Cantor Confusion
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From: William Hughes on 18 Oct 2006 07:33 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1161008572.469763.93200(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > jpalecek(a)web.de schrieb: > > ... > > > > 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. > > > > > > > The fact that you cannot compute a list of all reals does not mean that > > > there is no list of all reals. There is one, but it is not possible to > > > publish this list. > > > > You are seriously wrong. > > You should have noted that this was an ironic reply. But in order to > avoid machines and undecidabilities: > The set of constructible numbers is countable. [The terms "constructable" and "computable" do not mean the same thing. The word you want is "computable"] With the definitions you use, the set of computable numbers is not countable. - William Hughes
From: William Hughes on 18 Oct 2006 07:37 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Alan Morgan schrieb: > > > > > > > > > > >> As I have inductively gone through the entire list of balls introduced > > > > >> into the vase and found that each of them has been removed before noon, > > > > >> why should stating that trivial fact be considered a joke? > > > > > > > > > >But you cannot go inductively through the cardinal numbers of the sets > > > > >of balls in the vase? They are 9, 18, 27, ..., and, above all, we can > > > > >show inductively, that this function can never decrease. > > > > > > > > You think that's bad? I have an even simpler situation! Add one ball > > > > at 1 minute to noon, another ball at half a minute to noon, another > > > > at 1/4 minute to noon, and so on. The number of balls in the vase before > > > > noon is always finite, but somehow, miraculously, at noon the number of > > > > balls in the vase becomes infinite. When, oh when, does that transition > > > > from finite to infinite happen? > > > > > > > > I submit that this is just as wierd a result as the original problem. > > > > > > Weird is that adding 9 balls instead of 1 per transaction leads to zero > > > balls. > > > Weird is that taking off 1 ball per transaction leads to all balls > > > taken off and no ball remaining, if the enumeration is 1,2,3,... but to > > > infinitely many balls remaining, if the enumeration is 10, 20, 30, .. . > > > This in particular is weird because there is a simple bijection between > > > 1,2,3,... and 10, 20, 30, ... > > > > > > > Right. No matter which balls you pick you are going to remove an > > infinite > > number of balls. So the number of balls you remove does not matter. > > Which balls you remove does matter. > > Fine. The question is, however, is this set "all balls" or how many > balls remain in the vase at noon? As there is more than one way to remove an infinite set of balls (remove all balls, starting with 1, remove just the even balls, remove just the primes, remove just the multiples of 10 ...) the answer to your question depends on exaclty which balls are removed. So the set removed may or may not be the set "all balls". - William Hughes
From: William Hughes on 18 Oct 2006 07:40 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > mueck...(a)rz.fh-augsburg.de wrote: > > > > > William Hughes schrieb: > > > > > > > > > > > No. The fact that everything that is true about the infinite > > > > > > must be justified in the finite, does not mean that everything > > > > > > that can be justified in the finite must be true about the > > > > > > infinite. > > > > > > > > > > > > You prove that something is true in the finite case. You > > > > > > do not justify your transfer to the infinite case. > > > > > > > > > > Who has ever justified such a proof? In fact that is impossible because > > > > > there is no infinity. Therefore all such "proofs" are false. But if we > > > > > assume the existence of the infinite, then the sum of the geometric > > > > > series is the most reliable entity at all. (Niels Abel: With the > > > > > exception of the geometric series no series has ever been calculated > > > > > precisely.) > > > > > > > > If you wish to assme that infinity does not exist, knock yourself > > > > out. However, if you are trying to show that the assumption > > > > that infinity does exists leads to a contradiction you need > > > > to justify the proofs that you make using that assumption. > > > > > > > > > > > > > > > > > The axiom of infinity > > > > > > > applies to the paths. They are nothing but representations of real > > > > > > > numbers. These exist according to set theory, therefore the paths > > > > > > > exist too. > > > > > > > > > > > > > > > > > > > Yes, but the question is not "do the paths exist?". > > > > > > > > > > There are two questions: Do the infinite paths exist and does the > > > > > geometric series wit q = 1/2 have a limit? I don't need any further > > > > > infinities. > > > > > > > > > > > > > No, there is a third question: "What is the connection between the > > > > infinite paths and the limit of the series?" You have only shown a > > > > connection between finite paths and partial sums. > > > > > > Wrong. The connection between finite paths and partial sums of edges > > > leads to > > > (1-(1/2)^n+1)/(1 - 1/2) edges per path. > > > > > > > And this is the only connection you have ever shown. You then > > take a limit and get 2. But you have never shown that this > > limit is connected to anything. > > Each path consists of aleph_0 edges. > > > The fact that you have a connection > > in the fnite case is not enough to show that the same > > connection holds in the infinite case. > > So you disagree with lim (1/2)^(n+1) = 0 for n --> oo. (At least in > this special case.) No. The limit is correct. What I disagree with is the meaning you assign to this limit. - William Hughes
