Why can no one in sci.math understand my simple point?
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From: Peter Webb on 17 Jun 2010 03:29 "Virgil" <Virgil(a)home.esc> wrote in message news:Virgil-6240F4.21454316062010(a)bignews.usenetmonster.com... > In article <4c1995e5$0$26118$afc38c87(a)news.optusnet.com.au>, > "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > >> > >> > It is proof that there is no countable set of all real numbers, since >> > any alleged such set is provably and constructably incomplete. >> > >> > Similarly, it is proof that there is no countable set of all >> > constructable numbers, since any alleged such set is provably and >> > constructably incomplete. >> >> I hate to disagree with you, because we are on much the same "side", but >> this is not correct. Cantor's proof shows that you cannot form a list of >> all >> Reals. This is not the same as the Reals being uncountable. > > If the reals were countable they would be listable, since such a list > would be a "counting" of them, so that NOT being listable implies NOT > being countable. That does not follow, and I have already provided a counter-example. Computable numbers are countable, but cannot be listed. > > An infinite set is defined to bee countable if and only if there is a > surjection from the set of natural numbers to that set. When such a > function is a bijection, it is commonly called a list. > Only if the bijection can be explicitly created. There are countable sets which cannot be listed, such as the countable set of computable Reals. > Since the set of reals is infinite but cannot be listed in this way, it > follows that the reals necessarily are NOT countable. By this (incorrect) logic, the computable numbers must also be uncountable. But they are not. >> >> You can use Cantor's diagonal construction to similarly prove that you >> cannot form a list of all computable numbers. However the computable >> numbers >> are in fact countable. You can't simply equate the two concepts; they are >> not exactly the same thing. > > For infinite sets, listability and countability are equivalent. No. Witness the infinite set of computable numbers. Countable but not listable. Cantor's proof shows that the set of Real numbers cannot be listed. It does not immediately follow that they are uncountable. Plenty of countably infinite sets cannot be listed. The set of computable numbers is one. The set of halting TMs is another.
From: WM on 17 Jun 2010 03:49 On 16 Jun., 23:43, MoeBlee <jazzm...(a)hotmail.com> wrote: > > Look simply at the results. > > If a theory says that there is an uncountable set of real numbers such > > each number can be identified as a computable or definable or > > constructable one, or in other ways, then this theory is provably > > wrong. > > Please tell me which EXACT, SPECIFIC theorem of ZFC "says" that "each > real number can be identified as a computable or definable or > constructable one, or in other ways". If you start with Cantor's list, then the diagonal can be identified. Do you mean that in ZFC Cantor's argument is wrong? Further: A number that cannot be put in trichotomy with others is not a number. Is such an unknown entity called a number in ZFC? > > > Reason: Cantor either proves that a countable set is uncountable or > > that a constructible/computable/definable number is not constructible/ > > computable/definable. > > Whatever Cantor proves, my question is specifically as to ZFC. I consider results only. If ZFC allows the conclusion that Cantor's diagonal argument is wrong, then ZFC may be right. Is it right? > > > What the theory internally may be able to prove or not to prove is, at > > least for my person, completely uninteresting. > > What interests you is entirely up to you, of course. But you've not > shown that ZFC is inconsistent (where 'inconsistent' is defined as > "having as a theorem some formula and its negation"). I have been told that ZFC yields this result: There are uncountable many real numbers. And by number I understand something that can be put in trichotomy with other numbers. If this is not the case, ZFC may be right. Others have told me that ZFC does not prove any uncountability at all and that the "standard model" is not a model of ZFC at all. If this is the case, ZFC may be right. Regards, WM
From: Virgil on 17 Jun 2010 03:54 In article <4c19cd2c$0$316$afc38c87(a)news.optusnet.com.au>, "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > Cantor's proof applied to computable numbers proves you cannot form a > computable list of computable numbers. Cantor's proof applied to Reals > proves you cannot form a computable list of Reals. To be correct, there is no computable list of ALL of the computable numbers, even though the set of computable numbers is e countable, but there are lots of possible computable lists of computable numbers. For example, for n in {1,2,3,...}, f(n) = 1/n is a computable 'list' of computable numbers. > > The property "that you cannot form a computable list" is not the same as the > property "is uncountable". For example, computable numbers have the first > property but not the second. Cantor's proof is about what can be expressed > in a list, and not directly about uncountable sets (which don't even get > mentioned in the proof). > > > > > > > > > > - Tim
From: Virgil on 17 Jun 2010 03:59 In article <4c19cef9$0$17178$afc38c87(a)news.optusnet.com.au>, "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > "Virgil" <Virgil(a)home.esc> wrote in message > news:Virgil-6240F4.21454316062010(a)bignews.usenetmonster.com... > > In article <4c1995e5$0$26118$afc38c87(a)news.optusnet.com.au>, > > "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > > > >> > > >> > It is proof that there is no countable set of all real numbers, since > >> > any alleged such set is provably and constructably incomplete. > >> > > >> > Similarly, it is proof that there is no countable set of all > >> > constructable numbers, since any alleged such set is provably and > >> > constructably incomplete. > >> > >> I hate to disagree with you, because we are on much the same "side", but > >> this is not correct. Cantor's proof shows that you cannot form a list of > >> all > >> Reals. This is not the same as the Reals being uncountable. > > > > If the reals were countable they would be listable, since such a list > > would be a "counting" of them, so that NOT being listable implies NOT > > being countable. > > That does not follow, and I have already provided a counter-example. > Computable numbers are countable, but cannot be listed. > > > > > An infinite set is defined to bee countable if and only if there is a > > surjection from the set of natural numbers to that set. When such a > > function is a bijection, it is commonly called a list. > > > > Only if the bijection can be explicitly created. There are countable sets > which cannot be listed, such as the countable set of computable Reals. > > > Since the set of reals is infinite but cannot be listed in this way, it > > follows that the reals necessarily are NOT countable. > > > By this (incorrect) logic, the computable numbers must also be uncountable. > But they are not. > > > >> > >> You can use Cantor's diagonal construction to similarly prove that you > >> cannot form a list of all computable numbers. However the computable > >> numbers > >> are in fact countable. You can't simply equate the two concepts; they are > >> not exactly the same thing. > > > > For infinite sets, listability and countability are equivalent. > > No. Witness the infinite set of computable numbers. Countable but not > listable. > > Cantor's proof shows that the set of Real numbers cannot be listed. It does > not immediately follow that they are uncountable. Plenty of countably > infinite sets cannot be listed. The set of computable numbers is one. The > set of halting TMs is another. One can create lists which contain all of the computable numbers , (or all of the halting TMs) but which, of necessity, list some other things as well.
