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From: Eckard Blumschein on 12 Apr 2005 07:22 On 4/12/2005 12:02 AM, Will Twentyman wrote: : >>>Eckard Blumschein wrote: >> >>>>>>http://iesk.et.uni-magdeburg.de/~blumsche/M280.html >> You are correct in that meanwhile nobody asks for why and how >> cardinality was introduced. > > This seems unlikely at best. Without looking at any notes, it seems > like cardinality gives a meaningful method of comparing the "size" of > sets and defining what we mean by "size". 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. Let me ask and answer the question why it was introduced: Cantor's thinking was strongly guided by his intuition. That was one reason why his unbelievable claims were so appealing. Cantor himself wondered after he got evidence for what already Albert von Sachsen (1316-1390) found out. A more intelligent man would have expected that the whole universe does not contain more points than any linear interval. 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. I quote Ebbinghaus whose introducing quotation (at the beginning of what he wrote on Mengenlehre) was taken from Lessing: "Given, there is a big useful mathematical truth which arose from a fallacy of an author ..." Obviously, Ebb. admitted that the introduction of cardinality was based on Cantor's fallacy. 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. > It is a natural extension of > the cardinality of a finite set. I wonder if cardinality of finite sets ever played any role except for embellishmet to transfinite cardinality. > > The underlying notion of infinite whole >> numbers is undecided between infinite and numbers mutually excluding >> each other. Cantor's thinking was correspondingly split. He did not >> decide between the meaning oo of what he called Maechtigkeit and later >> cardinality and his intention and pracice to use it as and like just a >> number that served as an infinite or even more than infinite measure of >> quantity. So it is different from infinity but definitely not more >> precise. It is just nonsense. > > 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. An infinite number is by no means more precise than infinity or any number. It is simply self-contradictory. > 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)). I know that Cantor handled his cardinalities like numbers. However, this is neither justified nor advantageous in any sense. > 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. >> As far as I know, the only decisive question is whether or not an >> infinite set is bijective to the set of natural numbers. In that case it >> is calles countable infinite, else non-countable. > > 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. >> Cantor got (in)famous just because he claimed to have revealed different >> levels of infinity. He was a bluffer. > > Most people Is this not possibly exaggerated. > seem to find it convenient as a measure of levels of > infinity. I do. Please tell me what operation or whatever it makes more convenient to you. I only know tremendous trouble with it. > If you don't care to think of it that way, just focus > on the definition as it applies to sets. 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. >> No. How did he define his infinite whole numbers? >> Instead of serious work, he published what M280 contains a link to. Read >> it and judge yourself. > > Unfortunately, it's in German and I can't claim I would be able to > reliably translate it. Google was not helpful. So it is perhaps German mathematicians turn to unmask Cantor by means of his original papers. Translation doesn't pay. I am ashamed to speak the same language as did Cantor. Rabbi Sachs of London shook my hands und forgave me to be a German. I hope this is valid for all wrong doings including Cantor's. >>>I don't think he was planning >>>on the results. >> >> He was keen to create the most unusual. > > Most unusual or most correct? I've seen several odd ways of looking at > things that are useful for easily getting a correct result. Would you regard someone correct who deliberately ignored Aristotele, Cauchy, Galilei, Gauss, Kronecke, Lebniz, and many others without any convincing argument for that. 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? >>>>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. >> 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. > If you don't accept that the reals are numbers (which I > think is your position), then the entire topic of conversation is moot. In that I follow Cantor's definition, and additionaly I accept that e.g. pi is an real number. However, the real numbers are something special in that they lost the property of ordinary numbers to be numerically identifiable. >> Do not deny compelling arguments. Neither my feeling nor yours matters. > > Which is why I work with the ZF version of set theory and accept the > various infinite cardinalities. You might feel that ZF or ZFC is appropriate. There is no tenable basis for that. >> Such map does not exist. There is no approachable well-ordered map of >> the reals. > > AC asserts (without proof and non-constructively) a well-ordering of the > reals. There is no bijective map between the reals and rationals > precisely because they have different cardinality. I do not like arbitrary definitions with the only purpose to conceal mistakes. 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. Eckard
From: Lee Rudolph on 12 Apr 2005 07:18 Gerry Myerson <gerry(a)maths.mq.edi.ai.i2u4email> writes: >In article <vcb4qec7718.fsf(a)beta19.sm.ltu.se>, > Torkel Franzen <torkel(a)sm.luth.se> 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. > >Fortunately, no real number has ever died, so the problem >of naming a successor has not arisen. Then what are all those cardinals doing in a conclave, eh? Eh? Lee Rudolph
From: Robert Low on 12 Apr 2005 07:34 Eckard Blumschein wrote: > I wonder if cardinality of finite sets ever played any role except for > embellishmet to transfinite cardinality. I used it only yesterday when deciding which packet of chocolate biscuits to purchase. -- Rob
From: Eckard Blumschein on 12 Apr 2005 08:04 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. First of all we should look at the attitudes. Dedekind wrote: "Numbers are a free creations of man." Cantor wrote (crawlingly to cardinal Franzelin): ".. ich wollte nur auf eine gewisse subjektive Notwendigkeit fýr uns hinweisen, aus Gottes Allguete und Herrlichkeit auf die tatsaechlich erfolgte (nicht a parte Dei zu erfolgende) Schoepfung, nicht bloý eines Finitum ordinatum, sondern eines Tranfinitum ordinatum zu schliessen." Hilbert wrote: "From the paradise created by Cantor... " Russell wrote: "The solution of the difficulties which formely surrounded the mathematical infinite is probably the greatest achivement of which our age has to boast!" Fraenkel wrote: "... final attack on infinity!" As a summary, I am always missing the due humbleness. Platonian thinking is perhaps more appropriate. One cannot force the reals to have the same properties as exhibited by ordinary numbers. Eckard
From: Eckard Blumschein on 12 Apr 2005 08:10
On 4/12/2005 1:18 PM, Lee Rudolph wrote: >>Fortunately, no real number has ever died, so the problem >>of naming a successor has not arisen. > > Then what are all those cardinals doing in a conclave, eh? Eh? Good idea, a conclave of all mathematicians because the concept of Cantorian cardinalities and all related nonsense has died. Eckard |