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From: Matt Gutting on 13 Apr 2005 09:35 Will Twentyman wrote: > > > 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. That's what keeps me reading, anyway. You have to have a blunt object to sharpen your axe on. Or something like that. Matt
From: David Kastrup on 13 Apr 2005 09:40 Matt Gutting <tchrmatt(a)yahoo.com> writes: > Eckard Blumschein wrote: > >> This was your statement. Where is your evidence for it or at least >> some justification? Simply tell me the successor of pi. > > What do countability and successorship have to do with each other? Uh, everything? > The rationals can be ordered in such a way that each one has a > defined successor. They can also be ordered in such a way that none > has a defined successor. This ordering has nothing to do with > whether the set is countable or not. It pretty much has everything to do with it. If you can make a subset of the rationals obey the Peano axioms, then this subset is in one-to-one correspondence with the natural numbers. If you can make the entire rationals obey the Peano axioms by choosing a different successor relation), then the rationals are in one-to-one correspondence with the naturals. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Eckard Blumschein on 13 Apr 2005 11:26 On 4/12/2005 11:29 PM, Will Twentyman wrote: >> All this just reflects Cantor's claim that one can attribute different >> quantities to infinity. Size means about the same as quantity, >> Maechtigkeit or cardinality. It would be meaningful if it was correctly >> founded. > > No, it is meaningful because it is precisely *defined*. I disagree. >> Obviously, Cantor was unable to grasp the notion infinity. >> Maybe his teacher Weierstrass is to blame for that. As known, W. spoke >> of infinite numbers. So he failed to make quite clear that infinity and >> numbers strictly exclude each other. > > It seems odd to accuse someone of not understanding infinity when he > worked with it so much. Cantor's papers reveal to what extent he understood it. >> Let me also ask and answer how it was introduced: >> Is cardinality really a big useful mathematical truth? I cannot confirm >> that. Anything started at December 7th, 1871 when Cantor presented his >> proof for the reals to allegedly be more than just countable. After the >> won war against France, this was a time of euphoria. What a miracle! >> More than infinite, and the best: Even the most posh people failed to >> refute Cantor's claim. Well, there was a lot of quarrel. Cantor himself >> named about 30 opponents of his theory, some of them very famous ones. >> When he got mentally ill this was taken an indication for the huge >> effort he made in order to create something epochal. The soap opera >> continued with Bertrand Russell, Zermelo, and many others who attached >> to the glory and took the attention away from the fact that a serious >> basis is missing. > > The usefulness of Cantors ideas appear in categorizing sets, as well as > defining some subsets of the reals with interesting measures. Now > whether there are applications for the other sciences, I don't know. Maybe, these categories and subsets of the reals are likewise junk. Any real scientific breakthrough has a record of very widespread benefits. >>>How can you say it is not more precise? >> >> >> Please read yourself how Cantor tried to answer objections. Would you >> call it precise if he was not even able to convincingly explain how he >> imagines his infinite whole numbers: even, odd or what? Would you call >> it precise if Cantor mentioned Aristotele and Spinoza and declared they >> were wrong without to explain why? Would you call it precise when Cantor >> admitted that an opponent was correct but then he veiled the difference. > > Yes, How do you imagine and explain infinite whole numbers? > no, no. However, the second no needs to be qualified, since his > work is a precise explanation of why. I would not consider Aristotle a > good choice for the foundation of modern mathematics. sometimes, I would rather trust in ancient than in certain modern mathematics. > His reasoning was > good, but even that has potential flaws. Of course. > >> An infinite number is by no means more precise than infinity or any >> number. It is simply self-contradictory. > > There is a difference between cardinality and infinite numbers. Which > are we talking about? Which do you object to? Neither requires the other. The close link between both is best to be seen from Cantor's original work and also from David Hilbert's speech in honor of Weierstrass in 1925. Transfinite cardinalities were thought like transfinite numbers, numbers in excess of infinity. > >>>You can use cardinality to >>>compare the sizes of N, P(N), and P(P(N)). >> >> I accept that one has the freedom to define card(N). >> However, the power set of N is not qualitatively related to it. >> If it is non-countable infinite, then it has the quality oo and also the >> quality to be uncountable, as also has P(P(N)). > > However, there are no bijections between any of those three sets, and > any mapping from N to P(N) or P(N) to P(P(N)) are limited to injections. > This can certainly be defined to be a qualitative ranking of the three > sets. Certain types of mappings either do or do not exist. I agree that it is impossible to map an uncountable set. >> I know that Cantor handled his cardinalities like numbers. However, this >> is neither justified nor advantageous in any sense. > > It is convenient for ranking them. If I have several classes of sets, I am only aware of two different infinite sets: countable ones and non-countable ones. I understand how Cantor fabricated even more than more than infinite numbers. > I > would like to have a notation that is strongly suggestive of the > existence or non-existence of a surjective (possibly bijective) map > between any two sets in the classes to be compared. Why? If uncountable numbers cannot be mapped, then I do not see any reason to conjecture anything. > The numbers give an > intuitively understood labeling system that will allow someone to > understand at a glance what types of maps can exist. Where do you not > see the advantage of such a labeling system? In the case of uncountable numbers, because they cannot be mapped in general. >>> You can use cardinality to compare the sizes of N, P(N), and P(P(N)). >>>oo does not distinguish >>>between them at all, and countable/uncountable