Cantor Confusion
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From: mueckenh on 8 Nov 2006 06:16 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > 1 > > > > 11 > > > > 111 > > > > ... > > > > > > > > The length of each column is omega. > > > > > > I assume that we are using ZFC as our logical system. > > > > > > Assuming by the "length of each column" that you mean the number of 1's > > > in each column, then the number of 1's in each column has cardinality > > > aleph_0. So, what you wrote is essentially correct. > > > > > > > The length of each line is less than omega. > > > > > > Each line has a finite number of 1's. Finite cardinality is less than > > > aleph_0. So, what you wrote is essentially correct. > > > > > > > The length o the diagonal is less than omega. > > > > > > The number of 1's in the diagonal is aleph_0. So, what you wrote is > > > false. In fact, I have no clue what you could possibly be thinking that > > > would lead you to make such an obviously incorrect statement. The length > > > of the diagonal is clearly the same as the length of the first column, > > > and you just wrote above that the length of each column is omega. > > > > The length of the diagonal is clearly not more than the length of any > > line. > > Do you disagree that the length of the diagonal is the same as the > length of the first column? No it is the same. I distinguish beween lines and columns. > > As for your statement that "the length of the diagonal is clearly not > more than the length of any line", the length of the first line is 1, > and this is less than the length of the diagonal, so your statement as > written is false. Maybe that my use of "any" was wrong. Please replace it by "every": the length of the diagonal is clearly not more than the length of every line. Regards, WM
From: mueckenh on 8 Nov 2006 06:19 Virgil schrieb: > In article <1162821418.935074.235580(a)e3g2000cwe.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > In article <1162557998.763485.221280(a)i42g2000cwa.googlegroups.com> > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > > > > > > No, something quite different was argued there. Namely that the > > > > > > > limit > > > > > > > *also* is the number of edges in the infinite tree (or somesuch) > > > > > > > requires transfinite induction. > > > > > > > > > > > > But is does not. > > > > > > > > > > Indeed, it does not. But I thought you were maintaining that it would > > > > > be? > > > > > > > > Good heavens! I should require transfinite induction? > > > > > > Sorry, I misread. You insist that the limit *also* is the number of edges > > > in the infinite tree. To prove that you need transfinite induction. > > > > Then you need transfinite induction to prove that the number of > > rationals in the case of an infinite set Q is countable. > > When one can construct explicit injections from Q into N, no sort of > induction is needed at all. When one can construct explicit injections from the set of edges into N, no sort of induction is needed at all. 1 2 3456 .... Regards, WM
From: mueckenh on 8 Nov 2006 06:25 Virgil schrieb: > In article <1162826490.608758.156730(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > > > "Whole number" is Cantor's name for his creation. > > > > > > > > > > > > My question is : Do you maintain omega > n for all n e N? > > > > > > > > > > Before I can answer the question, I need to know what you mean by the > > > > > words/terms. So, please define "omega", "N", and ">". Also, by > > > > > "maintain" do you mean that ZFC proves it? > > > > > > > > Do you keep on thinking that omega > n for all N, applying all > > > > words/terms as Kunen would understand them. > > > > > > Using Kunen's definitions, then ZFC proves that if n is a natural > > > number, then n < omega. Do you disagree that ZFC proves this? > > > > No. Therefore omega is larger than any finite natural number. > > Every natural number is finite. > > omega is not finite. > > > > Omega is the number of digits of the diagonal of > > > > 1 > > 12 > > 123 > > ... > > > > The number of digits of the first and any other column is omega. The > > number of digits of any line is less than omega. > > > > The length of the matrix is omega, as a maximum taken. > > > Does WM claim a line numbered omega? No. You claim omega lines where omega is actually a number, though not a natural one, but a number which can be larger than any natural. I prove that this is false by showing that in this case a number omega would be required by the simple bijection: 1 <--> 1 1,2 <--> 2 1,2,3 <--> 3 .... 1,2,3,... <--> omega 1,2,3,...,omega <--> omega+1 > > If so then what is the line number of the immediately preceding line? > If not then it is only a supremum on the line numbers and not an actual > maximum. > > > The width of the matrix is omega, as a supremum not taken. > > > > This small problem contradicts ZFC. > > No, it contradicts WM. It contradicts the square matrix which is required for Cantor's diagonal argument. Regards, WM
From: mueckenh on 8 Nov 2006 06:32 Virgil schrieb: > In article <1162828907.384237.5300(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > In article <1162563567.735020.246810(a)b28g2000cwb.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > Virgil schrieb: > > > > > > > > > In article <1162470874.593282.36250(a)b28g2000cwb.googlegroups.com>, > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > > WM merely repeated his automatic error several more times here. > > > > > > > > > > WM claims that a list in which the nth listed element is a string of > > > > > length at least n characters cannot produce a diagonal of length > > > > > greater that any finite number of characters. > > > > > > > > > > > > > The diagonal needs an element from every line, the n-th element from > > > > the n-th line. Therefore it cannot be longer than every line. > > > > > > The "diagonal"must be longer than every finite "line". > > > > But it cant, because each of its elements stems from a line. > > Which line(s) is the diagonal NOT longer than when line n is at least of > length n? > > > > > The only way it > > > will ever fail to be longer than some "line" is if that line is itself > > > infiitely long. > > > > Which is impossible, because each of its elements stems from a line. > > When for all n in N, line n is of length n, which is quite possible, > then for all n in N, the diagonal is of length >= n. > > That looks like an infinite diagonal to me. Actually infinitely long and potentially infinitely broad. > > WM is assuming that since every natural is finite, there must be a > largest natural, but in ZF and NBG, that is specifically false. Intermingling of actual infinity and potential infinity kept these systems alive, until now. Regards, WM
From: mueckenh on 8 Nov 2006 06:44
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? > > > 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. 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,...omega <--> omega+1 > If you don't define 'the width', then 'the width' is empty language and > is irrelevent. > > > The number of elements of the diagonal is assumed to be omega. > > That is wrong, because only the supremum is omega. > > 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. Regards, WM |