Cantor Confusion
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From: Virgil on 24 Oct 2006 14:57 In article <1161683454.085728.22020(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Therefore we write the limit. The limit of all n is omega. By what definition of "limit"? > But the *facts* are based on the number of transactions. The trick with > the vase is simply to force the infinite number of transaction into a > finite time interval. The only relevant question is "According to the rules set up in the vase problem, is each ball which is inserted into the vase before noon also removed from the vase before noon?" An affirmative answer confirms that the vase is empty at noon. A negative answer directly violates the conditions of the problem. How does "Mueckenh" answer?
From: Virgil on 24 Oct 2006 15:20 In article <1161684584.835474.311080(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > jpalecek(a)web.de schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > No. That does it not mean. Unless you agree that no real numbers do > > > exist. > > > > Come on, real numbers do exist. > > Where? The same place that naturals, ordinals and rationals exist, in the mind. And nowhere else. > > You cannot take the limit. By taking the limit, you say "All paths with > > however large, but FINITE length happen to behave somehow uniformly". > > Absolutely not. All finite segments lead to a sum of less than 2 edges. A that "sum" is irrelevant to the issue of bijections between the set of edges and the set of paths in an infinite binary tree, why does "Mueckenh"keep harping on it? It must be because he knows the is wrong. > > > There is nothing about infinite paths in the statement. Or otherwise, > > if there is a positive sum, there should be at least one positive term. > > What we need is the number of terms. We know it is omega. Therefore we > can sum up the geometric series. The aleph_0 terms 1 + 1/2 + 1/4 + ... > yield exactly the sum 2. > > 1 + 1/2 + 1/4 + ... = ... + 1/4 + 1/2 + 1. > > You know that in absolutely converging series infinitely many terms can > be replaced without changing the sum? WRONG! No term can be replaced by different term but the terms may be rearranged in the sense that if the values of f:N -> R are the terms of any absolutely convergent series and g:N <--> N is any bijection then the composite function n -> f(g(n)) also represents an absolutely convergent series with the same limit. > It is not important where we > start or end. It is important that we start with logic, which "Mueckenh" fails to do. > > Or what about recursion? The square root of 2 can be calculated by the > recursion f(n+1) = f(n)/2 + 1/f(n) > > For infinite paths we have the number of edges given by the recursion > f(n+1) = 1 + f(n)/2 with f(1) = 1 and n -> oo. The limit is 2, even for > arbitrary f(1). A very stable recursion. But it does not give the "number" of endless paths in an infinite binary tree. Each edge, being a left or right branching from a node, can be uniquely identified by a finite string of left and right branchings from the root node leading to and including it as the last branching. Each infinite path can, in a similar way, be uniquely identified with an infinite string of left and right branchings from the root node. Unless "Mueckenh"can show a surjection from the set of finite strings of {L,R} to the set of infinite strings of {L,R}, he fails.
From: Virgil on 24 Oct 2006 15:26 In article <1161685159.790898.5360(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Sebastian Holzmann schrieb: > .... > > Since for every finite subset of the set of \phi_n, there exists a model > > of ZFC - AoI where this subset is true (any model of ZFC - AoI should do > > the trick), by compactness, there exists a model of ZFC - AoI where > > _all_ \phi_n are simultaneously true. ("True" might not be the correct > > English word for it, please do someone correct me in this.) A set that > > has, for any n, at least n elements cannot be finite. Therefore it must > > be infinite. > > Further in this case it would be nonsense to introduce the axiom of > infinity. There is nothing in the logic of axiom systems that requires absence of redundancy in a set of axioms. It is considered to be moderately desirable, but is not at all considered to be necessary.
From: Lester Zick on 24 Oct 2006 15:31 On 23 Oct 2006 21:22:10 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: > >Lester Zick wrote: >> On 23 Oct 2006 08:48:07 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> >> wrote: >> >> > >> >Lester Zick wrote: >> >> On Mon, 23 Oct 2006 02:00:06 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> >> >> wrote: >> >> >> >> >In article <1161518242.756958.103660(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: >> >> >> >> [. . .] >> >> >> >> > Virgils definition, although not wrong, definitions >> >> >can not be wrong, does not lead to desirable results. >> >> >> >> Since when can definitions not be wrong? >> > >> >A definition is a statement by a person that a certain >> >symbol will be used by them to stand for some concept. >> > >> >What is there in such a statement that can be wrong? >> >> The concept? >> >> >Example: I will use the term "gleeb" to refer to an integer >> >which is divisible by 2. >> > >> >How can that statement be wrong? How would you >> >define "wrong" for such a statement? >> >> Self contradictory predicates defining the concept in the definition. >> Ex: "squircles are square circles" "x is an even, odd" "gleeb is a >> finite integer divisible by 0". > >All of those are perfectly valid definitions. Just because it >doesn't exist doesn't mean the concept can't have a name. Self contradictions can have names. They're just false definitions unless of course you want to argue that self contradictions can be true. As for validity I'd appreciate it if you could explain the difference between "valid" "correct" and "true". To me it would appear to be a distinction without a difference and the various terms would only be used to sanctify different kinds of subjects, definitions, concepts, etc. However then I would have to ask what designates a definition as distinct from a concept since both employ sequences of predicates. ~v~~
From: Lester Zick on 24 Oct 2006 15:32
On Tue, 24 Oct 2006 10:35:39 +0000 (UTC), Sebastian Holzmann <SHolzmann(a)gmx.de> wrote: >mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: >> >> Example: I will use the term "gleeb" to refer to an integer >> which is divisible by pi. >> >> This statement is wrong. And it is nonsense. And don't come up with the >> empty set. > >This statement is not wrong. The statement "There exists a gleeb" would >be wrong. Aren't you allowed to speak about non-existing entities? Of course you're allowed to speak of self contradictory subjects. They're just not true unless of course you want to argue that self contradictions can be true. ~v~~ |