Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Lester Zick on 14 Nov 2006 17:31 On 14 Nov 2006 03:47:17 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > >Lester Zick schrieb: >are. >> > >> >Try to construct as many numbers as you can using only 100 bits. Then >> >increase to 10^10 bits, then increase to 10^100 bits. More is not >> >available. I find this very convincing. >> >> I don't. It's a problematic argument at best. Based once again on a >> hypothetical finitude of the "physical" universe whatever that means. > >Here I cannot understand you. The accessible universe is finite, This is not true or at best problematic. I don't consider universe finite even though others do. But regardless I don't consider the argument relevant to the issue of numerical size or dimensions. >allowing for not more than 10^100 bits (a closer estimation would be >10^205, but that is irrelevant). Now, to express a number requires at >least one bit. What more is needed to see? The issue seems to be whether reduction to a finite number of bits or whatever is determinate of the number. Let's assume we have only room for 2 bits or 4 bits or whatever. Is that determinate of the numbering capacity of our thoughts and numbers represented in our thoughts? I don't think so. I see infinitesimal subdivision able to express itself to any degree necessary for the computation of relationships between infinitesimal ratios. In other words instead of extending out numbers infinitely all we're doing is subdividing unity. And this requires no further finite space than unity regardless of precision or extension. There is a credible argument related to infinity here when people try to claim that infinity comes into play beyond the natural numbers. Then any finitude to the universe would be a relevant consideration. But if we never venture beyond the natural numbers then we never confront that kind of infinity. And we can always state the concept with exactitude whether computational limits are realizable or not. ~v~~
From: Virgil on 14 Nov 2006 19:32 In article <1163503467.288576.181580(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > >> mueckenh(a)rz.fh-augsburg.de wrote: > > >> > Franziska Neugebauer schrieb: > > >> >> mueckenh(a)rz.fh-augsburg.de wrote: > > >> >> > Franziska Neugebauer schrieb: > > >> >> [...] > > >> >> >> Are there really three vertices in WM's "triangle"? > > >> >> > If finished infinities [...] > > >> >> Verbiage. > > >> > Yes. But, sorry to see, it is the fundament of modern mathematics. > > >> "Finished infinities" is your wording. > > > Precisely describing the fundament of modern mathematics. > > > > Cbjre Bs Oryvrs. > > Always insisting on having the last word? > No more than you are doing.
From: Virgil on 14 Nov 2006 19:36 In article <1163503932.205665.318490(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1163425753.113699.136190(a)b28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > >Everyone is > > > > welcome to choose their own axioms. > > > > > > That's mathematics? > > > > If one chooses a set of axioms no one else is interested in it is pretty > > much useless mathematics, but still mathematics. > > But to reject an axiom is philosophy? No. Merely lack of interest. > > > > > > Any definition of a "finished" anything which is based on the word > > "finished" itself, is circular at best. > > I did not intend to define it but explain that this expression is used > in current mathematics. Not by mathematicians, by and large, but only by those hanging on the periphery. > > > Do you claim to teach mathematics at your school? If so, at what level > > and with what content. > > No, I do not claim it, I simply do it. > Analysis I + II, Algebra I + II, for first and second semester. > History of mathematics, and history of the infinite for students of all > semesters. Poor students! > > > If you do not, then whatever you do claim to teach is irrelevant. > > > > Are any of your textbooks primarily mathematics texts? If not, what is > > their primary subject matter, not that it does matter. > > Primarily mathematical textbooks: > Algebra und Geometrie > Differential- und Integralrechnung > Kleine Geschichte der Mathematik > Die Geschichte des Unendlichen > And see my new book to appear this year: > Die Mathematik des Unendlichen (? 28.50) Poor students! > > Regards, WM
From: Virgil on 14 Nov 2006 19:43 In article <1163504232.149095.222920(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > While infinite collections in any physical sense are not possible, why > > are imaginary infinities, such as sets of numbers must be, unimaginable? > > You cannot imagine the set (that would require infinitely many > neurons), but you imagine a typical number with its properties and the > envelope. (That's why you think a set is more than the collection of > its elements.) WM may think that, but that is exactly what I do NOT think about a set. > > > > How is an imagined set with no members any more actual that an imagined > > set with every natural number as a member? To me they are both equally > > figments of the imagination. > > You are right. A set with no members is a squared nonsense. It is > convenient to assume its existence for formal reasons, but it does not > exist, as Cantor admits. If a paper sack with one apple in it can exist, why cannot an empty paper sack exist. If containers with objects in then can exist what prevents containers with no objects in them from existing? > > Cantor: It is useful to introduce a symbol which expresses the absence > of points. O means, that the set has no point, i.e., strictly speaking > it not present at all. (Es ist ferner zweckm??ig, ein Zeichen zu > haben, welches die Abwesenheit von Punkten ausdr?ckt, wir w?hlen dazu > den Buchstaben O; P == O bedeutet also, da? die Menge P keinen > einzigen Punkt enth?lt, also streng genommen als solche gar nicht > vorhanden ist.) > > > How is an imagined horse any more real that an imagined unicorn? > > > An imagined unicorn is real in your brain, at least those parts which > you can imagine. You will not be able to imagine the contents of its > stomach or all its hairs (Unicorns have hairs, because black holes have > hairs and unicorns live in lack holes.) But I am sure, you are not able > to imagine infinitely many unicorns. I can imagine infinitely many of them quite as easily as one of them. > > > I can more easily imagine an infinite set of finite naturals than one > > which must contain any infinite natural. > > You cannot imagine all the naturals. The envelope is without value. The value of an envelope is its contents, and I can, and have, imagined the set of all finite naturals. > > > > I cannot even imagine what an infinite natural (as distinct from an > > infinite ordinal) would be like. For such fantastic creations one should > > appeal to such as TO. > > No infinite number exists and, of course, no infinite natural exists. Depending on how one defines "number" one can quite easily have infinite numbers. For ordinal numbers, for example, it is quite easy, even necessary, to have infinite ordinals.
From: Virgil on 14 Nov 2006 19:46
In article <1163504325.434688.200920(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1163430459.318473.317960(a)i42g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > > If you add only one element to each column, you get the order type > > > omega + 1 for the length of he matrix. > > > > The "length", being a cardinality rather than an ordinality, is > > unaffected, since Card(omega) = Card(omega+1) > > The diagonal must have an order type. It must have the order type omega > and the order type omega + 1 simultaneously, because it maps lines on > columns. It maps from subset of the Cartesian product of the set of lines with the set of columns to the set of digits. Which is quite a different thing. |