From: WM on 17 Jun 2010 04:05
On 17 Jun., 04:37, "Peter Webb" <webbfam...(a)DIESPAMDIEoptusnet.com.au> wrote: > "WM" <mueck...(a)rz.fh-augsburg.de> wrote in message > > news:ad6b33fa-7e91-4f0e-a390-30ee1050284b(a)s9g2000yqd.googlegroups.com... > > > > > > > On 16 Jun., 13:15, "Peter Webb" <webbfam...(a)DIESPAMDIEoptusnet.com.au> > > wrote: > >> "WM" <mueck...(a)rz.fh-augsburg.de> wrote in message > > >>news:91a57f95-54ee-4f0a-8341-b2a7dc2f11de(a)h13g2000yqm.googlegroups.com... > >> On 16 Jun., 05:37, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > > >> > Virgil <Vir...(a)home.esc> writes: > >> > > But until you can determine which of those 10 cases, how can you > >> > > compute the number? > > >> > You can't. The number is computable nonetheless, in the sense that > >> > there > >> > exists an effective procedure for churning out its decimal expansion. > > >> > As noted, computability is a purely extensional notion. Recall the > >> > classical recursion theory exercise, which we find, in some form or > >> > other, in pretty much any text on the subject: > > >> > Let f : N --> N be a function such that > > >> > f(x) = 0 if Goldbach's conjecture is true, and 1 otherwise. > > >> > Is f computable? > > >> All computable, Turing-computable, nonetheless computable, definable > >> (in any useful, i.e., finite language), and by other means > >> determinable numbers form a countable set. > > >> ________________________ > >> All subsets of N are countable. Big deal. > > > All subsets of N that can be specified somehow belong to a countable > > set. > > As I said, all subsets of N are countable. Also as I said, big deal. No. You misunderstood. Please read carefully. The set of all subsets of N that can be specified somehow is countable. > > No, try to think in a rational way. Every real number that can be > > identified by definition, computation, and whatever notions else may > > be invented by set theorists to shore up their broken walls, all these > > numbers belong to a countable set. > > Yes. And so do all subsets of N that can be specified. > > > Therefore Cantor "proves" the uncountability of a countable set. > > No. Cantor's proof is not about a "set" of Reals; it is about a "list" of > Reals. But from this list he concludes that the set of reals that can appear as a diagonal, hence can be specified somehow, is uncountable. This conclusion is wrong > > You seem to not understand that something can be a set but cannot be > enumerated as a list, which is pretty much the whole point of Cantor's > proof. I understand that aleph_0 is considered to be a number. Then we can conclude that even sets that cannot be enumerated, can have cardinal number aleph_0. Because they are a subset of a set that can be enumerated. > > > > >> ____________________ > >> No, it proves there is a Real which is not on the list. > > > What is the interpretation? The reals that can be constructed, are > > uncountable. > > No. The interpretation is that you cannot form a list of all Reals, and any > list which purports to be so must miss at least one Real. The following list contains all reals (i.e. their finite names) that can be specified. This contradicts your claim. 1.1.a 0 2.1.a 1 3.1.a 00 .... Every line can be expanded to a (countable) infinity of an infinity of lines to cover every word in every language and every alphabet. There is no name missing. The list is counatble. > > > Another proof shows: The reals that can be constructed, are countable. > > So there is a contradiction. What is the reason: The assumption of > > finished infinity. > > No, the pretty obvious (and correct) conclusion is that you cannot form a > list of the constructable Reals. > > Note, again, that you are implicitly treating a list of Reals as the same > thing as a set of Reals; Cantor's proof is about Lists, not Sets. Cantor concludes about sets. That is wrong. Further there is a list given above that contains all constructable reals - though some more names. But it would be wrong to assume that a subset has larger cardinal number than its superset. > > > There are no infinite sequences of digits other than established by > > finite formulas as a_n = 1/n or pi or sqrt(2). > > Cantor's list with infinitely many infinitely long lines containing > > arbitrary sequences of digits does not exist (other than by finite > > formulas - but they are countable). In fact nobody has ever seen that > > chimera. A typical case of mass hysteria. How could mathematicians be > > taken in by that rubbish? > > Well, mathematicians don't confuse lists of Reals with sets of Reals. Why then do they claim that the set of reals is uncountable and conclude that 2^aleph_0 is larger than aleph_0? In fact, that is the error. Regards, WM |