Cantor Confusion
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From: mueckenh on 10 Jan 2007 06:50 William Hughes schrieb: > > The diagonal consists of line ends, of finished lines. > > But this does not make the diagonal a line, so the question > of whether the diagonal exists is not relevant. Wrong. > > > Every end of a line belongs to a line. However, these lines > > > can be different for different line ends > > > > Your assertion is wrong. There are never different lines required for > > different line ends. Different line ends are always elements of one > > single line. > > The assertion was that the lines can be different. This > is trivially true. Your statement that "Different line ends are > always > elements of one single line" is equivalent to the statement that > L_D exists. We know this is false. Then you should re-investigate your "knowledge". It is obvious for any given pair of natural numbers, that they belong (as indexes) to a single line. Induction supplies the proof for all natural numbers. Name two line ends which do not belong to one single line, or stop claiming you could do so. > "Different line ends are always > elements of one single line" is not true for the potentially > infinite set of all "Different line ends". Of course it is. The potentially infinite set is always a finite set. Don't you know that? Have you another definition of potential infinity? Do you think it is actual infinity? > > > L_D does not exist. > > > > Of course does it not exist, because the diagonal does not exist. > > > > Make up your mind. Your repeated claim is that L_D does exist. > My claim was that L_D exists as a fixed line IF the complete diagonal exists actually. (This was argued in order to disprove actual infinity.) For the potentially infinite diagonal, a last line also exists, but not as a fixed line. Potenial infinity is changing (growing) finity. > > > To spell it out clearly and herewith closing this discussion: > > Your error is to assume that more than one line could be necessary to > > supply two or more different natural numbers, elements (or indexes) of > > the diagonal. That is wrong. > > No. It is clear that given any two natural numbers, or indeed any > set of natural numbers you can write down, there exists a single > line which contains all the natural numbers you wrote down. But this > line > *can* depend on which set of natural numbers you write > down. Of course. And if you write down all natural numbers, then you write down L_D. If the set of all natual numbers is complete, then L_D is complete. If the set is potentially infinite, then L_D can change. It depends on which set of natural numbers you just have. Regards, WM
From: mueckenh on 10 Jan 2007 07:02 Franziska Neugebauer schrieb: > Since there is no largest element in "potentially" infinite sets (in > "actual/complete/finished", too) this sentence makes no sence at all. A potentially infinite quantity (set or not) is always finite. Therefore in a linearly ordered set here is a last element. Contrary to the claim of set theorists, a set is not fixed in reality. Avoiding this recognition maks present tansfinite set theory absolutely useless for any science except itself. (It is like a bad lantern which only enlightens its own ost.) > > > If it turns out, that the diaogonal has a larger element, then it > > turns out that a line containing this (and all smaller ones) does also > > exist. > > > In any case, a line containig all elements of the diagonal does exist. > > Proof? The existence of the diagonal (if existing) and the fact that a given set of natural numbers is always a subset of a natural number. > > To define what you are talking about: The diagonal of the EIT contains > > only digits 1. > > > > 1 > > 11 > > 111 > > ... > > But these digits can be indexded by natural numbers 1,2,3,... > > > > The diagonal therefore, contains natural numbers as indexes as I said. > > Can you define "contain as indexes" mathematically? > That should not be necessary in this newsgroup. The n-th digt carries the index n. Regards, WM
From: mueckenh on 10 Jan 2007 07:12 Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > >> > Franziska Neugebauer schrieb: > >> >> mueckenh(a)rz.fh-augsburg.de wrote: > >> >> > >> >> > The finite union of two or more finite trees is a finite tree. > >> >> > An infinite union of finite trees is the infinite tree. > >> >> > >> >> What is an infinite union? (Please give a definition) > >> > > >> > An infinite union of trees is the union of all (or nearly all) > >> > finite trees which reach from level 0 to level n where n eps N. > >> > >> 1. Please give a definition of "union of all finite trees". > > ==================================================== > > Definition: Denote the nodes of the tree by > > > > (0,0) > > (1,0) (1,1) > > (2,0a) (2,1a) (2,0b) (2,1b) > > ... > > (n,0a) (n,1a) ... > > > > The union of all trees up to the n-levels tree is > > > > {(0,0)} U {(1,0), (1,1)} U .. U {(n,0a) (n,1a) ...} which is obviously the same as {(0,0)} U {(0,0), (1,0), (1,1)} U .. U {(0,0), (1,0), (1,1),..., (n,0a) (n,1a) ...