Cantor Confusion
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From: Franziska Neugebauer on 21 Nov 2006 16:38 You have not commented on the first part of my posting: ,----[ <4562fc49$0$97261$892e7fe2(a)authen.yellow.readfreenews.net> ] | mueckenh(a)rz.fh-augsburg.de wrote: | | > Franziska Neugebauer schrieb: | >> > Name one single element of the diagonal which is not contained in | >> > a line (which contains this and all preceding elements). | >> | >> 1. Burden of proof weighs upon you who claimed "And there are lines | >> as long as the diagonal is.". The question remains: Which lines? | > | > The diagonal D is a subset of the union U of lines L_n. | > Therefore D - UL_n = empty set. | | Do I miscomprehend your sentence | | "And there are lines as long as the diagonal is."? | | Condider the following scenario: | | state 1: Given a parking space with two cars having their | wheels mounted. (each car has 4 wheels) | | state 2: Unmount all the wheels of both cars and put them on one | stack of wheels. (stack of wheels has 8 wheels) | | Proposition 1: "There are cars in state 1 which have as many wheels | as the stack of wheels in state 2 has." | | Proposition 2: "There is a car in state 1 which has as many | wheels as the stack of wheels in state 2 has." | | Now my interpretation: Claim 1 states that there are at least two (due | to the plural form "cars" and due to the "are" instead of "is") cars | each of which have the property of having as many wheels as the stack | of wheels has (8). That is wrong in my view. | | Claim 2 states that there is (at least) one car which has as many | wheels as the stack of wheels has (8). Which is wrong, too. | | Question 1: Do you agree with my interpretation? | | We frequently use formulas like Ex(p(x)) which is translated as "There | is an x having property p". | | Question 2: Is it correct that you mean by writing | "And there are lines as long as the diagonal is." you mean | "There is a line as long as the diagonal" is? I. e. in the sense of | Ex (p (x)) with x for line and p(x) "lenght of line equal length of | diagonal"? `---- mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> >> Do you want to posit, that there is some line identical to the >> diagonal? That is not (yet) proven. > > The number of elements of the diagonal is larger than all n: > E omega A n : omega > n. 1. n is not suitably constrained. For any n > omega the formula is omega > n is wrong hence the formula is wrong. 2. omega is asserted to exist so the existencial quantification is redundant. What you probably mean is An (n e (What set do you have in mind?) -> omega > n) Please correct. > The lines have only finitely many elements. > > In a linear order of finitely many elements n exactly the following is > implied: > If for every n there exists an line L(n) such that L(n) contains all > elements m <=n > then there exists a line L such that for every n, L contains all > elements m <=n . The constraint on n is missing here, too. > This is valid for the finitely many elements of every line. (How many > lines there are is irrelevant.) Proof? > All elements of the diagonal are elements of the lines UL(n). > Therefore all elements of the diagonal are elements of a finite line. > > Therefore: "E omega A n : omega > n" is false. "E omega (A n (omega > n))" is not (yet) a theorem in ZF. F. N. -- xyz
From: Virgil on 21 Nov 2006 16:44 In article <1164126395.211430.7520(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Tell me which point is not accepted: > 1) Every line which contains an index of the diagonal, contains all > preceding indexes of the diagonal too. > 2) Every index of the diagonal is in a line. > 3) In order to show that there is no line containing all indexes of the > diagonal, there must be found at least one index, which is in the > diagonal but not in any line. This one is flat out false, unless one assumes, a priori, a last line and a last member of the diagonal. It is certainly false in ZF or NBG, where such an assumption is also false. It is quite enough to show that for every index in the diagonal, except 1, there is some line not containing that index. Then no line can contain every index. > 4) There is no line with infinitely many indexes. > 5) Conclusion: There is no diagonal with infinitely many indexes. This is false too, as it depends on the truth of a false claim. > > Regards, WM
From: Virgil on 21 Nov 2006 16:48 In article <1164126570.219569.222590(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > Well, the same in set theory. In one form it has AC as a fundamental > > > > truth, in another version it is not a fundamental truth. What is the > > > > essential difference between the cases? > > > > > > > That there is not an "underlying plane" which it is related to. > > > > So there should be an "underlying plane", whatever that may mean? > > For geometry, it is required, yes, for arithmetic it is not. There are geometries for which there is no underlying plane. One dimensional geometry, for example. And various finite geometries and projective geometries have no underlying plane.
From: Virgil on 21 Nov 2006 16:53 In article <1164126713.968092.237570(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > You are not clear about what the numbers in > > > > your tree are. Are they the nodes? Are they the paths? Sometimes > > > > you say one thing other times you say something different. So to get > > > > proper understanding. What are the things that represent numbers? > > > > > > Infinite paths. > > > > I do not understand. You stated the nodes represent bits. > > Yes. That's the stuff numbers are built from. That is the stuff /numerals/ may be built from, but the numbers themselves need not be dependent on any particular representations. Is the number represented by "VIII" and different from the number represented by "8", or by "2^3" ?
From: Virgil on 21 Nov 2006 16:57
In article <1164126857.392845.209600(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Randy Poe schrieb: > > > > > > > > > > > I know that at most 10^100 digits of sqrt(2) can be determined, in > > > > > principle. > > > > > > > > In principle, if a is the sqrt(2) to 10^100 digits, then > > > > 0.5*(a + 2/a) is the sqrt(2) to 2*10^100 digits. > > > > > > > > What "in principle" prevents me from calculating 2/a, > > > > or adding it to 1, or taking 0.5*(a + 2/a)? > > > > > > Lack of bits to represent 2/a if a already requires all bits available. > > > > That means it's hard in practice. > > > > But what prevents 2/a from existing in principle? > > It is not "hard in practice", but it is impossible. There is no chance. > That means "in principle". Then WM's principles are much less coherent that anyone else's. > > No, in principle we cannot circumvent the uncertainty relations and we > cannot do what we know to be excluded in eternity. That is a meaningful > understanding of "in principle". If we accept your "in principle", then > in principle we can maintain the most ridiculous mess - and then we > arrive at the finished infinity. Good! Let's do it and be done with it. I much prefer our own "ridiculous mess" to WM's even more ridiculous one. |