does not distinguish >>>between the last two. >> >> That is true. The reason why there is no justification and no reason for >> this distinction is in principle the same as expressed by Hilbert's >> hotel. In so far the whole Cantorian concept is not even consequent. > > Hilbert's hotel only explains why the union of two countable sets is > countable. How is it relevent to the rest of this discussion? If the union of two countable sets is countable then this is repetitiously valid. P(P(N)) is the repetitious application of the power operation. If something is uncountable then this quality cannot be compared in grammatical sense. Aleph_2 sounds at least as amusing as double death penalty. I guess, it has no useful application if it is justifyable at all. IR+ is as uncountble as is IR no matter that IR* is a subset of IR. All numbers constituting IR+ can be mirrored into IR and vice versa. >> 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. > > RxR is aleph_1, so P(RxR) is aleph_2. P(RxR) is also the set of all > relations on R. This has potential impact on various topics in > analysis. I don't know off hand if it does, but it could. Hm. >> Please tell me what operation or whatever it makes more convenient to >> you. I only know tremendous trouble with it. > > Comparison of sets. Of infinite sets. In that case there is one reasonable comparison: either countable or not. Anything else seems to be speculative and of questionable use. >> I would not have any reason to complain if Cantorian set theory was >> satisfactory to me. I hope, overdue abandoning of Cantor's fallacious >> infinite numbers will enforce a more reasonable rebuild of set theory. > > It is obvious that you don't care for it from top to bottom. Are there > any particular definitions or axioms you object to? If it's just the > "infinite numbers" you object to, that will not remove cardinality. > Note: when you say oo+a=oo, you are talking about cardinality. Transfinite cardinals are infinite numbers. This would perhaps not cause much damage. I would like to put my finger squarely onto an other moot point. Cantor's introduction of numbers beyond any sense distracted attention from important peculiarities of the real numbers. All experts will agree that the continuum is something special. Cantor ascribed this enigma to the silly idea of more than infinitly many numbers while he did not question the identity of the real numnbers. His set theory treats real numbers as if they were just natural ones. I am not sure whether or not it is really justified to believe that all axioms are valid for the real numbers too. In particular, AC does not remedy the problems of lacking identity and missing successor. >> >> Would you regard someone correct who deliberately ignored Aristotele, >> Cauchy, Galilei, Gauss, Kronecker, Leibniz, and many others without any >> convincing argument for that. > > Perhaps not convincing to you, anyway. Definitely not convincing to Kronecker. I wonder if anybody would like to show that Cantor had any convincing argument against the other ones. > >> Would you regard someone correct who just followed his intuitive guess >> and therefore performed operations that were and are still incorrect >> except for the idea that one declines to decide whether the operand is >> infinite or a number? > > Now I don't have any idea what you're referring to. I get the sense you > feel there are no proofs associated with Cantors work. I dealt with most of his proofs in detail. The decisive one was his second diagonal argument using a suggestion by Paul du Bois Reymond. I already explained that this argument suffered from wrong interpretation. >>>>>>2) Why did he manage to find so much support? >>>>> >>>>>Because his results are consistent with the axioms and definitions he used. >>>> >>>>That is definitely not true. Read the original papers! >>> >>>See the note above regarding my skills with German. >> >> Well, you might deal with my arguments independently. > > I am attempting to do so. I suspect we disagree on the axioms or the > validity of the proofs. That or you simply think the definitions or > their popularizations are nonsense. No. We disagree because I found out that Cantor misinterpreted the fact that the reals are not countable. He assumed that one of two sets must have either a smaller or the same or a larger size than the other one. While this is correct for finite sets, it is wrong for infinite ones. > >>>>No. He just made the wrong assumption that the reals can be mapped. This >>>>cannot work despite of AC. >>> >>>If you accept that the reals are numbers, then they can be mapped to >>>*something*. >> >> Please check the basis for such confidence. Perhaps, you just reiterates >> what was told to you. You can certainly just draw a line as to include >> infinitely many reals. The unresolvable problem is: Nobody is able to >> resolve this line into all single reals. We know how Cantor understood >> mapping, cf. his diagonalizations. > > f(x)=x is a mapping from the reals to the reals. One already fails to map x itself in the sense of listening the reals one by one onto a something like a matrix or the like. >> In that I follow Cantor's definition, and additionaly I accept that e.g. >> pi is a real number. However, the real numbers are something special in >> that they lost the property of ordinary numbers to be numerically >> identifiable. > > The second sentence didn't make sense to me. Are you talking about some > reals being non-constructible? In order to understand the notion of infinity you have to abandon any idea that you can reach it by means of counting/numbers. Infinity is a different quality. >> Can you please send me a map of the reals? I guess: It does not work. >> Forget cardinality together with AC. Forget this dark German >> megalomaniac chapter in history of mathematics. > > A map is from a set to a set. For example: f(x)=x is a map from R to R. > What do you think a map is? In general, a map is a plan to look at, usually on a sheet of paper. Well maybe, I should call it a list in order to be understood correctly. List has also different meaning. So try to guess what I might call map or list. Eckard
From: Eckard Blumschein on 13 Apr 2005 11:30 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. Eckard
From: Eckard Blumschein on 13 Apr 2005 11:32
On 4/12/2005 10:57 PM, 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. I conclude that you did not read M280. E. |