Why can no one in sci.math understand my simple point?
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From: Sylvia Else on 17 Jun 2010 23:37 On 18/06/2010 10:40 AM, Transfer Principle 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. Well, if he's not working in ZFC, then he cannot make statements about ZFC, and he should state the axioms of his system. > > Else's post, therefore, is typical of the posts which seek > to use ZFC to prove Herc wrong. Part of the problem, as others have noted, lies in determining what it is that Herc thinks he's proving. His assertions become inpenetrable as an increasing function of their distance from the start of his posting. Sylvia.
From: Virgil on 18 Jun 2010 00:47 In article <4c1ae6b7$0$18229$afc38c87(a)news.optusnet.com.au>, "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > > > > Since by definition, "listability" = "countability", Cantor's proof of > > unlistability proves uncountability. > > Really? Where did you get that from? > > The computable Reals cannot be listed. > > Therefore according to you they are uncountable. > > But they aren't. > > Maybe your definition needs a little work? Consider the set of computable numbers, S. According to http://en.wikipedia.org/wiki/Countable_set, one definition of such a set, S, being countable is that there is a injective function from S to N, which is equivalent to there being a surjective function from N to S. Since you object to there being any bijection from N to any superset of S, you must equally be rejecting any surjection from N to S and rejecting any injection from S to N, since from any such bijection such a surjection is easily derived. So in what sense do you claim that the the set S of computable numbers is countable? It is certainly not in any sense that I am aware of.
From: |-|ercules on 18 Jun 2010 01:03 "Sylvia Else" <sylvia(a)not.here.invalid> wrote > On 18/06/2010 10:40 AM, Transfer Principle 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. > > Well, if he's not working in ZFC, then he cannot make statements about > ZFC, and he should state the axioms of his system. Can you prove from axioms that is what I should do? If you want to lodge a complaint with The Eiffel Tower that the lift is broken do you build your own skyscraper next to the Eiffel Tower to demonstrate that fact? > >> >> Else's post, therefore, is typical of the posts which seek >> to use ZFC to prove Herc wrong. > > Part of the problem, as others have noted, lies in determining what it > is that Herc thinks he's proving. His assertions become inpenetrable as > an increasing function of their distance from the start of his posting. heheh. That's my favorite Simpson's quote, "The potential for mischief varies inversely with one's proximity to the authority figure.". Herc
From: Peter Webb on 18 Jun 2010 01:10 "Virgil" <Virgil(a)home.esc> wrote in message news:Virgil-6672C4.22473617062010(a)bignews.usenetmonster.com... > In article <4c1ae6b7$0$18229$afc38c87(a)news.optusnet.com.au>, > "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > >> > >> > Since by definition, "listability" = "countability", Cantor's proof of >> > unlistability proves uncountability. >> >> Really? Where did you get that from? >> >> The computable Reals cannot be listed. >> >> Therefore according to you they are uncountable. >> >> But they aren't. >> >> Maybe your definition needs a little work? > > Consider the set of computable numbers, S. > > According to http://en.wikipedia.org/wiki/Countable_set, > one definition of such a set, S, being countable is that there is a > injective function from S to N, which is equivalent to there being a > surjective function from N to S. > > Since you object to there being any bijection from N to any superset of > S, you must equally be rejecting any surjection from N to S and > rejecting any injection from S to N, since from any such bijection such > a surjection is easily derived. > > So in what sense do you claim that the the set S of computable numbers > is countable? > http://en.wikipedia.org/wiki/Computable_number "Although the set of real numbers is uncountable, the set of computable numbers is countable". Easily proved. All computable numbers can be generated by a finite TM (by definition), and there are only countable finite TMs (as these can easily be enumerated). > It is certainly not in any sense that I am aware of. That a set of Real numbers cannot be listed is *not* the same thing as the set is uncountable. Cantor's proof does *not* demonstrate that Reals are uncountable, it just proves there can be no explicit enumeration of them. This is easily seen by comparison with a diagonal proof that all computable Reals cannot be listed - this doesn't immediately mean they are uncountable, as they aren't.
From: Tim Little on 18 Jun 2010 01:30
On 2010-06-18, Ross A. Finlayson <ross.finlayson(a)gmail.com> wrote: > The rationals are well known to be countable, and things aren't both > countable and uncountable, so to have a reason to think that > arguments about the real numbers that are used to establish that > they are uncountable apply also to the rationals, the integer > fractions, has for an example in Cantor's first argument, about the > nested intervals, that the rationals are dense in the reals, so even > though they aren't gapless or complete, they are no- where > non-dense, they are everywhere dense on the real number line. As your sentence is less than coherent, I will merely point out that it is generally poor form to use 9 commas in a single sentence except when listing items. I will grant that parody often benefits from abuses of ordinary sentence structure, such as, for example, and not in any way showing that these are the only possible forms, sentences, like this one, which are convoluted to exhibit, by way of meandering, that they imply that mental processes, of the original writer, that is, which may be, perhaps, less than clear, and so in some way, to some readers, humourous. - Tim |