Cantor Confusion
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From: Daniel Grubb on 9 Nov 2006 14:50 >> Since the length of the diagonal equals the length of the first column, >> you are also saying that there is some line whose length is greater than >> or equal to the length of the first column. Is that correct? >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? This seems to be the place where you disagree with everyone else. You make an assertion, but you do not give a proof of that assertion. Everyone else says you are wrong in that assertion and use the current example as a counter-example. --Dan Grubb
From: Virgil on 9 Nov 2006 15:24 In article <1163069363.908618.240110(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > > > In article <1162825586.068932.25310(a)m7g2000cwm.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > ... > > > > > n e N was implied. If you can compare two entities in some > > > > > dimension > > > > > like length or weight , then they are of the same sort with respect > > > > > to > > > > > this dimension. > > > > > > > > Perhaps. But we do not talk here about length or weight. We are > > > > talking > > > > mathematics. > > > > > > How else lengths and weight could be compared if not by mathematics? > > > > Weights by scales. Lengths by putting them alongside. No mathematics > > involved. > > Comparison of numbers by size is one of the first tasks of mathematics. > > > But at least weight is not a mathematical entity. Length is, > > but not in the way you see it. > > The numbers counting the units of weight or length are mathematical > entities. > > > > > > Of course. But the usual set theory without yet specifying any > > > > > models > > > > > starts from this point. [everything is a set] > > > > > > > > Not when I followed the courses on set theory. > > > > > > Why didn't you? > > > > I did. Why do you think I did not? > > Because you do not know that in ZFC everything is a set Irrelevant and unproven. > becauses you > do not know that limits are inside set theory. Irrelevant and unproven. > And because you do not > know why Cantor called the numbers of the first class countable by > numbers of the second class. Here are some explanations of both topoi > by Cantor himself. Those who cannot think for themselves are compelled to argue from authority. Besides which, "topos" and "topoi" have specific mathematical meaninigs inappropriate here.
From: Virgil on 9 Nov 2006 15:33 In article <1163070703.249948.129150(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueck...(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > Oh no? The lengths of the columns of the list > > > > > > > > > > 0.1 > > > > > 0.11 > > > > > 0.111 > > There are aleph_0 lines and aleph_0 columns. Both infinities > > are exactly the same. > > Why then is there a column with aleph_0 terms but no line with aleph_0 > terms? If you fill in all spaces with 0's then every line HAS as many terms as any column has terms. > Is that really *exactly* the same? Close enough! > > > > The fact that no line contains all the columns does not change > > the number of columns. > > > > The fact that no line contains all the columns and the first column > > contains all the lines does not change either > > the number of columns or the number of lines. > > That is an assertion. But it is wrong. Then how does it change anything? > Two sets which are distinct by one element, are two different sets. But if both are countably infinite they have the same size. eovind one element from among infinitely many does not affect size. > > > > Yes we have a change. The sets C and L change. However, since > > the supremums of C and L are identical, the supremums do not change. > > But after the change the lines have infinitely many indexes while the > columns have only finitely many. Not by my count. >What about the iterpretation as > finitely many infinite numbers? What about it? > > > > Maybe. But a diagonal cannot exist in a domain where no lines reach. > > > > Correct, however there is no such domain. To say that the > > diagonal has aleph_0 1's does not mean that the diagonal has > > an index aleph_0. It does not. The diagonal only has finite indexes. > > But it has infinitey many indixes So it has infinitely many finite indices. So what? > > > Given any finite index there is a line that reaches that index. > > Of course. Could it be that everything stating the existence of > infinitely many lines is nonsense, considering the fact that infinitely > many terms do not exist in any line? Not in ZF, whatever may happen in the confusion of WM.
From: Virgil on 9 Nov 2006 15:49 In article <1163071764.333438.152230(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > No. You are confusing the length of the columns with the set of line > > indexes contained in each column. These are not the same. > > Explain how in a square matrix (squarity proven by the existence of a > diagonal) the columns can be longer than the lines for *every* line > and every column. Just lucky, I guess. > > > > The set of line indexes contained in the first column is |N. > > The length of the first column is the size of |N, aleph_0. > > > > The set of column indexes is |N. > > > > The bijection defined by the diagonal connects the line indexes > > with the column indexes. Finding a bijection between |N and |N > > is not hard. > > Of course not, because you can do that for finite indexes only. Can WM find any line or any column that has anything OTHER than a finite index? He must then explain how it came by that impossible to achieve index when every index except the first must be an immediate successor of another. > But > finding that there is a column with aleph_0 indexes while there is no > ine with aleph_0 indexes shows that the idea of the existence of > aleph_0 is false. Except that it is not individual lines or columns but sets of indices which are to be compared and those sets are identical. Every row index is also a column index and vice versa. Can WM name any exceptions? > If there were aleph_0 lines with aleph_0 a (non-natural) number, then > we would need a last line. WM might, but no one else would. >Therefore "the fact" that there are aleph_0 > lines is not a fact. Maybe not in WM's world, but in the world of actual mathematics it is > It is simply the result of quantifier exchange to > say E a A n : n < a instead of A n E a : n < a. And those who claim these statement logically equivalent are wrong. One can show that for a and n members of, say, N "for all n there is an a such that n < a" is true merely by considering a = n+1. One can equally show the falsity of "there is an a such that for all n, n < a" by considering n = a.
From: Virgil on 9 Nov 2006 15:52
In article <1163072778.801997.82010(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > It is necessary to distinguish between the lengths of the > > lines and columns and the indexes of the lines and columns. > > > > Let the set of line indexes be LI. > > Then LI is just |N. For every natural number n we have a line > > n. There is no last line so there is no line infinity. > > > > Let the set of column indexes be CI. > > Then CI is just |N. For every natural number > > n there is a column n. > > > > Let the set of column lengths be CL > > Then CL has only one element, aleph_0. > > The supremum of CL (also the maximum) is > > aleph_0. > > There are two sets the bijection of which has not yet been considered > but sould be: > A = The set of lengths of initial segments of the (first) column. > B = The set of lines. > Here we can set up a bijection, expressing the lines by the natural > indexes of the elements: > > 1 1 > 2 1,2 > 3 1,2,3 > ... ... > n 1,2,3,...,n > ... ... > omega 1,2,3,... > > which shows that omega or aleph_0 does not exist. It may seem to show it to WM, but there are more things in heaven and earth that are drempt of in his philosophy. |