} End of definition. ================================================= > > > > Example: The union of the one-level tree and the two-levels tree is > > > > {(0,0), (1,0), (1,1)} U {(0,0), (1,0), (1,1), (2,0a), (2,1a), (2,0b), > > (2,1b)} > > 1. "Definition By Example" is considered Bad Pratice. The upper part contains the definition. To support a definition by an exampe is good practice in most text books. > 2. The union operators are not "evaluated". How do they evaluate? Read a book on set theory. > 3. Why don't you take the standard graph theoretical approach? > (Hint: graph G = (V, E), V: set of vertices, E: set of edges). Because it veils the conradiction raising from the tree. > 4. What is "infinite" in "infinite union of trees which reach from > level 0 to level n e N"? I can only spot finitely many trees having > finitely many levels. Then you think that there is no infinite union of naural numbers? Spot a natural number which is not, as a level, in the tree. > > >> 2. Please give a definition of "union nearly all finite trees". > > > > Union of all trees with a finite number of exceptions. > > Of what? Please define exception. In German: "Ausnahme". (It is usual to use "fast alle" in mathematics of infinite sets. Just look it up in a book.) Regards, WM
From: William Hughes on 10 Jan 2007 07:23 mueck...(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > The diagonal consists of line ends, of finished lines. > > > > But this does not make the diagonal a line, so the question > > of whether the diagonal exists is not relevant. > > Wrong. > > > > > Every end of a line belongs to a line. However, these lines > > > > can be different for different line ends > > > > > > Your assertion is wrong. There are never different lines required for > > > different line ends. Different line ends are always elements of one > > > single line. > > > > The assertion was that the lines can be different. This > > is trivially true. Your statement that "Different line ends are > > always > > elements of one single line" is equivalent to the statement that > > L_D exists. We know this is false. > > Then you should re-investigate your "knowledge". It is obvious for any > given pair of natural numbers, that they belong (as indexes) to a > single line. Induction supplies the proof for all natural numbers. > > Name two line ends which do not belong to one single line, or stop > claiming you could do so. > > > "Different line ends are always > > elements of one single line" is not true for the potentially > > infinite set of all "Different line ends". > > Of course it is. The potentially infinite set is always a finite set. > Don't you know that? Have you another definition of potential infinity? > Do you think it is actual infinity? > > > > > L_D does not exist. > > > > > > Of course does it not exist, because the diagonal does not exist. > > > > > > > Make up your mind. Your repeated claim is that L_D does exist. > > > My claim was that L_D exists as a fixed line IF the complete diagonal > exists actually. (This was argued in order to disprove actual > infinity.) > For the potentially infinite diagonal, a last line also exists, but not > as a fixed line. Potenial infinity is changing (growing) finity. > > > > > To spell it out clearly and herewith closing this discussion: > > > Your error is to assume that more than one line could be necessary to > > > supply two or more different natural numbers, elements (or indexes) of > > > the diagonal. That is wrong. > > > > No. It is clear that given any two natural numbers, or indeed any > > set of natural numbers you can write down, there exists a single > > line which contains all the natural numbers you wrote down. But this > > line > > *can* depend on which set of natural numbers you write > > down. > > Of course. And if you write down all natural numbers, then you write > down L_D. > If the set of all natual numbers is complete, then L_D is complete. > If the set is potentially infinite, then L_D can change. It depends on > which set of natural numbers you just have. L_D cannot change. L_D is a line. A line cannot change.. You are saying that for every maximum integer, n_m, that can be shown to exist. a line L_M(n_m) which contains all integers up to and including this maximum (i.e. all integers that have been shown to exist at this point) can be shown to exist. No one is disuputing this. However, you cannot find a single line L_D that contains every L_M(n_m) that can be shown to exist. - William Hughes
From: Andy Smith on 9 Jan 2007 23:44
Or the not very elegant Peano IIb: 1) All natural numbers form an ordered set bounded by alpha (aka 0) and omega (aka infinity). 2) All natural numbers have a successor and a a predecessor; 3) alpha has no predecessor, omega has no successor, and neither are natural numbers. 4) For any natural number n there exists another natural number strictly bounded by n and (omega-n). This has some dodgy consequences I think. not least that this does now define the infinite integers. omega = -1 (because omega*2 = omega -1 (in binary ...1111*10 = ...1110 =omega - 1). The universe operates in 2's complement arithmetic! I think I might wait a decade or two before mentioning this elsewhere. |