Why can no one in sci.math understand my simple point?
in [Logic]
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From: Virgil on 18 Jun 2010 16:00 In article <86419e27-de46-4972-83e8-3c09d037f867(a)u7g2000yqm.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 17 Jun., 21:59, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > I never said a real number can be defined by an infinite sequence. > > All finite words belong to a countable set. If you exclude infinite > words (sequences) then there is no chance for uncountability. > > > > > A real number > > > can be defined only by a finite word. But there is no diagonalization > > > over finite words. > > > > Without even commenting on what you mean or whether it is true, it > > does not refute that the formalized argument is first order logic > > applied to axioms and incontrovertible > > Incontrovertible is religion. Because its adherents exclude > refutations from their perception. Something being "Incontrovertible" in FOL means you can't controvert it that while playing by the rules of FOL. AS far as I am aware, there are no definite set of rules of logic like FOL (first order logic) for religions. > > > You've not said what "wrong" assumption I've "started with". > > The possibility of an infinite sequence of infinite sequences that can > be completed in order to obtain a completed "anti-diagonal" sequence. That possibility is a consequence of an axiom set like FOL plus ZFC. And what is possible within such an axiom set in not constrained by whatever other axioms WM wishes to impose, nor even by WM's views on "reality". > > > All of this business of yours does not refute what is simply > > introvertible, that a formal proof exists in the manner I've > > mentioned. > > There may be a proof. But as the result is wrong the proof is not > worth much. Since it has not been proven wrong in FOL + ZFC, or whatever other system it was proved in, the proof stands. There is a form of pure mathematics which operates much like games, in that one sets rules and then plays within those rules. For this sort of mathematics, those like WM who insist on repeatedly breaking those rules are viewed as cheaters, and deserve to be. > > > > > The translation of these notions into your "incontrovertible" theory > > > is the weak point. > > > > NO, you did not listen to what I said. I did NOT say anything about an > > incontrovertible THEORY. Rather, I said it is incontrovertible that a > > certain finite sequence of finite sequences of symbols exists. > > But this finite sequence leads to the result that an uncountably > infinite sequence of infinite sequences exists. And that is wrong. WM's assertion of error conclusion is cheating. > > > > > > > Does ZFC not prove that all constructible numbers are countable? > > > > I don't know. What is the definition IN to set the rules for otherTHE LANGUAGE of ZFC of > > 'constructible number'? > > > > Anyway, I have no idea how you think that bears upon what I just > > wrote. > > So there seems to be a gap in ZFC. But it is easy to prove that in > fact there are only countably many constructible numbers. WM is free to set whatever rules he wants for his games, but is not free to override the rules that others have set for their games. And that form of "cheating" is exactly what he is forever trying to do.
From: Virgil on 18 Jun 2010 16:10 In article <9fd092f4-caed-4b3d-8a7d-3e97fc9e62f3(a)x21g2000yqa.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > But if there are more than countably many real numbers, then there are > most of them so called moonlightr numbers. We do not know anything > about them. How can they obey trichotomy? In every sufficiently complex system there are propositions that are true but which cannot be proven true within that system and propositions than are false but cannot be proven false within that system. So that there is no problem in real mathematics in having real numbers x and y and being unable to determine whether x < y, or x = y, or x > y, even though it is known that one of them must be true and the other two false. WM mistakenly believes that his mathematic rules form the only mathematics games possible, but we find the rules to his game silly enough not to want to play by them. And WM finds that he cannot play by our rules.
From: Daryl McCullough on 18 Jun 2010 16:16 Peter Webb says... >Cantor's diagonal proof does *not* show the Reals are uncountable; it just >proves the much weaker statement that "the Reals cannot be listed". Those two statements mean the *same* thing! A "list" of objects is just a function that maps each natural number to an object. To say that a set is listable is just to say that there exists a list that contains all objects in the set. And that's exactly what it means to say that a set is countable. -- Daryl McCullough Ithaca, NY
From: WM on 18 Jun 2010 16:30 On 18 Jun., 03:42, "|-|ercules" <radgray...(a)yahoo.com> wrote: > "Transfer Principle" <lwal...(a)lausd.net> wrote > > > > > > > On Jun 17, 6:56 am, Sylvia Else <syl...(a)not.here.invalid> wrote: > >> On 15/06/2010 2:13 PM, |-|ercules wrote: > >> > the list of computable reals contain every digit of ALL possible > >> > infinite sequences (3) > >> Obviously not - the diagonal argument shows that it doesn't. > > > But Herc doesn't accept the diagonal argument. Just because > > Else accepts the diagonal argument, it doesn't mean that > > Herc is required to accept it. > > > Sure, Cantor's Theorem is a theorem of ZFC. But Herc said > > nothing about working in ZFC. To Herc, ZFC is a "religion" > > in which he doesn't believe. > > > Else's post, therefore, is typical of the posts which seek > > to use ZFC to prove Herc wrong. > > To say the list of computable reals DOES NOT contain every digit (in order) of ALL possible > infinite sequences > > is to say this list does not contain every digit (in order) of PI. > > 3 > 31 > 314 > ... Pi is constructable and computable and definable, because there is a finite rule (in fact there are many) to find each digit desired. But as there are only countably many finite rules, there cannot be more defined numbers. Therefore matheologicians have created undefinable "numbers". It is impossoible to know anything of such a "number". Therefore they are not numnbers and, moreover, they cannot be generated as anti-diagonal numbers in Cantor lists. Hence, the "monnshine numnbers" in fact do not help to save set theory. They are nothing but nonsense but addicts of set theory selectively exclude every impression that could wake them up. It is impossible to convert lost set theorists, but it may be possible to save some souls. Regards, WM
From: WM on 18 Jun 2010 16:37
On 18 Jun., 05:22, "Peter Webb" <webbfam...(a)DIESPAMDIEoptusnet.com.au> wrote: > > Since by definition, "listability" = "countability", Cantor's proof of > > unlistability proves uncountability. > > Really? Where did you get that from? From Cantor. > > The computable Reals cannot be listed. > > Therefore according to you they are uncountable. Also according to Cantor. He said (in a letter to Hilbert) that uncompuatble numbers are nonsense (he actually said infinite definitions are nonsense, but that is the same) and of course he is right. 1906, 8. Aug. Cantor to Hilbert Lieber Freund. .... „Unendliche Definitionen" (die nicht in endlicher Zeit verlaufen) sind Undinge. > > But they aren't. > > Maybe your definition needs a little work? No. There is no uncountability. That needs to be understood. Uncomputable numbers are not numbers, because nobody knows what they are. Regards, WM |