Cantor Confusion
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From: mueckenh on 12 Nov 2006 12:27 William Hughes schrieb: > omega 1,2,3,4... > > omega+1 2,3,4 ...,1 > > omega+2 3,4,5 ...,1,2 > > 2*omega 1,3,5... 2,4,6... omega = 1,3,5,... omega = 2,4,6,... omega, omega = 2omega 1,3,5,..., omega, 2,4,6,...,omega = 2omega + 1 > > It can however be a term in another sequence representing > omega+1. There is no single sequence which represents > omega+1. > > > but omega is the first part of it, omega = 2,3,4,... > > > > This dilemma is what I have been trying to explain for years now. > > > > The explanation is that if we do not insist that an ordinal be > represented by an initial seqment of ordinals, then there > is no unique representation of an ordinal. I we do not insist, then by definition omega = 1,2,3,... = n, n+1, n+2, .... omega + 1 = 1,2,3,..., omega. 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. Regards, WM
From: mueckenh on 12 Nov 2006 12:30 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. > > > > 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? > > 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. Regards, WM
From: David Marcus on 12 Nov 2006 12:43 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > > What is wrong with my reasoning? > > > > > > Nothing. Wrong is only the assumption that "goes on forever" could be > > > considered a finished infinity, i.e., could be denoted by a fixed > > > cardinal number being larger than any natural number. > > > > I'm not sure what you mean. In particular, I'm not sure what > > "considered", "finished" (or "finished infinity"), "denoted", and > > "fixed" mean here. > > I see. But recently you used the word "completed infinity". I don't think I ever said that. Do you have a quote? > Be sure that my finished infinity means the same. > > > > If you look at the diagonal, you always see that it is only there where > > > lines are. Therefore you will always see that it is of finite length. > > > "It goes on forever" does not mean actually infinite length. The > > > lengths of the lines also increase from line to line forever. > > > Nevertheless, an infinite length will never be reached. > > > > To me, "infinite length" just means "goes on forever". To you, the two > > phrases have different meanings, it seems. So, what do you mean by the > > phrase "infinite length"? > > "Goes on forever" is a property of the set of natural numbers, as well > of the elements n as of initial the segments 1,2,3,...,n. Yes. > It does not > make you believe that there are infinite natural numbers n. If "infinite natural number" means a natural number which is larger than every natural number, then I don't believe there are infinite natural numbers. > Why does it > make you believe that there is an infinite initial segment 1,2,3... ? Sorry. I don't understand. "1,2,3,..." is the set of natural numbers. You just wrote a few lines above that the natural numbers "go on forever", and I agreed to it. You seem to be asking me why I believe the set of natural numbers goes on forever. But, we just agreed that was true. So, what are you asking me? -- David Marcus
From: mueckenh on 12 Nov 2006 12:46 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. > > > > 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! > 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. > > > > His second falsehood is when he states that a set of first cardinality > > > (meaning sets of cardinality aleph-0) can only be counted with use of > > > numbers of the second class (meaning omega and larger). And I think that > > > especially this quote has lead Wolfgang Mueckenheim astray. Also see > > > my discussion about this quote with Dave Seaman. The falsity is apparent > > > if you realise that quote means that every set with cardinality aleph-0 > > > has an omega-th element. > > > > No explicitly stated but implied. > > Explicitly stated. See my discussion with Dave Seaman where I give the > quote, and where there is a long discussion about the meaning. I hoped to settle this question with my quote. > > > Cantor explained it (as I posted > > recently and repeat it here): > > 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. > > > 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. > > 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. > > > In the examples above, we have no omega. Introducing it in fact as > > number which follows on all (even) natural numbers, we get > > 2, 4, 6, ..., 2nu, ..., omega, 1, 3, 5, ..., (2nu + 1), ..., 2omega > > That is false. > > That ordered set has ordinality 2*omega + 1. What is the problem? That each omega can be substituted by 1,2,3,... or say a,b,c,... yielding 4 omega. Regards, WM
From: David Marcus on 12 Nov 2006 12:48
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > omega 1,2,3,4... > > > > omega+1 2,3,4 ...,1 > > > > omega+2 3,4,5 ...,1,2 > > > > 2*omega 1,3,5... 2,4,6... > > omega = 1,3,5,... > omega = 2,4,6,... > omega, omega = 2omega > 1,3,5,..., omega, 2,4,6,...,omega = 2omega + 1 > > > It can however be a term in another sequence representing > > omega+1. There is no single sequence which represents > > omega+1. > > > > > but omega is the first part of it, omega = 2,3,4,... > > > > > > This dilemma is what I have been trying to explain for years now. > > > > > > > The explanation is that if we do not insist that an ordinal be > > represented by an initial seqment of ordinals, then there > > is no unique representation of an ordinal. > > I we do not insist, then by definition omega = 1,2,3,... = n, n+1, n+2, > ... > omega + 1 = 1,2,3,..., omega. > > 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. The dots in {1,2,3,...,n} have a different meaning than the dots in {1,2,3,...,omega}. Don't take the notation that literally. The ordered set {1,2,3,...,omega} would perhaps be better written in this context as {1,2,3,...,n,...,omega}. You can't deduce things from notation. -- David Marcus |