Cantor Confusion
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From: mueckenh on 23 Nov 2006 14:02 William Hughes schrieb: > > > Therefore we can define bijections between potentially > > > infinite sets. > > > > We can never prove hat a bijection fails, like in Cantor's argument. > Real numbers are represented by potentially infinite sequences. Wrong. Without actual infinity, there are no irrational numbers. Think about it. > A list of reals is a function that takes an element of the potentially > infinite > set of natural numbers and returns a potentially infinite sequence. > The diagonal number is a potentially infinite sequence. Wrong. Ask your set therorist companions. > > Defintion. B is a g-subset of a potentially > infinite set A if B is a set or a potentially infinite > set and any element of B is an element of A. We can > now contruct potentially infinite power sets, and > the usual theory follows. > > You are confusing potentially infinite with computable. > While it is true that a computable set is either > finite or potentially infinite, it is not true that a potentially > infinite set is computable. Try to inform you. Cantor is a very competent source. A potentially infinite variable is a variable which is always finite but can grow without bound. A potentially infinite set does not exist. Regards, WM
From: mueckenh on 23 Nov 2006 14:04 William Hughes schrieb: > Eckard Blumschein wrote: > > Why do you not simply write instead: > > Something potentially infinite is countable because it is not thought to > > exhibit the impossible property to include all elements of something > > actually infinite? > > Because this is nonsense. By defininition a set A is countable > if and only if there is a bijection between A and the natural numbers. > Because of the way we have defined bijection and potentially > infinite set it does not matter at all whether or not A is actually > infinite when deciding if such a bijection exists. All you may have defined is actual infinity = potential infinity. A good example for a false definition. > "One cannot eat the cake and have it" is not much of an > argument. Do you have any other argument why you cannot > have bijections between potentially infinite sets. Because they are never complete. Try to inform you. Cantor is a very competent source. A potentially infinite variable is a variable which is always finite but can grow without bound. A potentially infinite set does not exist. Regards, WM
From: mueckenh on 23 Nov 2006 14:08 William Hughes schrieb: > There is no difference between > saying "a potentially infinite set exists" and saying "an > actually > infinite set exists". > > or not? If not, do you have anything even remotely relevent to say? > > > This semi-coherent ramble has nothing whatsoever to do with > the question of whether there are any real (as opposed to cosmetic) > differences between potentially infinite and actually infinite sets. Until then, no one envisioned the possibility that infinities come in different sizes, and moreover, mathematicians had no use for âactual infinity.â The arguments using infinity, including the Differential Calculus of Newton and Leibniz, do not require the use of infinite sets. [Thomas Jech, Set Theory Stanford.htm, Stanford Encyclopedia of Philosophy] [Brouwer] maintains that a veritable continuum which is not denumerable can be obtained as a medium of free development; that is to say, besides the points which exist (are ready) on account of their definition by laws, such as e, pi, etc. other points of the continuum are not ready but develop as so-called choice sequences. [Fraenkel, Abraham A., Bar-Hillel, Yehoshua, Levy, Azriel: Foundations of Set Theory, North Holland, Amsterdam (1984) p. 255] There are at least two different ways of looking at the numbers: as a completed infinity and as an incomplete infinity. [Edward Nelson: Completed versus incomplete infinity in arithmetic, Princeton] A viable and interesting alternative to regarding the numbers as a completed infinity, one that leads to great simplifications in some areas of mathematics and that has strong connections with problems of computational complexity. [Edward Nelson: Completed versus incomplete infinity in arithmetic, Princeton] Wenn man die verschiedenen Ansichten, welche sich in bezug auf unsern Gegenstand, das Aktual-Unendliche (im folgenden Kürze halber mit A.