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From: Will Twentyman on 13 Apr 2005 11:27 Eckard Blumschein wrote: > On 4/12/2005 11:46 PM, Will Twentyman wrote: > > >>>Cantor was mislead by his intuition. >>>I do not attribute the difference between countable and non-countable to >>>the size of the both infinite sets. >> >>What do you view the difference between them to be? > > The difference resides in the property of each single real number > itself. Cantor assumed his list represents all real numbers. Actually, > nobody can provide any list of real numbers, not even two subsequent of > them can be named. Back up. First of all, real numbers are not a necessary part of the discussion regarding the difference between countable and non-countable. We could be comparing N to P(N) just as easily as N to R. Second, a single real number is not something that you can discuss countably issues with. Third, Cantor assumed the list precisely to prove it doesn't exist. He knew there was no list of real numbers. The assumption of such a list was the first step in a technique called "proof by contradiction". Fourth, the non-existence of such a list is the reason why the reals are uncountable. >>>Actually, infinity is not a quantity but a quality that cannot be >>>enlarged or exhausted. Whether or not an infinite set is countable >>>depends on its structure. The reals are obviously not countable because >>>one cannot even numerically approach/identify a single real number. >> >>No, that is NOT the reason the reals are not countable. > > > This was your statement. Where is your evidence for it or at least some > justification? Simply tell me the successor of pi. Definition: a set A is said to be countable if it is finite or if there exists a bijection between A and N. Theorem: R is not countable. Proof: assume R is countable, then there exists a bijection f mapping N to R. Consider a real number r specified as follows: if the nth decimal digit of f(n) is 5, the nth decimal digit of r is 1, otherwise the nth decimal digit of r is 5. r cannot be in the image of f, because for all n, f(n) differs from r at the nth decimal digit. This is a contradiction, therefor R is not countable. The successor of pi could be pi+1, but that is not relevant to the reals being uncountable. -- Will Twentyman email: wtwentyman at copper dot net
From: Will Twentyman on 13 Apr 2005 11:29 Matt Gutting wrote: > Will Twentyman wrote: > >> Don't worry, I think several of us had already figured all that out. >> I think Eckard's problem is simply that he doesn't understand the >> concept of definition or proof. Intuition may inspire a line of >> reasoning, but is never a substitute for proof. There seem to be some >> insightful responses to his nonsense, though. > > That's what keeps me reading, anyway. You have to have a blunt object to > sharpen your axe on. Or something like that. I hope to be one of the better responses. I consider it good practice, anyway. That's why I respond to people like Eckard and Herc... James if he posts some math. -- Will Twentyman email: wtwentyman at copper dot net
From: worldsofsolution on 13 Apr 2005 11:34 Are you saying the bijection between N and Q preserves the natural order of Q? Phrased another way, are you saying, if f is the bijection, and f(n) = 1/2 then f(n+1) is the successor of 1/2, in the sense that f(n+1) is not only greater than f(n) but f(n+1) is the next rational number after 1/2? How could that be? I've always understand it to mean that f(n)<f(n+1) but f(n+1) is not the next rational number after 1/2 which does not make sense.
From: Randy Poe on 13 Apr 2005 11:34 Eckard Blumschein wrote: > On 4/12/2005 10:59 PM, Will Twentyman wrote: > > >> IN, (Q: countable infinite > >> IR: non-countable infinite > >> > >> The reals are non-countable because of their structure that does not > >> allow to numerical approach/identify any real number. They are however > >> not of larger, equal, or smaller size as compared to the rational ones. > > > > Why would an engineer prefer less precision over more? > > Engineers contempt elusive precision. When I taught physics to engineering students it was a continuing struggle to explain to them why they should not copy down every digit their calculator provided when giving solutions to problems. - Randy
From: David Kastrup on 13 Apr 2005 11:35
Will Twentyman <wtwentyman(a)read.my.sig> writes: > Definition: a set A is said to be countable if it is finite or if > there exists a bijection between A and N. > > Theorem: R is not countable. > > Proof: assume R is countable, then there exists a bijection f mapping > N to R. Consider a real number r specified as follows: if the nth > decimal digit of f(n) is 5, the nth decimal digit of r is 1, otherwise > the nth decimal digit of r is 5. r cannot be in the image of f, > because for all n, f(n) differs from r at the nth decimal digit. This > is a contradiction, therefor R is not countable. > > The successor of pi could be pi+1, but that is not relevant to the > reals being uncountable. Of course it is. If I have a set and a mapping with the following properties: The set contains a dedicated member. For each of its members, the set contains the mapping of the member. Different members have different mappings. The dedicated member is not the mapping of any other member. A set containing the dedicated member, and containing for each of its members also its mapping, contains all members of the given set. So if I can find _any_ successor relation with the given properties, this is completely relevant to the reals being countable, since it puts them into a one-to-one correspondence with the naturals. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum |