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From: Eckard Blumschein on 13 Apr 2005 12:40 On 4/13/2005 2:10 PM, David Kastrup wrote: > You are babbling pseudophilosophical sophistic hogwash. That is simly > irrelevant to the math. Please indicate if you will have anything to contribute. Experts do not use this language. > Cantor showed a strict ordering of surjectability... Sorry, the last word seems to be your own creation. Just google for it and adapt to understandable terminology. > Cardinalities are just names for a relative > ordering of sets based on surjections. As far as I understood the criterion for two sets to have the same property to be equinumerous = equipollent = equivalent = having the same cardinality is said to be bijection, i.e. surjection (onto) and also injection (1 to 1). >> Cantor misinterpreted it by claiming that there are more real numbers >> than his list contained. > > Nonsense. Cantor never made such a list. At first he assumed that all reals are represented in his list. Then he showed that at least one number is not contained in his list. > And he actually _proved_ that no such list can be created. Yes. However he interpreted this the wrong way round: He did not correctly conclude that the reals can not at all be completely listed but imagined his list like a measure of infinity and suggested that there are more than infinitely many reals. E.
From: Will Twentyman on 13 Apr 2005 12:33 Eckard Blumschein wrote: > On 4/12/2005 9:03 PM, Matt Gutting wrote: > >>I'm not sure what you mean by "Convergency invites to restrict to a finite >>number of coefficients". Do you mean that to say "this sequence converges >>to the real number r" is to say that "r can be represented as a number >>which begins with the digits of one of the elements of this sequence"? >>That is true. However, what r *is* and what r is *approximated by* are two >>different things, and mathematicians keep that fact in mind. In this sense, >>the real numbers are not fictitious. > > The real numbers exist mathematically in the sense they are fictions. Funny, they can also be precisely defined several ways. They may not have a counterpart in the real world, but they certainly exist in the mathematical sense of being precisely defined. >>Perhaps I misinterpreted what you meant by saying "A part of mathematics would >>go slippery..." Would you mind explaining that? > > > Just an example. Children at school must not be taught Cantor's nonsense > infinite whole numbers. There are no infinite numbers. I would never suggest teaching children anything involving infinity until they are ready for a calculus class. What makes you think we would want to deal with such a tricky subject at that age? -- Will Twentyman email: wtwentyman at copper dot net
From: Eckard Blumschein on 13 Apr 2005 12:53 On 4/13/2005 2:22 PM, Arthur Fischer wrote: > Eckard: > > Just out of curiosity, could you provide your definitions for the > following concepts: > > - finite set > - infinite set > - countable set > - uncountable set > - non-countable set > - enlarging a set > > > Of course, mathematically precise definitions would be preferable, and > dictionaries do not, in general, provide for such definitions. Regrettably my time is limited. The pertaining definitions are easily available except for the last one. I would just like to try and comment on "enlarging a set" in case of infinite sets: This simply does not work. What about uncountable and non-countable I considered this two variants of translation to "ueberabzaehlbar". Fortunately, I very rarely found "over-countable". E.
From: Will Twentyman on 13 Apr 2005 12:41 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. And if the precision is not elusive, but right there in their grasp? -- Will Twentyman email: wtwentyman at copper dot net
From: Eckard Blumschein on 13 Apr 2005 12:57
On 4/13/2005 2:53 PM, Dik T. Winter wrote: > In article <425D0A7B.50309(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > ... > > 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. > > Isn't sqrt(1), sqrt(2), sqrt(3), sqrt(4), ... a list of real numbers? Of course I meant a complete list of real numbers regardless on what base: e.g. decimal or hexadecimal E. |