Cantor Confusion
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From: G. Frege on 24 Jan 2007 05:17 On Wed, 24 Jan 2007 08:43:43 +0100, G. Frege <nomail(a)invalid> wrote: >> >> If that's what he means, then I'll agree he could be close. It isn't a >> proof, but as a heuristic it is OK. >> > Though by using (almost) the same heuristic he might conclude that > rational numbers aren't countable too. Well... > Or even better: Consider a countable subset of the set of real numbers containing only irrational numbers. Then the argument (the heuristic) of the OP might lead him to the conclusion that this subset is not countable. F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 24 Jan 2007 05:21 On Wed, 24 Jan 2007 10:16:18 GMT, Andy Smith <Andy(a)phoenixsystems.co.uk> wrote: >>>> >>> If you found an eternal clock ticking [...] e.g. once a second, wouldn't >>> you say that it cannot have been ticking for all of negative time, >>> otherwise it would have already made an infinite [...] number of ticks? >>> >> Sure it will have made an infinite number of ticks already, you are >> completely right here. So what? >> > Doesn't that imply that you can have an infinite integer? > No. Why? > > It has done an infinite number of ticks, then infinite plus 1, plus 2 etc., all > distinguishable infinite integers? > Sorry, can't parse this. Please try to get that straight: /infinite/ is NOT an integer. Hence infinite + 1, infinite + 2, etc. doesn't' make any sense. F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 24 Jan 2007 05:25 On Wed, 24 Jan 2007 11:11:26 +0100, G. Frege <nomail(a)invalid> wrote: >>> >>> So I think he's reached the (correct) conclusion that you >>> can't denumerate the reals (in [0,1]) using naturals, >>> albeit in a somewhat clumsy way of saying it. >>> >> Yes! Thank you. >> > *sigh* > > Yes, you reached a correct "conclusion"; but by a faulty reasoning. > That's certainly not something to be proud of (at least not in > mathematics). > For example, you might consider a _countable_ subset of the set of real numbers containing only irrational numbers. Then your argument might lead you to the conclusion that this subset is not countable. (Which is a wrong conclusion, since the considered set is countable by definition.) F. -- E-mail: info<at>simple-line<dot>de
From: G. Frege on 24 Jan 2007 05:47 On Tue, 23 Jan 2007 14:16:50 -0700, Virgil <virgil(a)comcast.net> wrote: >> >> Unless there is an infinite number the number of [natural] numbers >> [...] cannot be infinite. (W. M�ckenheim) >> I have pondered about that question some time. How about the following argument? Assume (for the sake of the argument) that in the set of _all_ natural numbers, denotes by IN*, there were infinite natural numbers (though I don't have the slightest idea how such numbers would look like). Now let's construct a set IN the following way: IN := {n e IN* | n is a finite number}. Then IN is the set of all finite natural numbers. It MUST be, since it is constructed that way. You might think of it the following way too: IN := IN* - {n e IN* | n is infinite}. IN is the set of _all_ natural numbers /minus/ (set theoretic minus) _all_ the infinite natural numbers*. Hence IN is the reminder: the set of all natural numbers that are not infinite, hence finite. Now it's easy to see that IN (in contrast to WM's claim) is still _infinite_, though only containing _finite_ natural numbers. This shows/proves that his claim ("argument") is false. F. -- E-mail: info<at>simple-line<dot>de
From: mueckenh on 24 Jan 2007 06:16
On 24 Jan., 04:36, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > In article <1169578564.772090.181...(a)v45g2000cwv.googlegroups.com> imaginator...(a)despammed.com writes:... > About Wolfgang Mückenheim. > Actually I am getting the impression that what he presents here is not > what he presents in the mathematics courses at his University. (A > Fachhochschule is called a University in the Netherlands.) The official name is "University of applied Scciences" > I received > his book and I found no errors in the first four chapters I did read (of > the ten in all). I am only a bit unlucky about his distinction between > (indeed) actual and potential infinity, but that can be clarified later. See chapter 8. In short: Potential infinite never ends. So Cantor's diagonal is never completed. It remains unproven for ever whether there is a number created which is not in the list. Actual infinity is somewhere finished, so that can be decided whether the diagonal is finished and whether this finished diagonal belongs to the list of finished numbers. > Also his statement (in chapter 3) that irrational numbers only exist > due to actual infinity is based on the representation of numbers in some > integral base. That appears (to me) a bit shortsighted (*). On the other > hand, the introduction of rationals, irrationals, algebraic numbers and > whatever is clear (not strict, but clear enough for the intended audience). > Also his explanation of the Peano axioms is correct. So unlike some crank > books, this book does not contain serious errors in the first four > chapters. That may come as a surprise to some (it did to me). > > (*): The reason appears to be that irrationals can only be given by a rule > about how to compute it. But I think that: > 0.142857142857... > is also nothing more than a rule how to compute it. I agree. This sequence does exist as little as does 3.1415... But contrary to the latter, the first number might have an existing representation in base 142857 (and others, where trichotomy applies) as 0.111.... Nevertheless even that might reasonably be doubted. (See chapter 10 and the present discussion.) Anyhow, Dik, thanks for the fair report. If you notice misprints or errors in the first 8 chapters, please notify me. (Errors which you might encounter in the last two chapters are no errors.) Regards, WM |