Cantor Confusion
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From: Virgil on 26 Jan 2007 18:24 In article <1169823828.864775.196160(a)s48g2000cws.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > I defined level n and defined the bijection of levels and natural > numbers. That is enough. That claim is false. > > > > > It is irrelevant for the consideraton of trees. > > > > It is highly relevant to show where your reasoning is malfunctioning. > > No. You only try to huddle around, avoiding any concrete discussion. On the contrary, it is WM who keeps avoiding the concrete. > You cannot make me believe that your intelligence is insufficient to > understand this discussion. It is insufficient to allow him, or anyone to understand completely what is incomplete. > > > > > In particular we can prove by induction that the tree, formally called > > > T(oo), > > > > 1. Define T(oo). > > Has been done. > > > 2. Prove that the "formally called" entity T(oo) is really a tree. > > T(oo) is a tree by definition. Not by any mathematical definition, since it has no explicit set of edges. > > Look at my tree. Where is it? We do not see it here! > > Define "ens[e]mble of all natural numbers". > > > With pleasure (though it is not simple). > > The "ensemble of all natural numbers" is a somehing which cannot be > named without raising wrong impressions. If there is a thing there at all it can be named, unless WM attributes some magic to the process of naming of things so that, like the Hebrew "name" of God, it cannot be expressed. > We can only describe it by its > effect. If you have this entity, then you have every natural number. If > you loose this entity, then you loose nothing but natural numbers. Let us call this entity Peano. Does Peano contain all the natural numbers? Does Peano have to wave his wand for a natural number to come into being? If Peano disappears, do all the natural numbers disappear with him/her/it? If there can be "every natural number" what prevents the existence of a set of all natural numbers? Did Peano cast a spell to prevent such set from existing?
From: MoeBlee on 26 Jan 2007 18:42 On Jan 26, 3:40 pm, "Gerard Schildberger" <Gerar...(a)rrt.net> wrote: > major events (World War I, ... II) "..." between I and II ? What World War came between World War I and World War II ? Was that World War I I/II ? MoeBlee
From: Gerard Schildberger on 26 Jan 2007 18:55 | MoeBlee wrote: |> Gerard Schildberger wrote: |> major events (World War I, ... II) | "..." between I and II ? | | What World War came between World War I and World War II ? | | Was that World War I I/II ? No, My use of the elipses was to indicate missing words, in this case: (World War I, World War II). I thought was obvious, but one gets fooled all the time. ____Gerard S.
From: mueckenh on 27 Jan 2007 08:33 On 25 Jan., 23:33, Virgil <vir...(a)comcast.net> wrote: > Induction can possibly > prove that all the members of V* have some property, but can prove > nothing about V* itself. We can boil down the discussion about trees to the following simple question, considering only one path, for instance the path p on the outmost left hand side of the tree. This path p (in terms of nodes) is the union of all paths of finite trees with length n, n in N. Therefore all the path-*lengths* in the union are natural numbers. Notwithstanding the question whether there are infinitely many paths in the union or not: If the union path p is infinite, then at least one of the paths in the union must be infinite. Is this so? > > All I require is that a property holds (or not) for all the natural > > numbers or all the levels or nodes or edges of trees which can be > > enumerated by natural numbers. > Then you best conclusion is that it holds for all natural numbers but > not that it holds for N. Precisely that is what I want and what I do! Regards, WM
From: mueckenh on 27 Jan 2007 08:40
On 26 Jan., 02:21, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote:> > A path-length which has no upper bound may be sufficent to be called > > infinite. > > Sorry, a path-length is something fixed, so the wording is inadequate. > But if you call a path-length infinite you do not satisty (1), because > there you associate path-lengths with natural numbers. We can boil down the discussion about trees to the following simple question, considering only one path, for instance the path p on the outmost left hand side of the tree. This path p (in terms of nodes) is the union of all paths of finite trees with length n, n in N. Therefore all the path-*lengths* in the union are natural numbers. Notwithstanding the question whether there are infinitely many paths in the union or not: If the union path p is infinite, then at least one of the paths in the union must be infinite. Is this so? > > Bu it shows that the *set* of paths of the union is a proper subset of > > the union of all *sets* of paths. > > And, again, that is fundamentally wrong. It does not show anything of > that kind. The union of all sets of paths from finite trees contains > only finite paths. That is pretty basic set theory. If none of the > sets used in the union contains an infinite path, there is also not an > infinite path in their union. That is set theory. But it is wrong. The union of all finite paths of length n is infinite, but not actually. It has no upper bound. > > That is what I proved. In the union of the *sets* of paths in the > > finite trees there is no infinite path. > > And that is what I am stating all along, but you are arguing against. I know that it is correct. But it implies that the union of all finite paths is not infinite. This requirement can only be met by potential infinity. > > > Hence the subset which is in > > T(oo) also does not contain an infinite path. > > Indeed. That subset is empty. Because by your definitions *none* of the > finite paths is a path in T(oo). Indeed, also, *none* of the paths in > some T(m) is a path in T(n) when n > m. But, also by your definitions, > T(oo) does contain paths. Or do you now claim that T(oo) does not contain > paths at all? I claim that there is no actual infinity. T(oo) is a potentially infinite tree (i.e. finite without an upper bound). > This reasoning > is *exactly* the same as stating that N is finite. Correct! N is a potentially infinite set. No other form of infinity is possible, as we observe in T(oo). > P_C is the set of finite sets P_C. You mean finite sets P(k), I assume. > Now P_C is indeed a countable set. Its elements are P(1), P(2), etc. We can form the union of all P(k) to get the set of all paths in all finite trees. > A subset of P_C is something like { P(1), P(2) } with as elements > *sets* of paths. However, P is a set with as elements paths. So P > can *not* be a subset of P_C. It is a subset of the union set P(1) U P(2) U ... Regards, WM |