Cantor Confusion
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From: William Hughes on 8 Nov 2006 07:16 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > If he considers omega to be not the maximum but only the supremum of > > > the set of lines, then we agree that actual infinty does not exist. > > > > No. > > > > You seem to be saying that: > > > > if the set of lines does not contain a line of maximum length > > then the set of lines does not exist. > > > > However, my claim is that: > > > > the set of lines does not contain a line of maximum > > length > > and the set of lines exists. > > That means: All its elements exist? > > > > It is not true that a set cannot exist unless it has a maximum element. > > It is true, at least if all of its elements do exist. But in order to > force you to understand that simple truth, consider the bijection > between lines and columns. All lines do exist, all columns do not > exist. > 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. Let the set of line lengths be LL. Then LL is |N. LL does not have a maximum. However it does have a supremum, aleph_0. The bijection is between LI and CI, not between LL and CL. - William Huhges
From: MoeBlee on 8 Nov 2006 16:03 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > > > David Marcus wrote: > > > > > Obviously, I could be wrong, but I think WM means map it on a line of > > > > > the list. He seems to think that because we construct the diagonal from > > > > > the list, the diagonal must be one of the lines in the list. Why he > > > > > thinks this, I have no clue. > > > > > > > > You mean map the diagonal (or the range of the diagonal, or whatever) > > > > onto one of the finite sequences that is in the range of the infinite > > > > sequence of those finite sequences? I.e., map the diagonal onto a > > > > member of the range of S? A 1-1 map? If so, yes, I would share your > > > > bafflement as to why we should think there is such a mapping or what > > > > contradiction there is in there not being such a mapping. > > > > > > There is a mapping of the diagonal on a (each) column. > > > > Okay, if you want to put it that way. > > > > > There is no mapping of the diagonal on any line. > > > > Okay. > > So there is a difference between the natural numbers (initial segements > of the first column) and the natural numbers (lines). There is not a > bijecton N <--> N? I already asked you please to state exactly which of these you are claiming: (1) That my proof is not indeed a proof in Z set theory. That is, that there is something in my proof that is not first order logic applied to the axioms and previousl proven theorems of set theory. (2) That set theory is inconsistent. That is, that there is a sentence P and ~ P such that both are theorems of set theory. (3) That my proof does not correspond with your own mathematical notions. As I said, I don't care about (3). As to (2), then I'd like to see what you claim to be a proof in set theory of a sentence P and a sentence ~P. As to (1), then I'd like you to say what in my proof is not justified by first order logic applied to the axioms of set theory. > > > The diagonal cannot have more elements than the width of the matrix is. > > > > I answered that already. If you define 'the width', then it turns out > > to be equal to omega, which is just what the length of the diagonal is. > > The diagonal is assumed to exist such that each of its digits exists, > actually. This is established by the mapping on a column. No it is not. > But it cannot > be established by the mapping on any line. You should recognize that > the following bijection between columns and lines shows a > contradiction, because one element is missing: > 1 <--> 1 > 1,2 <--> 2 > 1,2,3 <--> 3 > ... > 1,2,3,...n <--> n > ... > 1,2,3,... <--> omega 1, 2, 3 ... is NOT a line. > 1,2,3,...omega <--> omega+1 1, 2, 3 ... is the sequence defined by f(n) = n+1. So 1, 2, 3, .. omega is the union of f with {<w w>} ('w' stands for omega). And it is NOT a contradiction that 1, 2, 3, .. omega is equinumerous with omega+1. And nothing about this shows anything in my proof that is not first order logic applied to the axioms of set theory nor any contradiction in set theory. Enough with your silly games. Please refer to (1) - (3) in this post and tell me already which one(s) you claim! > > I answered that already. And the cardinality of the diagonal is not > > assumed, but is proven, to be omega. > > It is *assumed* by stating the axiom of infinity. Without this > assumption the length was not omega. It is PROVEN from the axiom of infinity. Of course I we can't even assert the EXISTENCE of a denumerable sequence such as in this problem without adding an axiom to Z-axiom_infinity. And the axiom of infinity is an axiom of Z set theory. I said my proof is in Z set theory. That includes the axiom of infinity. Your complaining that I use the axiom of infinity is a STUPID and RUDE complaint, since you are now complaining that I am using something that I said FROM THE BEGINNING that I am using. I never said I could prove things about denumerable sets without my using the axiom of infinity. I said clearly, FROM THE BEGINNING of my proof that it is a proof in Z set theory. For you NOW to complain that I'm using an axiom of Z set theory is STUPID and RUDE since I would not have bothered to even post a proof about denumerable sequences and talk with you about it for so long if I accepted any condition that I can't use the axiom of infinity. Sheesh!!! MoeBlee
From: MoeBlee on 8 Nov 2006 16:14 To mueckenh(a)rz.fh-augsburg.de: P.S. About a couple of weeks ago you presented an argument about trees, and as you presented your argument, as you described it, it was clearly reasonable to regard you as intending that as a proof in set theory (it would have been UN reasonable to think the contrary). But later you switched, so that your argument was not to be taken as in set theory. Now, when I say that I am giving a proof in set theory, and after I discuss that with you while having reminded you of that so many times, you switch not your OWN argument this time, but you switch to make the terms of MY argument subject to criticism for USING AN AXIOM OF SET THEORY! I'm really very curious what satisfaction you get from such games you play. There can't be any intellectual satisfaction in such mindless games. So what is it get from them? MoeBlee
From: Virgil on 8 Nov 2006 17:56 In article <1162980145.465598.88740(a)m7g2000cwm.googlegroups.com>, mueckenh(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 > > > ... > > > > > > are not omega? I was under the impression that there are aleph_0 lines > > > with less than aleph_0 columns. > > > > Be careful with your quantifiers. The fact that every line has less > > than aleph_0 columns does not mean there are less than aleph_0 > > columns, it just means that no line has all columns. > > It just means that not all columns are there. Which columns does Wm claim are missing? > There is no actual > infinity. If any columns are missing the any rows that need those columns must be missing too, which means that if some columns are missing then some rows are missing. > Interesting is that all columns have aleph_0 lines. So all lines are > there. And for every potential column, there is some line that has that column, so all the potential columns must be actual. > > You should at least be able to see the difference between the actuality > of the aleph_0 lines which are present, which are realized, and the > aleph_0 columns which are not realized and therefore, are not present. I see that if any column is absent then any row which requires that column will be equally absent. And for every potential column, there is a row that makes it actual. > This supremum is a thing which is not realized by the number of columns > but which is realized by the numer of lines (if the notion of finished > infinity is correct). What ma ybe the reason for this difference? The reason is a short circuit in WM's brain which makes him see what is not there and not see what is.
From: Virgil on 8 Nov 2006 17:59
In article <1162981140.884767.222030(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > The set of lengths of columns, C, consists of the single element > > omega. > > This is a maximum taken by every length of a column. > > > The set of widths of lines, W, is the set of all natural > > numbers. > > W has more elements than C, but every element of W is > > smaller than every element of C. > > Correct. > > > > The supremum of W is omega. The width of the > > matrix is the supremum of the set of widths of the lines, i.e. > > omega. Thus the width of the matrix is equal to the height of the > > matrix. > > But this supremum is not taken. > > The diagonal connects width and length by a bijection. The element d_nn > of the diagonal maps the n-th line on the first n elements of the first > column. As long as n is a natural number, this is no problem. Only for > aleph_0 the diagonal has to map a not existing maximum on an existing > maximum. This is a hard task. (But it occurs only if aleph_0 is assumed > to exist.) WM is used to swallowing camels and straining at gnats. |