Cantor Confusion
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From: Virgil on 11 Jan 2007 15:02 In article <1168532635.566074.155260(a)k58g2000hse.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > > No. But the property of being the greatest line can and does change. > > > > > > > > Irrelevant. The question is whether L_D exists. > > > > > > It does. Does the tallest man exist? When did it start, when did it > > > cease? > > > > Recall: > > > > L_D is a line that contains every element that can > > be shown to be in the diagonal. > > > > > > The "tallest man" is something that can change. L_D is a line. > > A line cannot change. The analogy is not valid. > > L_D is the name of a line, like a championship title. And there are some championships in which no one holds a title. So there can be a name for a line which does not exist. > > > > L_D is a line. A line is fixed. L_D does not > > exist. > > It is the line containing the whole set. Which line does not exist. > > > > > > > > > Therefore you cannot assume actual infinity. > > > > > > > > You now admit that it is not possible to find L_D, > > > > > > In actual infinity (everything including L_D being fixed) it is not > > > possile to find L_D. > > > > And it is also not possible to find L_D in potential infinity. > > Let L_D go from 1 to oo. Then L_D is a variable with domain N.
From: Virgil on 11 Jan 2007 15:06 In article <1168532945.180245.116750(a)77g2000hsv.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > > Definition: Denote the nodes of the tree by > > > > > > > > > > (0,0) > > > > > (1,0) (1,1) > > > > > (2,0a) (2,1a) (2,0b) (2,1b) > > > > > ... > > > > > (n,0a) (n,1a) ... > > > > > > > > > > The union of all trees up to the n-levels tree is > > > > > > > > > > {(0,0)} U {(1,0), (1,1)} U .. U {(n,0a) (n,1a) ...} > > > > > > which is obviously the same as > > > > > > {(0,0)} U {(0,0), (1,0), (1,1)} U .. U {(0,0), (1,0), (1,1),..., > > > (n,0a) (n,1a) ...} > > > > > > End of definition. > > > > This presumes falsely, that every tree has the same nodes and edges and > > thus that all trees are subtrees of some ur-tree. > > But suppose I have a family of trees in which no two trees share any > > nodes or edges, what is the union of such a family? > > Suppose I have a set of initial segments {o}, {a,b}, {1,2,3}, in wich > no two segments share any element. What is the union of such a set? > Both, question and answer are completely irrelevant for the present > discussion. On the contrary, they point up a few of the many unspoken assumptions required by your arguments. > > > > > Books on set theory would not require the union of a set of disjoint > > trees to be a single tree. > > But would allow for the union of such trees with identical nodes. Only when such allowances are made explicit and described completely. > > > Then there isn't any one L_D, there are endlessly many of them, in fact > > an infinite sequence of them, for no sooner is one created than it is > > eclipsed by its successor. > > If you create with staying power. (Numbers are created, Dedekind said.) > > > But in ZFC and NBG, the completed diagonal exists > > alas, ZFC and NBG do not exist. In the world of mathematics, they have firmer existence that WM does.
From: Virgil on 11 Jan 2007 15:13 In article <1168534195.028560.224600(a)i39g2000hsf.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > 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. Perhaps WM's is, but nobody else need be hampered by the many strait jackets WM chooses to impose on himself.
From: Franziska Neugebauer on 11 Jan 2007 15:21 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. F. N. -- xyz
From: Virgil on 11 Jan 2007 15:26
In article <1168534541.442501.315540(a)k58g2000hse.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > >> >> > 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? > > >> > > >> My posts and my ideas do not take place *in* maths. They take place > > >> in the real _physical_ world. > > > > > > So does the contents of your posts and ideas, in part mathematics. > > > > This does not invalidate the fact that in mathematics there is no notion > > of time. > > A disadvantage or error which need not perpetuate in eternity (et > ultra). Triangles are eternal, like all other mathematical structures. It is only our attentions to them that are governed by time. > > >And hence in mathematics there are no temporal processes. This > > said "always" is not a intra-mathematically defined notion. > > There are mathematicians greater than you and any living set terrorists > which are convinced of the opposite. We who work with time-free mathematics do not fear the chronic depredations of you anti-set terrorists > > Time is important in mahematics. Time is important in physics, chemistry, engineering, etc., for a variety of reasons. That only makes it of marginal importance in mathematics in those areas of math used by those sciences and engineering. |