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From: David Kastrup on 12 Apr 2005 07:59 Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > On 4/12/2005 10:06 AM, Torkel Franzen wrote: >> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: >> >>> Since it is >>> impossible to completely write down all infinitely many numerals of just >>> one single real number, it is also impossible to name its successor. >> >> The impossibility of naming the successor of a real number is indeed >> the central flaw in today's mathematics. Little can be done about it, >> I'm afraid. > > After some musing, I see the reason already in the impossiblity to > numerically approach just any single real number. > > Is there really a central flaw in today's mathematics? If so, then I > would rather suspect Cantor's concept of cardinality as the primary > source of scores of subsequent mistakes. Priceless. You consider Cantor at fault for the reals not being in one-to-one correspondence with the natural numbers (as that is what having a transitive and grounded successor relationship all about) 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. > 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. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Barb Knox on 12 Apr 2005 08:00 In article <425BAF89.60801(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >On 4/12/2005 12:02 AM, Will Twentyman wrote: [snip] >> P(N) and P(P(N)) are also standards. This is where aleph_0, aleph_1, >> aleph_2 get started. Both aleph_1 and aleph_2 are uncountable, but they >> are different cardinalities. > >While I know these expressions, I wonder if aleph_2 has found any use in >application. >The countable infinite (IN, (Q ) makes sense to me, and the >non-countable infinite (IR) too. Anything else has to provide evidence >against the suspition that it is pure phantasmagora. 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? [snip] -- --------------------------- | BBB b \ Barbara at LivingHistory stop co stop uk | B B aa rrr b | | BBB a a r bbb | Quidquid latine dictum sit, | B B a a r b b | altum viditur. | BBB aa a r bbb | -----------------------------
From: Matt Gutting on 12 Apr 2005 09:10 Eckard Blumschein wrote: > On 4/11/2005 9:26 PM, Matt Gutting wrote: > > > >>But you seem to be claiming that real numbers (or at least some of them) >>are not quantities. > > > Rational and even natural numbers are thought to be embedded into the > real ones. This is one reason while the majority of mathematicians keeps > sticking on Cantor's guess that there are more reals than rationals. > However, we should be aware that rationals can be approximate the > continuum as close as one likes. So the theoretical difference between > the rationals and the reals including all embedded numbers is just given > by the theoretically complete loss of approachable identity for the > latter, no matter whether they are irrational or embedded like e.g. > 0.999999999999999.... 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. 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. > You need infinitely many real numbers in order to constitute just a > single urelement (mathematical atom). It is this fascinating peculiarity > of the reals I am mainly interested in. Weyl spoke of continuum sauce. > Stifel spoke in 1544 of fog. > > > >>You are assuming that comparison requires a quantitative measure. > > > Doing so, I refer to a quantitative comparison. > > > >>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. If one does not *need* quantities (numbers) to compare even finite sets, why insist on having numbers (and remember, we agree that infinity is not a number) to express the sizes of infinite sets? > > > >>>Look into a good dictionary for infinity. >> >>I have (Webster's Collegiate, which I regard as a good dictionary). I saw >>nothing even remotely involving addition of quantities to infinity. > > > It should tell you that infinity cannot be enlarged. > It doesn't. Why should it? 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. 2 an indefinitely great number or amount. 3 (a) the limit of a function when its value tends to become numerically larger than any preassigned value (b) a part of a geometric magnitude that lies beyond any part whose distance from a given reference position is finite (c) a transfinite number (as aleph-null) 4 a distance so great that the rays of light from a point source at that distance may be regarded as parallel" I don't interpret any of this as telling me that infinity cannot be enlarged. > > >>>No that is not the reason. oo+oo also equals oo. >> >>If, as you say, infinity is not a number (a direct quote from you), how can one >>add it to anything (including itself)? > > > 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"? > > > >>>>And what, precisely (*very* precisely) do you >>>>mean by "cannot be enlarged"? >>> >>> >>>The property to have no limit. >> >>That doesn't help to clarify the statement to me. > > > Ordinary people understand it like a barrel without bottom. > It is not just as wide as you like in the sense you may choose any huge > number but there is no chance ever to say that's it. It is not a number. > All numbers are finite. > > > >>>It is. >> >>No, you seem to be inferring that fact from the description of infinity as >>something "inexhaustible". If you define infinity as "something that cannot >>be enlarged", then you are speaking of a different concept than that which >>mathematicians refer to by the name "infinity". > > > 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. > > Eckard >
From: Willy Butz on 12 Apr 2005 09:35 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. Best wishes, Willy
From: Eckard Blumschein on 12 Apr 2005 11:35
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. Eckard |