Why can no one in sci.math understand my simple point?
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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
From: WM on 18 Jun 2010 16:41
On 18 Jun., 05:25, Sylvia Else <syl...(a)not.here.invalid> wrote: > On 18/06/2010 4:27 AM, WM wrote: > > > > > > > On 17 Jun., 15:56, Sylvia Else<syl...(a)not.here.invalid> wrote: > >> On 15/06/2010 2:13 PM, |-|ercules wrote: > > >>> Consider the list of increasing lengths of finite prefixes of pi > > >>> 3 > >>> 31 > >>> 314 > >>> 3141 > >>> .... > > >>> Everyone agrees that: > >>> this list contains every digit of pi (1) > > >>> as pi is an infinite digit sequence, this means > > >>> this list contains every digit of an infinite digit sequence (2) > > >>> similarly, as computable digit sequences contain increasing lengths of > >>> ALL possible finite prefixes > > >>> the list of computable reals contain every digit of ALL possible > >>> infinite sequences (3) > > >> Obviously not - the diagonal argument shows that it doesn't. > > > There is no diagonal element for a list of finite lines. > > The list of computable reals is not a list of finite lines. It is. Every real number that is defined is defined by a finite word (definition or formula). It is impossible to define a number by an infinite sequence, because the sequence never ends and the definition is never known. A finite word W can define an infinite sequence S up to every desired n. W ==> S But the reversal of the implication is not true (as usual). S ==> W is impossible and wrong. Regards, WM |