Cantor Confusion
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From: mueckenh on 14 Oct 2006 16:27 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > Han de Bruijn wrote: > > > > Set Theory is simply not very useful. The main problem being that finite > > > > sets in your axiom system are STATIC. They can not grow. > > > > > > Set theory provides for capturing the notion of mathematical growth. > > > Sets don't grow, but growth is expressible in set theory. If there is a > > > mathematical notion that set theory cannot express, then please say > > > what it is. > > > > Obviously the notion of "rational relation" as used in the binary tree > > cannot be expressed by mathematical notion: > > Consider the binary tree which has (no finite paths but only) infinite > > paths representing the real numbers between 0 and 1. The edges (like a, > > b, and c below) connect the nodes, i.e., the binary digits. The set of > > edges is countable, because we can enumerate them > > > > 0. > > /a \ > > 0 1 > > /b \c / \ > > 0 1 0 1 > > ............. > > > > Now we set up a relation between paths and edges. Relate edge a to all > > paths which begin with 0.0. Relate edge b to all paths which begin with > > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > > a is inherited by all paths which begin with 0.00, the other half of > > edge a is inherited by all paths which begin with 0.01. > > So each finite path of length N is related to > > 1 + 1/2 +1/4 + ... + 1/2^N > > edges > > > Continuing in this manner in infinity, > > > we get a limit which may or may not be related to anything. This geometric sum is defined if any infinity is defined. > > > we see that every single infinite path is > > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related to any > > other path. > > No, the statement that what holds for finite paths also > holds for infinite paths needs proof. > Your provide none. Who provides, if not I? Operating with the infinite can only be justified by the finite" (Hilbert). That I do. 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. Aren't you a bit ashamed to defend such an ambiguous theory by such naive and obviously arbitrary arguments? Regards, WM
From: mueckenh on 14 Oct 2006 16:28 Randy Poe schrieb: > Han de Bruijn wrote: > > > I merely note that there is no requirement in the problem that > > > the limit be the value at noon. > > > > The limit at noon - iff it existed - would be the value at noon. > > Wrong. That is a flat out incorrect statement showing a > fundamental misunderstanding about what limits mean. > > A CONTINUOUS function at x0 has the property that the > limit of f(x) as x->x0 is f(x0). But not all functions are > continuous. And you are in charge of determining which functions are continuous and which are not? The function f(t) = 9t is continuous, because the function 1/9t is continuous. Regards, WM
From: Han.deBruijn on 14 Oct 2006 16:29 Virgil schreef: > In article <267fc$452f5def$82a1e228$15540(a)news1.tudelft.nl>, > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > Virgil wrote: > > > > > In article <b8869$452f4a39$82a1e228$32738(a)news2.tudelft.nl>, > > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > >> > > >>How about: I'm not interested in "the end". I don't know where the end > > >>is. And I don't care as well. As long as the end is somewhere where it > > >>causes a uncertainity which is acceptable for my purpose. > > > > > > But what if "the end" isn't anywhere because there isn't one? > > > > > > As soon as you posit an end, you run into problems. You would be much > > > better off saying that all such questions about an end to the naturals > > > are unanswerable, and stick to what you can explicitly construct. > > > > Both approaches run into problems. Either you accept infinities, either > > you accept a little bit of Physics: uncertainity and inexactness. Guess > > you know what my choice is. Guess I know what your choice is. > > I do not object to the constraints of the mathematics of physics when > doing physics, but why should I be so constrained when not doing physics? Because (empirical) physics is an absolute guarantee for consistency? Han de Bruijn
From: Virgil on 14 Oct 2006 16:29 In article <1160814180.593933.263730(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > My purpose was to explain to you why your unreflected assumption is > > > It is not > > > contradictory to say that in a finite set of numbers there need not be > > > a largest. It seems that this false assumption is one of the basic > > > reasons for set theory. > > > > With any common meaning of "numbers" short of complexes, it is > > contradictory in mathematics, whatever it may be in "Mueckenh"'s > > philosophy. > > While it may not be possible to determine which of that finite set of > > numbers is largest, there has to be one. > > Why? > Because a particle has to have a certain speed even if you cannot > determine it? > Or is there an axiom? > Which one is it? Because one cannot have a set whose membership is so vaguely indeterminate as to necessarily have an ordering but necessarily not have an ordering as well.
From: Virgil on 14 Oct 2006 16:31
In article <1160814355.022151.205320(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > Apply your knowledge to the balls of the vase. > > > > Which knowledge tells me that at noon each and every ball has been > > removed from the vase. > > You are joking? 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? |