Cantor Confusion
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From: imaginatorium on 15 Oct 2006 01:52 Tony Orlow wrote: > stephen(a)nomail.com wrote: > > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >> Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>> David Marcus schreef: > >>>> Han de Bruijn wrote: > >>>>> I'm not interested in the question whether set theory is mathematically > >>>>> inconsistent. What bothers me is whether it is _physically_ inconsistent > >>>>> and I think - worse: I know - that it is. > >>>> What does "physically inconsistent" mean? Wouldn't your comments be > >>>> better posted to sci.physics? Most people in sci.math are (or at least > >>>> think they are) discussing mathematics. > >>> ONE world or NO world. > > > >> Sorry. No idea what you mean. Do you have anything to say about > >> mathematics or would you prefer we all just ignore you? > > > > I am guessing that he means that if all of mathematics does > > not bow to his will, he would prefer mathematics not to exist > > at all. You are either with Han, or you are against him. > > His holy jihad can know no compromise. > > > > Stephen > > You fellers sure are dense. There is, of course, another possibility... >... Han's simply saying that the universe is > consistent, or it doesn't exist at all. Clearly, it exists, so obviously > it's not self-contradictory. Right: consistent with (physical) universe => not self-contradictory Is there some sort of proof for the reverse implication? You have been told, probably hundreds of times now: mathematics is the study of consistent abstract systems. If you don't like that, push off to sci.physics, or alt.crystal-gazing. Brian Chandler http://imaginatorium.org
From: stephen on 15 Oct 2006 03:01 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >>> Han.deBruijn(a)DTO.TUDelft.NL wrote: >>>> David Marcus schreef: >>>>> Han de Bruijn wrote: >>>>>> I'm not interested in the question whether set theory is mathematically >>>>>> inconsistent. What bothers me is whether it is _physically_ inconsistent >>>>>> and I think - worse: I know - that it is. >>>>> What does "physically inconsistent" mean? Wouldn't your comments be >>>>> better posted to sci.physics? Most people in sci.math are (or at least >>>>> think they are) discussing mathematics. >>>> ONE world or NO world. >> >>> Sorry. No idea what you mean. Do you have anything to say about >>> mathematics or would you prefer we all just ignore you? >> >> I am guessing that he means that if all of mathematics does >> not bow to his will, he would prefer mathematics not to exist >> at all. You are either with Han, or you are against him. >> His holy jihad can know no compromise. >> >> Stephen > You fellers sure are dense. Han's simply saying that the universe is > consistent, or it doesn't exist at all. Clearly, it exists, so obviously > it's not self-contradictory. Where there is a conflict between notions, > where there is paradox, there is also explanation. One has to choose the > most consistent path, while integrating the widest array of ideas possible. But there may be more than one most consistent path. We can never know what "reality" really is. We can make models, and our models can make predictions, but if two models make the same predictions there is no way to determine which one is actually "true", or to determine if either is actually "true". So if that is what Han means, then it is a rather pointless statement, like most of his and your objections. And of course it is particularly pointless given that nobody is claiming that the balls and vase problem has anything to do with reality. Stephen
From: jpalecek on 15 Oct 2006 05:40 mueckenh(a)rz.fh-augsburg.de napsal: > Dave L. Renfro schrieb: > > > Peter Webb wrote (in part): > > > > >> This is a complete red herring. There is no question that > > >> the Real generated by Cantor's proof is computable (r. e,) > > >> if the original list is, [...] > > > > mueckenh(a)rz.fh-augsburg.de wrote (in part): > > > > > Of course. That's why the diagonal proof only proves the > > > existence of numbers which belong to a countable set i.e. the > > > set of constructible reals. This proof proves in essence that > > > the countable set of constructible real numbers is uncountable. > > > A fine result of set theory. > > > > You're overlooking Peter Webb's hypothesis "if the original > > list is". You need to have a list (x_1, x_2, x_3, ...) such > > that the function given by n --> x_n is computable. Thus, > > before you can conclude what you're saying (which sounds like > > a metalogic "proof by contradiction" to me, but no matter), > > you need to come up with a computable listing of the computable > > numbers (or at least, show that such a listing exists). > > One cannot compute a list of all computable numbers. By this > definition, > (1) the computable numbers are uncountable. > (2) There is no question, that the computable numbers form a countable > set. > This is a contradiction. It is not necessary to come up with a list of > all computable numbers. There is no contradiction. The fact that you cannot compute a list of all computable reals does not mean that there is no list of all computable numbers. There is one, and it is not computable. This has nothing to do with countability/uncountability. Stop mixing recursion theory with set theory.
