Cantor Confusion
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From: Virgil on 7 Feb 2007 13:42 In article <1170851785.038841.94140(a)h3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 6 Feb., 13:28, "William Hughes" <wpihug...(a)hotmail.com> wrote: > > > > > Let E be the sequence of the sets F(e_i). There are two statements. > > > > i. For each of the sets F(e_i) in the sequence E, there is > > an element of F(e_i) which is greater than the cardinal > > number of F(e_i) > > > > ii There an element of one of the F(e) that is greater > > than the cardinal number of E. > > > > i is true, ii is false, E is a counterexample. > > E is not an example, because it is self contradictory. How would that, if actually the case, prohibit its being a counterexample? > > The potentially infinite set of even numbers is *constructed* by its > segments > > {2,4,6,...,2n} Except that what is merely potential cannot be a set, at least according to any serious notion of how a set must behave. For something to be a set in any standard set theory, its membership must be totally determinate, not vague and ambiguous as WM would have it for his "potentially infinite sets" whose membership is so indeterminate. > > Every time we increase n by 1 we increase 2n by 2. This cannot be > avoided. Therefore it is impossible to have for finite natural numbers > > lim[n-->oo] |{2,4,6,...,2n}| > 2n The above makes no sense, as the same variable appears both bound and free. > > But there are no other than finite natural numbers. But more than finitely many of them. A fact which WM keeps overlooking. > > This simple truth Which is not a simple truth at all but a nonsense expression. AS for WM, mathematically speaking, the truth is not in him.
From: Virgil on 7 Feb 2007 13:56 In article <1170852009.005792.107360(a)p10g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 6 Feb., 13:47, Franziska Neugebauer <Franziska- > Neugeba...(a)neugeb.dnsalias.net> wrote: > > mueck...(a)rz.fh-augsburg.de wrote: > > > It can be proven. {p(0), p(1), q(1), p(2), q(2), r(2),s(2) ... } > > > contains {p(0), p(1), p(2), ... } > > > > What does contain mean? > > The tree is a set of nodes. It contains paths as subsets. > > The special meaning of "contain" is irrelevant Special meanings are the essence of mathematics. Those who so casually dismiss them are, in their hearts, anti-mathematicians. > as long as we refer to > a unique meaning Which will be precisely one of those "special meanings" that WM has just declared irrelevant! > > No. A paths *is* > > > > a) a sequence of nodes, or > > b) a sequence of edges. > > > Yes to a and b. But not every set of nodes or edges is a path. So? > > > > You have simply stated the M�ckenheim axiom > > > > X is not finite -> there must be an x in X which is infinite > > > > which is alas not part of ZFC or any other modern set theory. > > > I did not start off with this assumption. It has been a part of WM's system ab initio. I showed: > 1) The union of finite trees contains the union of finite paths. Except that WM's "unions" and mathematical "unions" are not at all the same thing. So that what WM means by his statement (1) is mathematically undefined. > 2) The union of finite trees is the whole tree. See above! "Union" is mathematically meaningless here. > 3) The whole tree contains all paths. If by "the whole tree" WM means a non-finite tree, then his whole tree does not contain any finite paths at all. > 4) The union of finite paths contains all paths. See above! "Union" is mathematically meaningless here. > 5) This union is countable. The set of all paths ( as maximal chains of linked nodes) of a non-finite complete binary tree has been repeatedly proved to be uncountable. So that WM is, as usual, trying to sneak his views past by using carefully undefined notions.
From: Virgil on 7 Feb 2007 14:09 In article <1170852164.478408.105110(a)h3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 6 Feb., 20:46, Virgil <vir...(a)comcast.net> wrote: > > > WM must be ghost ridden to see them where they are not. > > > > The union over {{2,3},{5,7}} is {2,3,5,7}, which is not a member of > > {{2,3},{5,7}}. > > Again such a ridiculous example. The example shows that the union over a set need not be a member of the set, so that when WM claims otherwise, he must support his claim with a valid proof. And WM has never supported any of his claims with valid proofs. > > The union over {{1}, {1,2}} is a member of {{1}, {1,2}} . WM has not, and cannot, establish that the union over a non-finite set of naturals must be a member of that set being unioned. > The union over all paths is a member of all paths. The WM-union over all paths is the set of all nodes in the tree, which is not a path at all. > > > > Which requires a path with no end. > > > > > No. > > > > Then name that alleged end of the union of all paths! > > I do not assert that acual infinity exists. Therefore I need not name > an actually infinite path. Either the required path has an end or does not have an end. Tertium non datur! So when WM denies that the path does not have an end, he is claiming that it does have an end. And is required to provide evidence for that claim or allow it to be rejected. > > > > > In particular because there cannot be such a path. > > > > In mathematics there can be such a path, regardless of WM's > > misrepresentations. > > > > > But if you > > > assume its existence, then you see that the reals are countable or > > > that identical nodes yield different path systems, an idea which > > > presumably only you can utter. > > > > I do not assume it, I prove it, > > It is nonsense and remains nonsense to assume that a given tree has > different path systems according to... according to what? That WM assumes nonsense does not obligate sensible people to accept it. > According to > your emotional condition? My "emotional" condition at least allows me to take logic into account, which WM's emotional condition does not. > And if you prove this nonsense then you have > found a contradiction in set theory. If you think so, the prove your claim. Except that WM has been, at least so far, incapable of providing a valid proof of anything.
From: Virgil on 7 Feb 2007 14:12 In article <1170852299.543791.43100(a)m58g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 6 Feb., 20:48, Virgil <vir...(a)comcast.net> wrote: > > In article <1170754974.135681.22...(a)p10g2000cwp.googlegroups.com>, > > > > > > The simplest reason is that omega - n = omega for all n in N. > > > > > > Where did you get that from? Reference? EB? > > > > > You could even figure it out by yourself. > > > > If one needs to adopt WM's crazy assumptions in order to come to WM's > > crazy conclusions, we are all better off without them. Both his > > assumptions and his conclusions > > Excuse me, these were Cantor's assumptions and conclusions. Not the crazy ones I am referring to. For instance, one of WM's crazy assumptions is that a set of naturals which is not finite must contain a natural which is not finite. Does WM claim that Cantor ever assumed or concluded that?
From: David Marcus on 7 Feb 2007 14:17
Han de Bruijn wrote: > William Hughes wrote: > > > And since the limit is not a finite number the fact that there > > is a contradiction for all finite numbers does not mean that > > there is a contradiction for the limit. > > In my calculus class, the fact that the "limit is not a finite number" > just always meant: the limit _does not exist_, i.e. there IS NO limit. > > Why does contemporary mathematics employ two measures for the infinite? > Why doesn't the left hand know what the right hand (calculus) is doing? Because in mathematics, conclusions depend on definitions. If you change the definition, you can get a different conclusion. It can be convenient to use different definitions in different contexts. I would have hoped you would have learned this by now. -- David Marcus |