-U. bezeichnet), im Laufe der Geschichte geltend gemacht haben, übersichtlich gruppieren will, so bieten sich dazu mehrere Gesichtspunkte dar, von denen ich heute nur einen hervorheben möchte. Man kann nämlich das A.-U. in drei Hauptbeziehungen in Frage stellen: erstens, sofern es in Deo extramundano aeterno omnipotenti sive natura naturante, wo es das Absolute heiÃt, zweitens sofern es in concreto seu in natura naturata vorkommt, wo ich es Transfinitum nenne und drittens kann das A.-U. in abstracto in Frage gezogen werden, d. h. sofern es von der menschlichen Erkenntnis in Form von aktual-unendlichen, oder wie ich sie genannt habe, von transfiniten Zahlen oder in der noch allgemeineren Form der transfiniten Ordnungstypen ï¨ï¡ï²ï©ï±ïï¯ï©ï ï®ï¯ï¨ï´ï¯ï© oder ï¥ï©ï¤ï¨ï´ï©ï«ï¯ï©ï©ï aufgefaÃt werden könne. [G. Cantor, Gesammelte Anhandlungen, p. 372 ] Daà das sogenannte potentiale oder synkategorematische Unendliche (Indefinitum) zu keiner derartigen Einteilung Veranlassung gibt, hat darin seinen Grund, daà es ausschlieÃlich als Beziehungsbegriff, als Hilfsvorstellung unseres Denkens Bedeutung hat, für sich aber keine Idee bezeichnet; in jener Rolle hat es allerdings durch die von Leibniz und Newton erfundene Differential- und Integralrechnung seinen groÃen Wert als Erkenntnismittel und Instrument unseres Geistes bewiesen; eine weitergehende Bedeutung kann dasselbe nicht für sich in Anspruch nehmen. [G. Cantor, Gesammelte Anhandlungen, p. 373] Trotz wesentlicher Verschiedenheit der Begriffe des potentialen und aktualen Unendlichen, indem ersteres eine veränderliche endliche, über alle Grenzen hinaus wachsende GröÃe, letztere ein in sich festes, konstantes, jedoch jenseits aller endlichen GröÃen liegendes Quantum bedeutet, tritt doch leider nur zu oft der Fall ein, daà das eine mit dem andern verwechselt wird. [G. Cantor, Gesammelte Anhandlungen, p. 374] Summary: It is incoherent ramble to believe that the potential infinite yields order types or bijections. (free translation after Cantor) Regards, WM
From: mueckenh on 23 Nov 2006 14:10 Han de Bruijn schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > [ exposition of WM's binary tree snipped: just look it up ] > > > If you don't understand this simple and clear exposition, then there is > > no hope that you will be able to think any further. > > Affirmative. It's not a difficult argument altogether. Thank you, Han. Regards, WM
From: William Hughes on 23 Nov 2006 14:56
mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > Therefore we can define bijections between potentially > > > > infinite sets. > > > > > > We can never prove hat a bijection fails, like in Cantor's argument. > > > Real numbers are represented by potentially infinite sequences. > > Wrong. Without actual infinity, there are no irrational numbers. Think > about it. > Piffle. An irrational number is represented by a potentially infinite sequence that does not repeat. > > A list of reals is a function that takes an element of the potentially > > infinite > > set of natural numbers and returns a potentially infinite sequence. > > The diagonal number is a potentially infinite sequence. > > Wrong. Ask your set therorist companions. Piffle. How can a definition be wrong? > > > > Defintion. B is a g-subset of a potentially > > infinite set A if B is a set or a potentially infinite > > set and any element of B is an element of A. We can > > now contruct potentially infinite power sets, and > > the usual theory follows. > > > > You are confusing potentially infinite with computable. > > While it is true that a computable set is either > > finite or potentially infinite, it is not true that a potentially > > infinite set is computable. > > Try to inform you. Cantor is a very competent source. A potentially > infinite variable is a variable which is always finite but can grow > without bound. A potentially infinite set does not exist. > Piffle. Finite sets exist. A potentially infinite set is defined in terms of finite sets. (Since I am not using Cantor's definitions I fail to see the relevence). - William Hughes |