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From: Eckard Blumschein on 12 Apr 2005 11:42 On 4/12/2005 2:00 PM, Barb Knox wrote: > I don't understand: here you appear to accept the distinction between > countably and not-countably infinite, yet your main point in this thread > seems to have been that there is only a single "oo" that cannot be added to > or otherwise extended. How do you reconcile those 2 views? 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. 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. Eckard
From: Willy Butz on 12 Apr 2005 11:33 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. Thanks for pointing that out, Eckard. In my previous posting I forgot to mention that it is not possible ordering cardinalities, according to Eckard. This is another important topic in discussions with him. Best wishes, Willy
From: Torkel Franzen on 12 Apr 2005 11:41 Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > Cantor was mislead by his intuition. No, no! He was misled by a little furry creature who twisted his ears this way and that, and finally convinced him to gwak forth all this nonsense about infinities. We must counter the activities of these insidious furry creatures. We must be on our guard for further intrusions.
From: Eckard Blumschein on 12 Apr 2005 11:57 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. > 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. >
From: Eckard Blumschein on 12 Apr 2005 12:27
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. > > 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... >>>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? > It doesn't. Why should it? Mabe you should try a dictionary for science and technilogy. > 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. > 2 an indefinitely great number or amount. A definition should not 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. Eckard |