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From: Matt Gutting on 12 Apr 2005 14:32 Eckard Blumschein wrote: > On 4/12/2005 1:59 PM, David Kastrup wrote: > > >> You consider Cantor at fault for the reals not being in >>one-to-one correspondence with the natural numbers > > > No. I consider him intending and deliberately drawing the wrong > conclusion from his second diagonal argument. To me it is clear that > there is no one-to-one correspondence between reals and rationals. > Cantor was mislead by his intuition. So he tried to show that there are > more reals as compared to the "size" of the set of the rationals. I > argue that such comparison lacks any basis. Infinity is a quality, not a > quantity. > > >>(as that is what >>having a transitive and grounded successor relationship all about) > > > Intitively nobody objects that there is an ascending order of the reals > too. The structural peculiarity of the reals is, however, such that one > can not even numerically identify any single real number, not to mention > a successor. > I'm not sure exactly what you mean by "numerically identify". If you mean that there is no symbolic representation for any real number, then you are quite obviously wrong, since e (for example) is a finitely long symbolic representation of a single, specific real number. If you mean that there is no finitely long decimal expansion for a real number, this is certainly true of some real numbers, but not all of them. And the same is true for rationals. (Besides which, any rational, and thus a considerable part of the reals, can be represented with a finite expansion given an appropriate base.) > > >>when that is exactly what he was showing in the first place. >> >> >>>As a summary, I am always missing the due humbleness. >> >>You are also missing a clue. > > > If you did not yet get the point you might look into M280. > > >>>Platonian thinking is perhaps more appropriate. One cannot force the >>>reals to have the same properties as exhibited by ordinary numbers. >> >>Congratulations. Exactly this, and nothing else, is what Cantor's >>diagonal proof was all about. > > > You are joking. No, he's not. Perhaps indeed you are missing the point. Matt > > >
From: Matt Gutting on 12 Apr 2005 15:03 Eckard Blumschein wrote: > On 4/12/2005 3:10 PM, Matt Gutting wrote: > > >>I'm not sure what you mean by "loss of approachable identity". The >>difference between the rationals and the reals is that every convergent >>sequence of reals converges to a real, while not every convergent >>sequence of rationals converges to a rational. > > > In other words, numerical representations of rational numbers do not > require infinitely many numerals. Convergency invites to restrict to a > finite number of coefficients. In that case you do not reach a real > number but are satisfied by an rational approximation instead. Real > numbers are fictitious. > Numerical representations of reals need not require infinitely many numerals either. And whether even a rational number requires an infinite number of digits to be represented depends on the method of representation chosen. 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. > >>One can easily construct the rationals, and the reals as well, in >>such a way that one needn't consider simpler number sets to be embedded >>in them. > > > Even embedded natural numbers cannot be numericall identified without > all infinitely many numerals e.g. 3,99999999999999999999999999999... I thought you just said that "numerical representations of rational numbers do not require infinitely many numerals"? Since 4 = 3.999999... is an integer and therefore a rational number, you appear to be contradicting yourself here. > > > >>>>It is possible to define >>>>"measure" to have meaning for the set of natural numbers. >>> >>> >>>Isn't a quantitative measure is a quantity to compare with? >>>Quantities can be expressed by means of rational numbers. >>>What rastional number does express the quantity of IN? >>> >> >>My point is that one can speak meaningfully of comparison of >>(in this instance) sets without requiring that each set be >>assigned a given quantity. I can compare the set A = {a,b,c} >>and the set B = {Nudel,Pudel,Strudel} by assigning a quantity to >>each, noting that each has three elements and concluding that they >>are the same size. I can also compare by seeing whether I can find >>(i) an element in A for every element in B, (ii) an element in B for >>every element in A, or (iii) both. In this case, (iii) is true, and >>I conclude that A and B are the same size, as before. > > > Did you refer to infinite sets? > > No, but neither did you. You were speaking here, as I understood it, simply of how one compares one set to another ("Isn't a quantitative measure a quantity to compare with?"). You made no mention, when speaking of quantitative measures, about whether you were comparing two finite sets or two infinite sets. My point is that one can describe a method of comparison between sets which (i) works for infinite sets exactly the way it does for finite sets, (ii) gives meaningful, well-defined comparisons for any two sets, (iii) does not refer to "quantity" in any way, and (iv) yields exactly the same results as a quantitative comparison does for those sets to which "quantitative comparison" applies. Having