An uncountable countable set
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From: Randy Poe on 13 Jul 2006 09:31 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1152736417.397578.308650(a)m79g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > > > Dik T. Winter schrieb: > > > > > > > > > > I proved in my special list even that the diagonal number is a > > > > > > > rational. > > > > > > > > > > > > I wonder whether it was a proof or just some handwaving. > > > > > > > > > > 0.0 > > > > > 0.1 > > > > > 0.11 > > > > > 0.111 > > > > > ... > > > > > replace 0 by 1. > > > > > > > > As far as I see the diagonal starts with 1.000... Am I right? > > > > > > I use only the digits behind the point: So the diagonal is 0.111... = > > > 1/9. > > > > So you imply additional 0's after your notation. I was not sure. > > > > > If there is no other outcome possible, I don't need a further > > > definition. 0.999... = 1 follows from the definition of (+,-,*,/) in > > > the real numbers. > > > > Pray tell me how. Somewhere else you stated that the representation > > 0.111... did not exist. What are you arguing? Either the representation > > 0.111... does exist or not. And if it does exist the definitions of the > > mathematical operations are not sufficient to give a meaning to it. > > > > Anyhow, how do you show that 1.000... - 0.999... = 0 with the definitions > > you are using? > > In "usual" mathematics we have 10 * 0.999... = 9.999... and from that > we get easily 0.999... = 1. > > I argue that 0.111... does not exist. If by "usual mathematics" you mean a mathematics in which the meaning of infinite decimals have not been defined, then we don't have 0.999... or 9.999... either. When you say 0.111... does not exist, do you mean (a) the notation is undefined, or (b) when you define it, you get a number that doesn't exist? If (a), then that is fixed by defining it, which can be done with limits. If (b), then you are arguing that 1/9 doesn't exist? - Randy
From: David Hartley on 13 Jul 2006 09:35 In message <1152785777.506058.199660(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de writes > >David Hartley schrieb: > >> The transpositions are relevant in so far as WM claims that Cantor wrote >> that a sequence of transpositions can not change the order-type. Either >> Cantor was wrong (or using a different idea of the limit), or WM is >> misinterpreting him, my German is not good enough to tell. > >Daraus folgt, da? solche Umformungen einer wohlgeordneten Menge die >Anzahl derselben unge?ndert lassen, welche sich auf eine endliche oder >unendliche Folge von Transpositionen von je zwei Elementen >zur?ckf?hren lassen, ... > >It follows that only such transformations of a well-ordered set leave >its (ordinal) number unchanged, which can be derived from a finite or >infinite sequence of transpositions of each two elements, ... "of each two elements". That is a strange expression in this context. ",each of two elements" would make more sense, although redundant unless he hadn't previously defined "transposition". If that is the case, it would seem Cantor was wrong, you should have no trouble believing that. >> In any case, >> it's a proposition WM needs to prove to complete his argument, and he >> hasn't done so. > >What should that be good for? You would have finally proven a contradiction within standard set theory. You've been trying to do that for many years, why give up now, when you're so close? All you need to do is prove that, in this quotation, Cantor was right! > >In the binary tree everyone can see that no path can split without two >additional edges. So it is by no means possible that there were more >paths than edges. Nevertheless the set theorists go through eyes wide >shut and do not care. > >Who would accept logic arguments if set theory was concerned? Most of us, if you would just provide one. -- David Hartley
From: Virgil on 13 Jul 2006 13:45 In article <1152208684.463235.35190(a)k73g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > > Moreover, > > > > different orderings of the same subset of transpositions will yield > > > > different results. In the formula I gave for the diagonal number, > > > > calculation of one digit does *not* depend on the calculation of > > > > another digit. > > > > > > And what is the consequence of this? > > > > That the calculation of the digits can be done in parallel? > > It is done in zero time. "mueckenh" seems unaware of the difference between a requirement of serial processing as exemplified in his algorithm and the option of "parallel" processing which Cantor's diagonal rule achieves. > > > How do you define the "limit"? And if you define that, is that "limit" > > also well-ordered? Those are things you have to prove. > > I define limit by: *Using all finite natural numbers* just as like as > Cantor does. Cantor does not need any "limit" process because his rule does a "parallel processing" for all naturals at one go.
From: Virgil on 13 Jul 2006 13:48 In article <1152604493.362026.129000(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1152450704.938408.69690(a)h48g2000cwc.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Virgil schrieb: > > > > What we have said, and what is quite true, is that for every natural > > > > there is a larger natural. > > > > > > Of course. And therefore it is impossible to exhaust all of them or to > > > find a set which is larger than all naturals together. > > > > But nowhere an attempt is made to exhaust them. There is only an axiom > > from which it can be derived that there is a set that contains them all. > > With that axiom, Cantor's argument is a proof. Without that axiom, > > Cantor's argument is meaningless. > > When Cantor's proof was published, there was not such an axiom. > > > You are not arguing against Cantor's > > argument, you are arguing against that axiom. > > If an axiom states the existence of 100 natural numbers below 20, it > has to be abolished in order to save mathematics.The axiom of > infinity, interpreted as you do, is such an axiom There is a greater need to save mathematics from the incompetence of such as "mueckenh" that to save it from ZF and the axiom of infinity.
From: Virgil on 13 Jul 2006 13:50
In article <1152785028.533153.97860(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > Franziska Neugebauer schrieb: > > >> > > >> You are using "to index" simultaneously for two different meanings. > > >> The first is "to index a specific position" the second is undefined. > > >> What does "0.1111 is indexed by the list number 4 = 0.1111" exaclty > > >> mean? > > > > > > The unary number n indexes every position 1,2,3,...,n. > > > > How exactly is that achieved? > > The unary number n provides the existence of every m with 0 < m =< n in > my list. All these m have been used to index the digits of n, so these > indexes, now in n, can serve to index further numbers. > > > >> What does "completely indexed" mean? > > > > > > All digits are indexed. > > > > All positions *are* indexed. There is a bijection between the index set > > N and the figures a_i: > > > > ... i ... > > ^ > > | > > v > > ... a_i ... > > That is correct for all a_i of list numbers = naturla numbers. That is > not possible for 0.111... . By the way, this is the very reason why > 0.111... is not in the list. In a sense, it "IS" the list. |