From: jpalecek on 15 Oct 2006 05:50 Albrecht napsal: > Dave L. Renfro wrote: > > Dave L. Renfro wrote (in part): > > > > >> Each sequence of real numbers omits at least one real > > >> number. (Sequence being a 1-1 and onto function from the > > >> positive integers to the real numbers.) > > > > Jesse F. Hughes wrote: > > > > > Er, are you sure about that definition of sequence? Seems to > > > me that sequence is any function from the positive integers > > > to the reals. > > > > > > With your definition, the conclusion would be: every onto > > > function N->R is not onto. This is true, of course, but is > > > more likely to confuse the "Anti-Cantorians" rather than > > > enlighten them. > > > > Ooops, you're correct ... I messed up. I knew I should have > > stayed out of one of these kinds of threads. I was curious > > how mueckenh would respond to my bringing up other situations > > (in "real life") where one argues that something can't happen > > by showing that something contradictory (or at least, something > > not desirable) would arise if we assumed it did happen. He (she?) > > seemed to have ignored my examples, though. > > > > And what's with this other thing I saw in the thread, where > > someone argued that the diagonal argument is not complete > > because the assumption that the list of real numbers containing > > all the real numbers wasn't allowed for? That's just bizarre. > > If this person is _really_ concerned about that issue (rather > > than trolling, as I strongly suspect), why isn't he raising > > the same issue for _every_ argument? > > > Your consideration is really strange to me. > I will try to explain my claim again: There is only a diagonal number > which is proveable not in the list if there is a real number which is > build up out of all the real numbers in the list. But for an infinite > list you can't end the diagonal number. You may have a sequence which > converge. But you never have a limit. > Cantor argues that you must not have the limit. But with the same idea > you can e.g. construct a kind of an diagonal (natural) number of any > list of natural numbers (this idea is from Russell Easterly). In > consequence the set of the natural numbers is uncountable... No. An infinite sequence of digits is not a natural number. An infinite sequence of digits after the decimal point (0.xxxx...) is a real number. > You can proof a lot of strange things with the idea of Cantor. But > these things are not very useful. > > > Best regards > Albrecht S. Storz > > > > For example, in proving > > that 4 + 3 = 7 (using three applications of the successor > > function), he should complain that we didn't consider the > > case where 4 + 3 isn't 7, and when we consider this possibility, > > the proof fails. More generally, he should find fault with > > every proof of statements of the form "If P, then Q", including > > the arguments that he is using to support his position. > > > > Now that I think about it, don't a lot of these anit-Cantor > > arguments take the same form as what they're criticizing? > > They argue that something about diagonalizing is incorrect, > > because if it were correct, then [insert their attempt to > > obtain a contradiction]. > > > > Dave L. Renfro
From: William Hughes on 15 Oct 2006 09:34
mueckenh(a)rz.fh-augsburg.de wrote: > Dave L. Renfro schrieb: > > > Peter Webb wrote (in part): > > > > >> This is a complete red herring. There is no question that > > >> the Real generated by Cantor's proof is computable (r. e,) > > >> if the original list is, [...] > > > > mueckenh(a)rz.fh-augsburg.de wrote (in part): > > > > > Of course. That's why the diagonal proof only proves the > > > existence of numbers which belong to a countable set i.e. the > > > set of constructible reals. This proof proves in essence that > > > the countable set of constructible real numbers is uncountable. > > > A fine result of set theory. > > > > You're overlooking Peter Webb's hypothesis "if the original > > list is". You need to have a list (x_1, x_2, x_3, ...) such > > that the function given by n --> x_n is computable. Thus, > > before you can conclude what you're saying (which sounds like > > a metalogic "proof by contradiction" to me, but no matter), > > you need to come up with a computable listing of the computable > > numbers (or at least, show that such a listing exists). > > One cannot compute a list of all computable numbers. By this > definition, > (1) the computable numbers are uncountable. > (2) There is no question, that the computable numbers form a countable > set. > This is a contradiction. It is not necessary to come up with a list of > all computable numbers. Nope. You are mixing two approaches.. A set X is countable if there exists a surjective function,f, from the natural numbers to X There are two possibilities A: you allow arbitrary functions f B: you allow only computable functions f In case A, the computable reals are countable [ but you also have arbitrary reals, and the set of reals (computable and arbitrary) is uncountable.] In case B. the computable reals are not countable (i.e. there is no list of computable reals) You cannot arrive at a contradiction by taking one result from case B (the computable reals are uncountable) and one result from case A (the computable reals are countable). - William Hughes |