found such a comparison, I see no reason not to use it for infinite as well as finite sets. > > >>It doesn't. Why should it? > > > Mabe you should try a dictionary for science and technilogy. > The dictionary I cited makes specific reference to definitions of words used in a specialized or technical sense. > > >>Here is the definition (I think I can plead "fair use" on this one): >> >>"Infinity: >>1 (a) the quality of being infinite (b) unlimited extent of time, space, >>or quantity: boundlessness. > > > This is perhaps the original meaning and best expressed like something > that cannot be enlarged. > Apparently that was not the opinion of the editors, who (I suppose) spend a great deal of time considering how something can best be expressed. I don't see an immediate connection between "boundlessness" (something that cannot be exhausted or limited) and "something that cannot be enlarged". > >>2 an indefinitely great number or amount. > > > A definition should not contain the defined expression. I believe your native language is German? You may be confusing "infinite" (in this sense, perhaps "unbegrenzt") and "indefinite" ("unbestimmt"). The definition does not, in this case, contain the defined expression. > > ... > > >>>Do not mistake oo+a=oo like an equation of the same sort as it is valid >>>for numbers. It is just a formalized description of the fundamental >>>property to be infinite. >> >>You had not made that clear. In that case, why use "oo + oo" when it >>apparently means the same as "oo + a"? > > > IR is not larger than IR+. > > >>>A part of mathematics would go slippery when it would leave the original >>>meaning of oo that is still in use in other parts. >> >>There are a number of words that are used in mathematics in a different >>sense than they are used in common language - "function", "rational", >>"real","sheaf","bundle","ray"... >> >>It can be confusing for non-mathematicians who do not use mathematics >>rigorously to discover that mathematics adds to or changes the meanings >>of these everyday words. This confusion is not necessarily detrimental >>to mathematics; in many ways, it is a necessary consequence of using a >>fundamentally imprecise tool like language to describe a mathematically >>precise situation. > > > I know that. > Perhaps I misinterpreted what you meant by saying "A part of mathematics would go slippery..." Would you mind explaining that? Matt > Eckard >
From: Dave Rusin on 12 Apr 2005 15:26 In article <vcbu0mbx6es.fsf(a)beta19.sm.ltu.se>, Torkel Franzen <torkel(a)sm.luth.se> wrote: >Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > >> You did nor by chance refer to Bertrand Russel who was, according to >> Lavine, responsible for applying Cantor's basic idea to the reals? > > No, no! The furry evil creatures! Look out for them. That would be the Trolls, surely?
From: Will Twentyman on 12 Apr 2005 16:57 Willy Butz wrote: > Eckard Blumschein wrote: > >> [lot of weird theory about infinity and cardinalities, mixed with non >> mathematical stuff] > > > I don't want to enter this discussion, as I conducted identical > discussions with Eckard before in de.sci.mathematik in several threads. > I just want you to be aware that exactly the same discussion was running > over several thousands of postings, involving some dozens of people > within the last couple of months in the German math newsgroup. > > Basically there are only a couple of arguments, Eckard repeats again and > again: > - I don't understand the facts. Nevertheless I know better. > - oo is not quantifiable. This is a dogma. > - oo+a=oo. This is true, in whatever context. This is a dogma as well. > - Cantor made a mistake => mathematics is untenable in general. > - in my paper M280 (to be found on my homepage, but not in any > recognized scientific journal) I stated the contrary. > - all mathematicians are dazzled by Cantor and other insane people, and > they are not able to think on their own. > - for any intelligent person it should be obvious that ... > Mathematicians just don't admit that in order to not sacrifice their > beloved discipline. > > Anyway, I wish you a tremendous discussion on the fact that > cardinalities are nonsense and unnecessary, infinity is a word that is > not available in mathematics as it describes something unreachable, > Cantor is not authorized to define anything that may quantify infinite > sets, the set of real numbers is uncountable as for any given number > there is no successor, Cantor's diagonal arguments are pointless, there > are only two cardinalities of inifinite sets, namely countable and > uncountable, ... - don't laugh, we went through all that in > de.sci.mathematik. 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. -- Will Twentyman email: wtwentyman at copper dot net
From: Will Twentyman on 12 Apr 2005 16:59
Eckard Blumschein wrote: > On 4/12/2005 3:35 PM, Willy Butz wrote: > >>... there are only two cardinalities of inifinite >>sets, namely countable and uncountable, > > > I feel guilty for suggesting such restriction to aleph_0 and aleph_1 > Forget any use of the notion cardinality. > Instead, I am suggesting to distinguish just two different possibilities > of infinite sets: > > 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? -- Will Twentyman email: wtwentyman at copper dot net |