Cantor Confusion
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From: Franziska Neugebauer on 10 Nov 2006 09:42 William Hughes wrote: > The length of the set of natural numbers is the supremum of the > lengths of the initial segments. Since when do sets have a length? F. N. -- xyz
From: Daniel Grubb on 10 Nov 2006 11:44 >> >The diagonal of actually infinite length omega cannot exist without the >> >existence of a line of actually infinite length omega. >> >> Why not? Do you have a proof of this claim? >The diagonal of a matrix is defined as consisting of elements of this >matrix. For a diagonal longer than every line (or every column) this >is impossible. Let me restate the situation: We have the 'matrix' 1 2 3 4 5 6 7 8 9 10 etc and you argue that the length of the diagonal cannot be infinite. Of course, the first problem is that this is not a square matrix. If you want to make it into a square matrix, you can make it into this matrix: 1 0 0 0 0 0 .... 2 3 0 0 0 0 ... 4 5 6 0 0 0 ... ..... ..... In which case, it is a square matrix and every line has infinite length. So the diagonal, which has infinite length, has the same length as every row and every column. An alternative is to look at the incomplete, triangular array that we started with and notice that in triagular arrays the length of the diagonal is always at least as large as the length of each row. In the situation where there are finitely many rows, the length of the diagonal is the same as the number of rows. The sticking point is whether the length of the diagonal is the same as the length of some row even when there are infinitely many rows. You seem to be claiming the answer is yes. While it is clear that the length of the diagonal is at least the length of any row, it seems clear that the length of the diagonal is actually longer than the length of each row in this case since it is at least as long as the next row down. Since there is always a 'next row down', the length of the diagonal is longer than the length of every row. --Dan Grubb
From: mueckenh on 10 Nov 2006 12:39 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. > 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. O course such positions cannot be enumerated or indexed by natural indexes. This means, they are not defined at all. > > > > Indeed. To prove countability you do not need transfinite induction. > > > But my remark "you are wrong" was to your statement: > > > "The mapping N -> {edges} has been established". > > > See J. H. Conway, On Numbers and Games, for a clear exposition about the > > > difference between the union of all finite trees and the infinite tree. > > > > Whatever Conway may say, there is no difference between enumerating all > > rational numbers in the well-known scheme, starting a the corner of an > > infinite square matrix, and the edges of the tree. In both cases you > > have a system with a limit which is countably infinite. > > 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. > > 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?) > > > Perhaps you see a difference between the union of all finite numbers n > > and the set N? > > Apparently you are not interested in what Conway did write, otherwise you > would understand how ridiculous this comment is. 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? 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). Regards, WM
From: mueckenh on 10 Nov 2006 12:42 Virgil schrieb: > > Why don't you try to find out how it could be presented in set theory? > > Or why that cannot be done? > > We have already seen why it cannot be done in ZF or NBG. It contradicts > the requirements of those systems. > > > You have seen nothing but the fact that it contradicts the *results* of those systems. Regards, WM
From: mueckenh on 10 Nov 2006 12:46
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? > Nor, is it the defintion of > omega in modern mathematics. And, since the set of mathematicians who > know the mathematical meaning of the words "limit", "number", "grow" is > probably empty, your final sentence seems vacuous. > You carried out a survey? Or is that your guess? I know some mathematicians who know what a mathematician should understand by "limit", "number", "grow". > Your changing "finished entity" to "finished infinity" is a bit of a > stretch. Regardless, Hrbacek and Jech are clearly making a philosophical > comment, not discussing mathematics. They were discussing mathematics. Philosophy belongs to mathematics. It is not uncommon to meet a PhD in mathematics, who, in Germany, has the title "doctor of philosophy". It seems to me that you have a very restricted horizon, something Hilbert called Standpunkt. Regards, WM > So, it remains true that "finished > infinity" does not have a mathematical meaning. If you wish to use the > term in a mathematical discussion, it is incumbent on you to define it. This term has been used in a mathematical discussion by Hrbacek and Jech. I don't see why I shouldn't do the same. Regards, WM > No, that is still wrong. That's not what the word "definition" means in > Mathematics. Which mathematics do you allude to? Regards, WM |