{1,2,3,...,n}| = n. > Therefore no initial segment of natural numbers can be counted by an > unnatural number like omega. This leads to the problem |{1,2,3,...}| = > omega and |{1,2,3,...,omega}| = omega + 1. So we have |{1,2,3,...,a}| = > a or a + 1, corresponding to the kind of a. Obviously something has > been lost. perhaps it is WM's mind? WM is conflating cardinality with ordinality. The cardinality of a set is the smallest ordinal which can be put in bijection with the set, disregarding any order relation that the set might have defined upon it. The ordinality of a well ordered set (no other kind has an ordinality) is the unique ordinal which is order-isomorphic to the given well ordered set. E.G. the naturals with their usual order have ordinality omega and with any ordering have cardinality equal to that of omega, whereas the rationals with their statndard ordering do not have any ordinality at all but have cardinalty equal to that of omega. It is customary use aleph_0 to denote the cardinality of omega. The cardinality of omega + 1 is the same as the cardinality of omega, as they may be bijected to each other by ignoring their orderings.
From: Virgil on 12 Nov 2006 14:36 In article <1163352641.523201.10860(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > > It is now to be shown how one is lead to the definition of the > > > > > > > new > > > > > > > numbers. And this definition, given by Cantor and translated by > > > > > > > myself > > > > > > > is given above. There is no point to draw but only to understand > > > > > > > Cantor's *definition* (or not). > > > > > > > > > > > > Okay, so for you there is no point to draw about your quote other > > > > > > than > > > > > > understanding Cantor's own writings. And, I'll add that in this > > > > > > particular instance, Cantor, as you translated, is not in conflict > > > > > > with > > > > > > the theorem of current set theory that omega is a limit ordinal and > > > > > > the > > > > > > first oridinal that is greater than all natural numbers. > > > > > > > > > > That is what I said. omega is a limit. In modern set theory there are > > > > > limits. > > > > > > > > Moe said that "omega is a limit ordinal". He did not say that "omega is > > > > a limit". The two statements are not the same. > > > > > > Moe said "Cantor, as you translated, is not in conflict with the > > > theorem of current set theory". Cantor said "omega can be understood as > > > a limit". > > > > Cantor said "understood". That is not the same as saying "is". > > Cantor also said: "is". Therefore it is clear that here he meant "is". > > > Regardless, Cantor's papers are very old and predate the modern > > formulations. > > That does not mean that the modern formulations are better. Nor does it mean that they are worse. But since mathematics generally retains what is found good, and dumps what is found wanting, the odds certainly favor the modern formulations as being better. > > > > > > Do you really believe the > > > > two statements are the same? Or, are you trolling? > > > > > > A bottle of beer is not a beer bottle. But both notions have to do with > > > beer and with bottles. Why is a limit ordinal called so if it is not > > > the limit of some set of ordinals? Why is the field of rationals so different from a field of oats? > > > > Because the notions of limit ordinal and of limit are related to > > intuitive ideas that are similar enough that people decided to use > > similar terminology to name them. However, you can't deduce anything > > from the name. You must use the precise, rigorous definition. Thinking > > that you can use the name is a common fallacy. > > > > The fact remains that a limit ordinal is not the limit of a set of > > ordinals. A limit ordinal is an ordinal other than zero that is not a > > successor ordinal. This definition is given by both Halmos and Kunen in > > their books on set theory. > > > > > > is rude to use a common word with a personal meaning and not point this > > > > out. If someone asks you for a "definition", etiquette and honesty > > > > requires you to say, "I'm sorry, but I do not know what you mean." > > > > > > In mathematics it is practical to give names to various particular > > > properties and objects, i.e., to define new properties. Mathematics > > > without definitions would be possible, but exceedingly clumsy. > > > > Yes, but the problem is that you fail to state what definitions you are > > using. > > > > > > The phrases "actually existing" and "cannot exist" are not defined. > > > > > Eetiquette and honesty requires you to say, "I'm sorry, but I do not > > > know what the definitions of these words are. And then you should > > > attach a list of words you know. It can't be too long. So I will look > > > whether there are words which could be used to explain "actually > > > existing" and "cannot exist". > > > > Amusing. I (and others) have repeatedly asked you what you mean. I have > > suggested that you use the terminology in Halmos's Naive Set Theory or > > in Kunen's Set Theory. If you don't like these books, then pick a > > different one. However, Cantor's papers are much too old to be used for > > a discussion today. > > No. The hidden errors can better be recognized at the roots. Then by all means let us pull WM up out of his pot and examine his roots to find the sources of all his errors.
