Cantor Confusion
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From: mueckenh on 11 Nov 2006 08:56 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Therefore we can compare the diagonal with every line. We find that the > > diagonal is longer than every line. > > 1 > 1 1 > 1 1 1 > 1 1 1 1 > ... > > The length of the diagonal is clearly the same as the length of the > first column. And, the first column "goes on forever", while each line > does not "go on forever". Therefore, for each line L, the length of the > diagonal is longer than the length of line L. Hence, "the diagonal is > longer than every line." > > 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. 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. > Each line contains a finite number of 1's, but the first column and the > diagonal contain an infinite number of 1's. This is clear from the > picture. Why should something that is clear from the picture be > "impossible"? "but the first column and the diagonal contain an infinite number of 1's" is not clear from the picture. Moreover it is clear from the picture that the length of the diagonal cannot surpass the lengths of all lines. Regards, WM
From: mueckenh on 11 Nov 2006 08:57 William Hughes schrieb: > > > Lines with finitely many indexes cannot exhaust the column with its > > infinitely many > > An infinite number can. A God can do everything he wants. Infinite numbers are of the same power? Can you apply your reasoning to the following matrix? 1 21 321 4321 .... What ordinal numbers have the following sequences? 3,4,5,...,1,2 3,4,5,...,omega,1,2 Regards, WM
From: mueckenh on 11 Nov 2006 09:05 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". Is transitivity unknown in your personal branch of mathematics? > 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? > > Further this (Cantor's) definition supports my definition of > > definition. > > Regardless of whether Cantor meant his remark to be a definition, you > can't change what the word "definition" means in modern mathematics. It Again: What you think or have learned or think to have learned from some books concerning modern mathematics need not be a law enacted by God or by a majority of mathematicians. Further mathematics has nothing to do with democracy. > 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. > 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". Regards, WM
From: mueckenh on 11 Nov 2006 09:24 Dik T. Winter schrieb: > In article <1163180340.117151.181950(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > There is no difference between enumerating all the rational numbers in > > > > the well-known scheme, starting at the corner of an infinite square > > > > matrix > > > > > > There is. You can assign numbers to edges that terminate at nodes. And > > > *that* numbering is different. > > > > 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? > 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? > > > > It numbers only those edges that are finitely > > > far away from the root. But that way you do not number all edges in the > > > infinite tree, because that contains edges that are *not* finitely far > > > away from the root. > > > > That is an interesting remark. The paths are isomorphic to the binary > > representations of real numbers. So you claim that there are positions > > in the binary representation of the real numbers that are infinitely > > far from the decimal point. > > 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. 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. > > > There is a difference, see above. You can only number edges that are > > > finitely far away from the root, but in that way you will only number > > > edges that terminate at nodes finitely far away from the root. As in > > > the edges there are nodes infinitely far away from the root you will > > > never number the edges that terminate there. > > > > Let us enumerate the nodes. > > 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. > > > 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. > > > I have experienced worse opinions. But let us finish with the polemics > > (if possible), because we are at the most important point. Please think > > over your argument: > > 1) Do you say that the nodes cannot be enumerated? > > Depends on how the tree is built. If it is built with terminating edges > the answer is no, if it is built with non-terminating edges the answer > is yes. Let it be built without edges at all. They are only guides for the eye. Here it is 0. 0 1 01 01 ......... > > > 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? Regards, WM
From: Franziska Neugebauer on 11 Nov 2006 09:27
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: [...] >> Are there really three vertices in WM's "triangle"? > > If finished infinities [...] Verbiage. F. N. -- xyz |