Cantor Confusion
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From: Dik T. Winter on 24 Jan 2007 22:05 In article <1169637406.690911.177640(a)j27g2000cwj.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > On 24 Jan., 04:36, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > About Wolfgang M�ckenheim. > > > 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" Here something like that is called a "Technical University". But, what is in a name. > > 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. I am not yet there. I just read chapter 5. > > (*): 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.) Indeed. You will see more remarks on this in my review. > 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.) You think so ;-). But indeed, I try to be fair. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 24 Jan 2007 23:24 Andy Smith wrote: > In message <q2cer25laem96n2mig6mvlc189jbjk6rnh(a)4ax.com>, G. Frege > <nomail(a)invalid.?.invalid> writes > >On Wed, 24 Jan 2007 09:27:59 GMT, Andy Smith > ><Andy(a)phoenixsystems.co.uk> 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). > > > If you want to index 2^n numbers, you require 2^n indices. You require > as many bits to define your indices as log(no of numbers,base 2). With > the reals defined as an infinite series of bits, you require an infinite > series of bits to define your indices. Bzzt. Not proven. Why doesn't the same argument prove that the set {sqrt(2)/n | n = 2,3,4,...} is uncountable? Each of the numbers in the set does not have a finite decimal representation. And, it the number of numbers is infinite, your formula log(no of numbers,base 2) is meaningless. > But if the indices are to be > natural numbers they cannot have an infinite number of bits; all natural > numbers are finite. > > If you prefer, there is a 1:1 correspondence between the countably > infinite set of bits used to define the reals and their indices > (established by systematically counting the reals bit0, bit 0 + bit 1, > bit 0 + bit1 + bit2, etc). But that implies that the indices are > infinite. > > Is that faulty reasoning (I am asking, not being snotty) ? Yep. -- David Marcus
From: David Marcus on 24 Jan 2007 23:25 G. Frege wrote: > On Tue, 23 Jan 2007 19:45:39 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > > >> People cannot conceive of an infinite past, [...] > > > > Why not? I believe that was the usual assumption before the Big Bang > > was discovered. > > Not really... Remember? > > "In the beginning God created the heavens and the earth. Now the earth > was formless and empty, darkness was over the surface of the deep, and > the Spirit of God was hovering over the waters." > > This happened about 6000 years before Christ's birth, or so. I suppose I should have qualified in which group of people it was "usual". -- David Marcus
From: David Marcus on 25 Jan 2007 01:33 mueckenh(a)rz.fh-augsburg.de wrote: > On 23 Jan., 20:20, imaginator...(a)despammed.com wrote: > > David Marcus wrote: > > > As for Cantor, his great idea was that it made sense to compare sets by > > > whether you could biject or inject them. Before him, people thought that > > > this idea didn't work. > > I don't understand what you mean by this - I thought that "before > > Cantor", people just assumed that the only size "beyond any finite > > size" would be "infinite". > > No, already Bolzano found that there are different infinities. But he > did explicitly exclude that a bijection is suitable to find out > anything useful about that topic. It iwas simply the personal opinion > and belief of Cantor. There are better concepts, for instance the > intercession. (see my book > http://www.shaker.de/Online-Gesamtkatalog/details.asp?ID=1471993&CC=21646&ISBN=3-8322-5587-7) Thanks, but we aren't interested in religious books. -- David Marcus
From: David Marcus on 25 Jan 2007 01:36
mueckenh(a)rz.fh-augsburg.de wrote: > On 24 Jan., 00:42, "David R Tribble" <d...(a)tribble.com> wrote: > > mueckenh wrote (2007-01-18): > > > > > 1. The union of all finite trees is an infinite tree. > > > 2. Every finite tree contains only a finite set of paths. > > > 3. The countable union of all paths of the finite trees is therefore the > > > countable union of all finite paths. > > > 4. The countable union of all finite paths is in the union of all finite trees. > > > 5. The "complete" tree containing all paths is identical to the union of > > > al finite trees, with respect to nodes and edges. > > > 6. Identical trees cannot contain different sets of paths. > > > 7. Therefore, both trees contain the same set of paths. > > > 8. Therefore the "complete" set of all path is countable. > > > 9. Therefore the set of all real numbers is countable. > > > 10. Therefore ZFC is inconsistent.Just wondering if you're still using this argument. > > > > If so, perhaps you can explain how you manage the leap > > from (8) to (9). It looks like you're talking about the union > > of all finite-length trees (whatever that means) > > {a,b,c} U {a,b,c,d,e,f,g,h,} = {a,b,c,d,e,f,g,h,} > > Ordering of the nodes is defined by the type of tree. > > > in (1) through > > (8), then at (9) it looks like you somehow conclude that the > > set of all finite trees is equivalent to the set of all reals. > > Paths in a tree are always to be understood as maximum paths. No path > ends before the tree has ended. Paths in the tree containing all nodes > are all we can use to represent a real number. > > 1) Every complete infinite binary tree T (containing all nodes and > edges) contains all paths. > 2) The union tree T(oo) of all finite trees is well defined (as I have > shown elsewhere) and yields the complete infinite binary tree > containing all nodes and edges: T = T(oo). > 3) The union of all finite trees includes the union of all nodes and, > with it, the union of all such subsets which are paths (because every > path is a well defined subset of the set of nodes if the structure of > the tree is well defined). > 4) The set of paths in T(oo) is a subset of the countable set of finite > sets of all paths in the finite trees. > 5) A countable union of countable sets is a countable set (according to > ZF with AC). > ==> The set of all path is countable. (==> The real numbers are > countable.) > > Going on, we can say: > > 6) T(oo) = T contains only finite paths. > 7) T(oo) = T contains all paths including all infinite paths. > ==> There are no infinite paths. (There are no irrational numbers.) > > Nothing further remains to say. If "nothing further remains to say", why post the identical stuff so many times? Wouldn't once be enough? And, since nothing further remains to say, are you now going to stop posting completely? -- David Marcus |