From: Lester Zick on 12 Nov 2006 14:44 On Sat, 11 Nov 2006 15:41:39 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Lester Zick wrote: >> On Fri, 10 Nov 2006 17:33:16 -0500, David Marcus >> <DavidMarcus(a)alumdotmit.edu> wrote: >> >> >mueckenh(a)rz.fh-augsburg.de wrote: >> >> David Marcus schrieb: >> >> > > >> >> > > Exactly this is done by Cantor's definition given above: "It is allowed >> >> > > to understand the new number omega as limit to which the (natural) >> >> > > numbers n grow". This is a definition, at least for those >> >> > > mathematicians who know what "limit, number, grow" means. >> >> > >> >> > I seriously doubt that Cantor considered it to be a definition. If he >> >> > did, he wouldn't have said, "It is allowed to understand". Regardless, >> >> > it is not a definition by modern standards. >> >> >> >> Who judges what modern standards are, in your opinion? >> > >> >A silly question. You haven't even bothered to learn the current meaning >> >of the word "definition". If you would actually read a textbook or take >> >a math course at a university, you might learn something. >> >> Curious coming from one whose definitions aren't demonstrably true. > >Care to define "demonstrably true"? Already have in the root post to the thread "Epistemology 201: The Science of Science". Your main problem seems to be a willingness to assume the truth of whatever you're talking without demonstration. You repeatedly use the term "true" in a collateral reply to Gene Ward Smith while unable to demonstrate the truth of what you claim to prove. 'Tis a puzzlement indeed for one trained in precise and exact meanings of modern mathematics. ~v~~
From: Virgil on 12 Nov 2006 14:55
In article <1163353613.745777.63980(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1163234597.643039.70300(a)f16g2000cwb.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > Indeed. Wolfgang Mueckenheim appears to think that Cantor's writings > > > > are > > > > still all true in modern mathematics. > > > > > > I think that much of his set theory is wrong but more of it is true > > > than of modern set theory. > > > > Yes, I know you do not like it, but I state something else. You are > > arguing against modern set theory using statements by Cantor. Even > > statements by Cantor that are not valid in modern set theory. > > > > > > The first falseness is his assumption that every set can be > > > > well-ordered. > > > > > > This assumption is as wrong as the official assumption made by Zermelo. > > > > Assumption? > > AC is taken to be an axiom. That is nothing but an arbitrary > assumption. Well-ordering is so easy connected with AC that we can > state: Well-ordering has been assumed. It has been shown that ZF with AC will be consistent provided ZF without it is consistent, so that its assumption on top of ZF is reasonable. > > > > > > We all know now that that can not be proven from first principles. > > > > > > According to modern mathematics nothing can be proven from first > > > principles. > > > > Do you know what that term means in mathematics? It means from a > > chosen set of axioms using ordinary logic. > > The chosing of the set is an arbitrary act. It is not reduced to first > principles in modern math. Cantor had first principles! Any "first principles" over and above pure logic are necessarily axioms ( statements assumed without proof). > > > Within a branch of > > mathematics the axioms in the first principles are the basic axioms > > used within that branch. So within geometry the basic axioms might > > be the postulates by Euclides with the exclusion of the parallel > > postulate. In that system the parallel postulate can not be proven > > from first principles. > > > > > Cantor considered well-ordering as a first principle, > > > Zermelo introduced it at a first principle = axiom. Cantor was wrong, > > > Zermelo was right? > > > > Cantor did state it without suggesting either that it was a first > > principle or something else. He just assumed it. And he was wrong > > with that assumption. > > He was as wrong as Zermelo, not a bit more. > > > > > At the most in the ridiculous axiom faith of "modern > > > mathematics". > > > > Oh. Whatever that means. > > It means you attitude: After a set of axioms has been chosen, we an be > sure to make true mathematics. We can be sure that what is deduced properly from a set of axioms depends only on them. A properly produced theorem in any axiom system can be false only if at least one of the axioms on which it depends is false. That is the great strength of an axiom system. > > > > For the non-German speaking, Cantor explains that > > the ordered set has ordinal > > 1, 2, 3, ... omega > > n+1, n+2, ..., 1, 2, ... n omega+n > > 2, 4, 6, ..., 1, 3, 5, ... 2*omega > > (Note, in current mathematics the last is noted as omega*2, I will use > > the old notation in the sequel.) He is right and indeed, > > 1, 2, 4, 8, ..., 3, 6, 12, 24, ..., 5, 10, 20, 40, ... > > has ordinality omega^2. A set of cardinality aleph-0 can be ordered > > according to every countable ordinality. > > That did he mean with countable by numbers of class II. > You see, he did not use an omegath element. He did not need one to say what he wanted to say. > > > > > There is no element omega. But, of course, if the set is to be counted, > > > then there must be a number omega following after all natural numbers. Not in standard mathematics, however screwed up WM's notions may be. The cardinality of {1,2,3,...} union {omega} is the same as that of either{1,2,3,...} or omega. > > > This is the error. .. > > > > No. The above statement is not implied by the letter of Cantor to > > Mittag-Leffler. It is explicitly stated elsewhere. And it is wrong. > > To "count" a set of ordinality omega you do not need omega. In most > > cases you need omega, but there is one exception. Can you find it? > > To "count" a set of ordinality a with ordinals you need only the > > ordinals smaller than a. (And note that in counting with ordinals > > you start at 0, because that is the first ordinal.) > > The latter statement is true in modern mathematics. But nonsense > nevertheless. > But your assertion "To count a set of ordinality omega > you do not need omega" is just my position. To count the natural > numbers, you need not omega, because every set of natural numbers is > cunted by natural numbers. Cantor's position, however, was the > opposite. To count a set of ordinality omega one only needs the cardinality of omega. |