Cantor Confusion
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From: Dik T. Winter on 23 Nov 2006 20:39 In article <1164281187.964067.115190(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > Cantor wrote: "It is remarkable that removing a countable set leaves > the plaine *being connected*." But it is not remarkable that removing > an uncountable set leaves the plaine being connected? What a foolish > assertion. But removing an uncountable set can leave the plane either connected or disconnected. What is remarkable about that? Remove the line x=0 from the plane. You remove an uncountable set and the result is clearly disconnected. What is remarkable is that when you remove a countable set the result is *always* connected. So to get an disconnected set you *must* remove an uncountable set, but not any uncountable set will do. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 23 Nov 2006 20:53 In article <1164126272.884271.43260(a)m7g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > I + I = II is very fine and reliable mathematics. Absolutely true. And > > > this approach can be put forward --- very far. > > > > What do you mean with those symbols? How do you define "+" and "="? > > What is the meaning of "I" and "II"? > > What do you mean with "There" or "exists" or "a"? > How do you define "set" and "which" and "has"? > What is the meaning of "no" and "elements"? > > I think these symbols and expressions deserve closer examination than > "+" and "II". Oh, perhaps. But now you are entering the field of philosphy end exiting the field of mathematics. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 24 Nov 2006 00:49 In article <1164308217.013182.144230(a)l12g2000cwl.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1164198530.509417.22070(a)b28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Virgil schrieb: > > > > > > > In article <1164143519.034139.154960(a)h54g2000cwb.googlegroups.com>, > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > Randy Poe schrieb: > > > > > > > > > > > > > > > > What about all contructible numbers or all words? They cannot be > > > > > > > mapped > > > > > > > on N though they are a countable set. > > > > > > > > > > > > The set of all finite words, the set of polynomials, or any > > > > > > countable set can be put in bijection with N. Where do you > > > > > > get your view that it "can't be mapped to N"? > > > > > > > > > > The list of all finite words cannot be constructed. > > > > > > > > It can be constructed inductively, which means that in ZF or NBG, it can > > > > be constructed. > > > > > > The natural numbers cannot be constructed inductively. Induction proofs > > > do not cover N, I was told. > > > > Then you were told wrong. > > Induction yields infinitely many numbers? Induction requires a set like the infinite set of finite naturals as found in ZF or NBG, for example. > So you can do by induction infinitely many steps? Two steps. > So, if you would count these steps, you would get an infinite number? These two steps together with the axiom of infinity and som other bits of ZF or NBG, gives you completion for an infinite set. > But if you would count the differences 1, you would not get an > infinite number? One doesn't have to in induction. > > > > > > The list of all > > > > > constructible numbers cannot be constructed. > > > > > > > > > > > > The list of all naturals can be constructed inductively, which means > > > > that in ZF or NBG, it can be constructed. > > > > > > It can be proven inductively that all initial segments of the set of > > > natural numbers are finite. So, if the induction proof concerns all n > > > in N, then N is finite. > > > > WM claims that a quality of members of a set must be inherited by the > > set itself. All that WM can prove is that every MEMBER of N is finite, > > which says nothing about N itself. > > I can prove that every step generating the next member n+1 from the > member n belongs to a finite set of steps. That at most only shows finite members, but does not require a finite set. > > The set of steps is finite and remains so for all n. That at most only shows finite members, but does not require a finite set. > > > > Those whose "feelings" tell them that they are Napolean Buonaparte have > > feelings as relevant to the structure of mathematics as EB's. > > You prefer to be Julius Caesar? You can have that one too. I prefer to be Virgil.
From: Virgil on 24 Nov 2006 00:59 In article <1164308389.620893.220340(a)l39g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1164197100.642614.53260(a)e3g2000cwe.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > MoeBlee schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > The countability of the set W of all the words of a finite alphabet is > > > > > proved by other means than the usual countability proof. The latter > > > > > consists in constructing a bijection with N, i.e., in constructing a > > > > > list. But it is impossible to construct a list of all words. It is > > > > > impossible to construct a bijection between W and N. > > > > > > > > WRONG. It is a theorem that from the set of finite sequences of members > > > > of a finite alphabet there is an injection into N. > > > > > > Nevertheless it is impossible to construct such an injection (as it s > > > impossible to construct a well order of the reals). > > > > Actually, it is quite trivial, Give each character in the alphabet of n > > characters a different integer value between 0 and n-1 inclusive, and > > represent the members of N in base n, and the bijection associates each > > word with the numeral, base n. > > By induction, at least in ZF or NBG or any system allowing induction, > > the construction is complete! > > Induction yields infinitely many numbers? In ZF there is a set, which we may call N, which (1) has {} as a member, and (2) for each x which is a member has union {x} as a member, and (3) is a subset of every set having properties (1) and (2). The principle of induction says that for any subset S of such an N, if S satisfies properties (1) and (2) then S = N. > > As far as I know, induction shows that you always have done a finite > number of steps. Then you don't know much. > By induction one can even prove that always a finite number of words > has been mapped on the natural numbers. By induction one can also map an infinite set of words onto the natural numbers. > > Now, after you have "completed" this finite list of words, take it and > construct a diagonal number. I prefer to take the infinite set of words. > This is always a finite number, as the list is finite. Maybe yours is, mine isn't. > Fine. > So we have proved that there is a finite word which was missing in the > "complete" list. You haven't proved anything, except your own incompetence at logic. > > I know what you think. Stay in your world. Stay in yours, which will exclude you from both expeditions into mathematics and expeditions into logic. > > Regards, WM
From: Eckard Blumschein on 24 Nov 2006 01:36
On 11/24/2006 2:25 AM, Dik T. Winter wrote: > > Mathematicians like you are hopefully aware of the trifle that the > > relation >= cannot be applied to the really real numbers. > > Oh. What are "really real numbers"? Those, like the imagined numerical solution to the task pi, which are just fictions. Those which are assumed as basis for DA2, > What you are missing is that mathematics provides an idealisation of > arithmetic (amongst others), and from that background provides > processes to do "real" calculations to get results in (amongst others) > the physical world. The whole field of numerical mathematics could > barely have been build without the idealised mathematical background. No no. I do not deny that numbers can come as close as you like to the ideal concept of infinity and continuum. Rational numbers already do that job. Regards, Eckard Blumschein |