Cantor Confusion
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From: Dik T. Winter on 11 Nov 2006 20:41 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? > > 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. 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. > At the most in the ridiculous axiom faith of "modern > mathematics". Oh. Whatever that means. > > 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. > 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. > 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.) > 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? And it is easily explained. There are two non-terminating countable subsets followed by a subset of one element. Each of the non-terminating countable subsets gives a term omega, and the subset of one element gives the 1. This is a quote from another article by me: > > I think this has to do with the problems of the conceptions of potential > > infinity vs. actual infinity that played an important role in mathematics > > before the early 1900's. Moreso because there were also religious issues. > > So Cantor crossed the border (after consultation with his religious leader > > and in strong opposition from him). The actual infinite was impossible > > (and unreligious) before that time, but he introduced actual infinity. > > No. There are hints on the actual infinite in the holy bible and by > Saint Augustin. Cantor corresponded with Cardinal Franzelin about that. > The Cardinal did not oppose to the idea. He agreed that God will know > Cantor's numbers "if they are not contradictory". The Cardinal only > disagreed with Cantor's "proof" of the actual infinite. Because Cantor > assumed that God had been forced to create it. Perhaps. I do not know the bible, neither what Saint Augustin did write. It is irrelevant for mathematics. I just was trying to find the reason. > > And so, apparently, he thought that if a set was actually infinite, > > it should have an actually infinite element, or somesuch. > > It should have an integer number counting its elements. Of course this > number had to appear in the sequence of numbers if it was a number. But > where? Obviously after all natural numbers. But then, what is the > number of 1,2,3,...,omega ? > It cannot be omega + 1, because, according to his equation above, > 2,3,4, ....,1 has already the number omega + 1. > Therefore omega cannot appear in the sequence. And, I think that Cantor would agree that 1, 3, 4, ..., 2 also has the number omega + 1. > So we have the irrevocable dilemma: > 1) omega is the number of countable many numbers like 1,2,3,... or > 7,8,9,... and as such a single term in the sequence following the > counted terms like 1,2,3,...,omega or 7,8,9,...,omega Lack of precision. Aleph-0 is the number of countable many numbers. Omega is the ordinal number of coutably many numbers with a particular ordering. And I fail to see why it should follow. >
From: Virgil on 11 Nov 2006 20:55 In article <1163253373.464057.260870(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > 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. The set of rationals { (n-1)/n:n in N} "goes on forever" in the sense that there is no end value IN THE SET beyond which the set does not go , but there is a value not a member of the set that it does not exceed. In the same way, the minimal of sets with members {}, and {{}}, and for each x also x union {x} is ordered by membership, with each being a member of all its successors, and there is not end to these successors. But there can be a set of which they are all members, which will of necessity not be a maximum, but can be a supremum. And in ZF or NBG there /must/ be such a set.
From: William Hughes on 11 Nov 2006 21:02 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > A diagonal consists of certain line elements. Therefore a supremum is > > > not sufficient. > > > > > > > Why is the supremum "not sufficient" to define the length of the > > diagonal. (Note, there is no line with an infinite index, so > > your answer should not depend on a line with an infinite index)? > > Every column has a fixed number omega of terms. You can add 1 further > term and you will get omega + 1 terms in every column. Every line has > less than omega terms. If you add 1 term to every line, the number of > terms in every line remains less than omega. What about the diagonal? > It has to satisfy both conditions, which is impossible. No. Adding one element to each line does not change the supremum of the lengths of the lines, so it does not change the length of the diagonal. The number of terms in each line, n, is less than omega. The number of columns is the supremum of the number of terms in each line. The number of columns is omega. The number of lines is omega. The number of columns is equal to the number of lines. Now add one term to each line. The number of terms in each line, n+1, is less than omega. The number of columns is the supremum of the number of terms in each line. The number of columns is omega. The number of lines is omega. The number of columns is equal to the number of lines. > > Therefore your conviction, expressed in many postings (it is possible > to have an infinite set each of whose elements are finite) is wrong. > > > > > > > The length of the diagonal will be longer than every > > > > line if and only if there is no line with maximum length. > > > > > > Therefore also the diagonal cannot have maximum lengh omega > n. > > > > > > > As there is no last line, there is no line with maximum length. > > > > > > Exactly. And therefore there is no "number" omega > > > > The length of the set of natural numbers is the supremum of the > > lengths of the initial segments. This supremum is omega. However, > > this supremum is not a natural number. > > But it is an ordinal number which can be increased by 1. And if > infinity in fact is finished, then it is a maximum. No. There is nothing that says that a set must have a maximum to be finished. - William Hughes
From: Dik T. Winter on 11 Nov 2006 21:06 In article <1163255058.569955.28440(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1163180340.117151.181950(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > I can assign numbers to edges that start at nodes. I can also assign > > > numbers to nodes. > > > > Yes. In the final tree what edge starts at 1/2? > > None. Every edge starts at a node which has the name 0 or the name 1. > If we enumerae the nodes instead of he edges, what would be different > according to Conway? Read his book. It is not overly large. > > As I stated already > > some time ago, there are two possible trees to build. One of the trees > > has edges that terminate at a node, the other tree has edges that start > > at a node and do not terminate. > > Look at Cantor's list. Consider its enumeration by natural numbers. > There are natural numbers which follow on an even number and there are > natural numbers which follow on an odd number. Are there two kinds of > lists? This makes no sense at all. Moreover, we were talking about a tree, not about a list. In the tree, when constructing from edges that do terminate, for each node there is an edge that terminates at that node, and for each edge there is a node where it terminates. On the other hand, when constructing from edges that do *not* terminate, for each node there are edges that emanate from that node. > > Wrong, again. You are still obsessed with the property that an infinite > > set of numbers must contain an infinite number. > > Not at all. You said that infinite tree contains edges that are *not* > finitely far away from the root. This is a gross mistake. Why? You are constructing a tree with non-terminating edges. This means that for each node there are edges that emanate from the node. You appear to think that the edges that emanate from the node labelled 1/3 are finitely far away from the root. If there is a node in your tree with that label, that statement is obviously false. I state also that there is in your tree *no* node with that label. > There are no > natural numbers which are infinitely far from the first one. But every > edge and every node of one single path can be enumerated. Yes, so what? > > Yes, go ahead. > > > Here are all the nodes of the path 0.000... enumerated by natural > numbers: 1,2,3,... > No node is infinitely far from the root, as you can easily check. > > And here are all the nodes of the tree enumerated: > > 1 > 2 3 > 7654 > 8 ... > > Also no one is infinitely far from the root. And so each node has a binary expansion of finitely many binary digits. What is the node number of 1/3? Of 1/5? > > > > The situation with the rationals is quite different, because in the > > > > matrix *each* rational is finitely far away from the root. > > > > > > Each node is finitely far from the root. (Does Conway really tell what > > > you reproduce here?) > > > > Well, my only advise is, read it. > > If he says so, then it wil not be a good idea to waste my time with it. Do you really think the node 1/3 is finitely far from the root in the tree? > > > 2) Do you agree that this implies: There are bit positions infinitely > > > far from the decimal point (or how this point may be called for binary > > > numbers). > > > > No. > > What then do you mean by infinitely far from the root? Your node 1/3 is infinitely far from the root because there is no finite (natural) number that can state the distance. BTW, in binary the point is called the binary point. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 12 Nov 2006 01:02
In article <1163252760.153565.112090(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > For every line there is always a next line longer than the first. > Nevertheless you do not believe that there are lines with infinite > length. If every line is no more that one column longer the one before it, And there is always such a line following every line, wherefore does adding one to such a finite make it infinite? |