From: Dik T. Winter on 18 Oct 2006 10:14 In article <J7B4p3.ItG(a)cwi.nl> "Dik T. Winter" <Dik.Winter(a)cwi.nl> writes: > In article <eh2fe1$j3e$1(a)mailhub227.itcs.purdue.edu> Dave Seaman <dseaman(a)no.such.host> writes: > > On Tue, 17 Oct 2006 02:48:55 GMT, Dik T. Winter wrote: > ... > > > One of the most serious errors can he found in the statement that > > > "to count sets of first cardinality you need ordinals of the second > > > class" > > > > Are you sure you are quoting him correctly? Cantor did say (in fact, > > this is a section heading in boldface): > > It was in an article explaining transfinite "counting". And I am quite sure > I did quote him reasonably correct. But the book is on my desk at work, so > I will try to find it tomorrow. Gesammelte Abhandlungen, Hildesheim, 1962, p. 213: ... der Unterschied ist nur der, da?, w?hrend die Mengen erster M?chtigkeit nur durch (mit Hilfe von) Zahlen der zweiten Zahlenklasse abgez?hlt werden k?nnen, die Abz?hlung bei Mengen zweiter M?chtigkeit nur durch Zahlen der dritten Zahlenklasse, bei Mengen dritter M?chtigkeit nur durch Zahlen der vierten Zahlenklasse u. s. w. erfolgen kann. or translated: ... the difference is only that, while sets of the first cardinality can be counted only through (with the aid off) numbers of the second class, the counting of sets of the second cardinality only through numbers of the third class, with sets of the third cardinality only through numbers of the fourth class, etc. From: ?ber unendlichen lineare Punktmannigfaltigkeiten, Nr. 6, sec. 15. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 18 Oct 2006 10:36
Dik T. Winter schrieb: > In article <1161019210.489120.290720(a)e3g2000cwe.googlegroups.com> "MoeBlee" <jazzmobe(a)hotmail.com> writes: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > A good, if no the best source to learn about the different meanings of > > > infinity would be Cantor's collected works. > > > > Set theory has advanced since Cantor. The best source to learn about > > current set theoretic definitions of 'infinite' is not Cantor. > > Indeed. There are some statements in that work that are false according to > current set theory (as I already did notice before). The works are a good > starting point when you want to talk about the history of set theory, but > that is all. > > One of the most serious errors can he found in the statement that > "to count sets of first cardinality you need ordinals of the second > class" > where (I may have first and second wrong), first cardinality means (in > current terminology) aleph-0 and second class ordinals means (in current > terminology) omega and larger (until omega^omega or somesuch). The > error is of course that there is a set of cardinality aleph-0 that can > be counted using finite ordinals only (the natural numbers). Numbers of first numberclass are the finite (natural) numbers (0), 1, 2, 3, .... To count them, you need a number of the second class, aleph_0 or omega. > > It is exactly the same error that is present in Tony's presentations and in > many of Wolfgang's presentations. In spite of Wolfgang's protestations, > the contents of the set N represent a potential infinity while the size > represents an actual infinity. N = omega = aleph_0. How should one of them have different character? According to Cantor, set theory deals with actual infinity, not with potential. Die weite Reise, welche Herbart seiner "wandelbaren Grenze" vorschreibt, ist eingestandenermaßen nicht auf einen endlichen Weg beschränkt; so muß denn ihr Weg ein unendlicher, und zwar, weil er seinerseits nichts Wandelndes, sondern überall fest ist, ein aktualunendlicher Weg sein. Es fordert also jedes potentiale Unendliche (die wandelnde Grenze) ein Transfinitum (den sichern Weg zum Wandeln) und kann ohne letzteres nicht gedacht werden. My translation: Every potential infinity requires a transfinity and cannot be thought without. Da wir uns aber durch unsre Arbeiten der breiten Heerstraße des Transfiniten versichert, sie wohl fundiert und sorgsam gepflastert haben, so öffnen wir sie dem Verkehr und stellen sie als eiserne Grundlage, nutzbar allen Freunden des potentialen Unendlichen, im besonderen aber der wanderlustigen Herbartschen "Grenze" bereitwilligst zur Verfügung; gern und ruhig überlassen wir die rastlose der Eintönigkeit ihres durchaus nicht beneidenswerten Geschicks; wandle sie nur immer weiter, es wird ihr von nun an nie mehr der Boden unter den Füßen schwinden. Glück auf die Reise! Very fine polemic - unmatched. Cantor was a master in all respects! Regards, WM |