Cantor Confusion
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From: Virgil on 11 Jan 2007 17:46 In article <1168551557.453822.22110(a)p59g2000hsd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > Virgil wrote: > > > > > In article <1168511743.391940.58070(a)77g2000hsv.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > [...] > > >> Conclusion: Every finite binary tree contains a finite set of path. > > >> The countable union of finite sets is countable. The set of paths is > > >> countable. > > > > > > The set of /finite/ paths in the union is countable. > > > But when one takes the union of sets of finite paths one only gets > > > finite paths in that union. There are no infinite paths in that union. > > > > > > The same thing happens with ordinals. When one takes the union of all > > > finite ordinals (like unary trees), there is no infinite ordinal IN > > > that union > > > > Absolutely right. > > But an infinite ordinal is that union! Then WM's infinite path IS the "infinite" tree, not a member of it.
From: Virgil on 11 Jan 2007 18:04 In article <1168551871.231366.88600(a)i56g2000hsf.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1168511743.391940.58070(a)77g2000hsv.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > > > > > > >> Since there is no largest element in "potentially" infinite sets (in > > > > >> "actual/complete/finished", too) this sentence makes no sence at all. > > > > > > > > > > A potentially infinite quantity (set or not) is always finite. > > > > > > > > There is no time in maths. > > > > > > So you write these letters and develop new ideas in zero time? There is > > > no existence outside of time. > > > > > > > > > Therefore in a linearly ordered set here is a last element. Contrary > > > > > to the claim of set theorists, a set is not fixed in reality. > > > > > > > > When your arguments (by the way: What exactly are your arguments?) > > > > > > Here you can read it: > > > > > > Theorem. The set of real numbers in [0, 1] is countable. > > > > > > Lemma. > > > Each digit a_n of a real number r of the real interval [0, 1] in binary > > > representation has a finite index n. > > > r = SUM (a_n * 2^-n) with n in N and a_n in {0, 1}. > > > > This is only true if N is actually infinite, but is quite false > > otherwise. > > It is always true. If N is not actually infinite, then there are no > actually infinite strings, i.e., in particular there are no irrational > numbers. The rational numbers with the usual addition and multiplications form a field, which must be closed under addition and multiplication. This cannot happen with a finite N. So WM would scuttle standard arithmetic to maintain his delusions. > > I corrected that already: 0 is not a natural number. Looks natural enough to me. There is nothing in mathematics which requires the naturals to start at 1 rather than 0, and there are many reasons in math, computer science, engineering and science why starting with 0 is more natural. > > > > The union of binary trees is defined as the union of levels. > > > > Not in general. > > But here. Give a precise definition then. For example, how does one form the union of a trinary tree with a decimal tree? > > > > The union of two or finitely many different finite binary trees simply > > > is the largest on. > > > > Not for disjoint trees. In that case the union is not a tree at all. > > But here. Not without a more precise definition of the WM-union of trees thatn has been so far presented. For example, it may be in finite binary trees that neither of two such trees is tree-isomorphic to any subtree of the other, What is the union then? > > > > The set of /finite/ paths in the union is countable. > > But when one takes the union of sets of finite paths one only gets > > finite paths in that union. There are no infinite paths in that union. > > The union is the set of infinite paths. No nodes? No edges? Just paths that were not in any of the original trees? > > > > The same thing happens with ordinals. When one takes the union of all > > finite ordinals (like unary trees), there is no infinite ordinal IN that > > union > > The union is the infinite ordinal. Which is not one of the objects being unioned, so there is no reason in that for the union of finite trees to be a tree itself.
From: Franziska Neugebauer on 11 Jan 2007 18:08 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> >> > This and your further questions are answerd by the following >> >> > texts. >> >> >> >> I did not pose any questions. I have informed you about the fact >> >> that there is no time and hence no temporal process _in_ math. >> > >> > Be informed then that mathematics is in time. >> >> From that it does not follow that there is a notion of time in >> mathematics. > > A failure Thanks again for confirming that there is no time im mathematics _now_. > need not and will not persist for ever in mathematics. (From > this you can see that time *is* in mathematics.) >> > [Brouwer] maintains [...] >> >> Please tell us, whether you want to discuss views of >> >> [ ] Cantor >> [ ] Brouwer >> [ ] Mueckenheim >> [ ] Cantor interpreted by Mueckenheim >> [ ] Brouwer interpreted by Mueckenheim. >> >> If more than one applies please tag each sentence of yours >> accordingly. >> > I do not remember having posted interpretations, but only original > quotes. I do not discuss _with_ or _instead_ _of_ or _against_ Cantor or Brouwer both of which are dead. If _you_ want to argue _you_ must rephrase them in your own words. > I do not want to discuss pot. infinity with you. (Therefore > sent Brouwer, Hilbert and Cantor to the front.) You accused others of not knowing what "potential infinity" means. So it is still up to _you_ (and not up to Cantor or Brouwer) to say what _you_ mean. > But the five alternatives do not differ at all with respect to the > definition of the potentially infinite. Then it must be even easier for you to rephrase it in your own words. F. N. -- xyz
From: Virgil on 11 Jan 2007 18:09 In article <1168552014.606918.26860(a)i39g2000hsf.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1168511880.370120.180940(a)p59g2000hsd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Here is the formal proof: > > > > > > Theorem. The set of real numbers in [0, 1] is countable. > > > > Your proof was neither formal nor valid. > > > > Among other things you invalidly assume an infinite the union of finite > > sets must contain an infinite object, rather than merely containing > > infinitely many finite objects. > > An infinite union of finite sets is an infinite object. Not necessarily, as infinite union of points can be a finite circle. If you mean 'infinite set' rather than 'infinite object', you are closer to the mark.
From: Franziska Neugebauer on 11 Jan 2007 18:10
mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> Virgil wrote: >> >> > In article <1168511743.391940.58070(a)77g2000hsv.googlegroups.com>, >> > mueckenh(a)rz.fh-augsburg.de wrote: >> [...] >> >> Conclusion: Every finite binary tree contains a finite set of >> >> path. The countable union of finite sets is countable. The set of >> >> paths is countable. >> > >> > The set of /finite/ paths in the union is countable. >> > But when one takes the union of sets of finite paths one only gets >> > finite paths in that union. There are no infinite paths in that >> > union. >> > >> > The same thing happens with ordinals. When one takes the union of >> > all finite ordinals (like unary trees), there is no infinite >> > ordinal IN that union >> >> Absolutely right. > > But an infinite ordinal is that union! There is no infinite number (ordinal) _in_ that union. Rephrased: There is no infinite _member_ in that union. F. N. -- xyz |