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From: Eckard Blumschein on 15 Apr 2005 11:02 On 4/13/2005 6:41 PM, Will Twentyman wrote: >> Engineers contempt elusive precision. > > And if the precision is not elusive, but right there in their grasp? This is basic to engineering, too. E.
From: Eckard Blumschein on 15 Apr 2005 11:20 On 4/13/2005 6:52 PM, Will Twentyman wrote: >> I conclude that you did not read M280. > > You conclude incorrectly. Thank you. I simply disagree with it at so many points > that I consider it unlikely that we can come to any agreement on it. After a hundred years Cantor has been idolized. > More specifically, reading it causes me to believe you do not understand > what Cantor was doing or how mathematicians reason. I can retrace his reasoning except for the cheeky idea to quantify the quality infinity. > I agree with you > that Cantor's original notation may not have been tidily presented, It was clever presented in a demagogic manner. > but > his fundamental concepts were sound Quantifying a quality is not a sound concept. > and have been formalized. I note > that you ignore ZF, ZFC, and other formalizations that may have > eliminated any "rough edges" on Cantor's terminology or exposition in > favor of the papers that serve as the basis for those works. Any > particular reason why? I see most of the paradoxes just superficially remedied. > I'll give one example where you are simply wrong: "Cýs infinite alephs > only distinguish between countable and uncountable sets." I am aware that they claim to manage much more. I referred to what they really are able to perform. > The infinite > alephs establish equivalence classes of sets which have a natural > partial ordering based on surjections, and which also subdivide the the > class of uncountable sets. Given it would be make sense to subdivide the class of uncountable sets. Do you expect these subdivition nearly equally fundamental as the division into countable and uncountable numbers? Eckard
From: Chris Menzel on 15 Apr 2005 11:53 On Fri, 15 Apr 2005 17:20:08 +0200, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> said: >> but his fundamental concepts were sound > > Quantifying a quality is not a sound concept. Except that's not what he did. He provided simple and clear definitions of "countable", "smaller than", etc which you either ignore or fail to grasp. You then replace these clear and simple definitions with vague and ill-formed "definitions" of your own creation. So all you've done is to change the subject. It only *looks* like you are disagreeing with Cantor, when in fact you are just speaking your own private, and quite irrelevant, language. Chris Menzel
From: W. Mueckenheim on 16 Apr 2005 09:22 Matt Gutting <tchrmatt(a)yahoo.com> wrote in message news:<1113486771.1fe45d99ab6181f3177e6f7915f8e9b8(a)teranews>... > Eckard Blumschein wrote: > >> > >>No one is saying that there are numbers which are infinite. > > > > > > Except for Cantor. > > > > He is saying that there are cardinalities which describe infinite sets, not > that there are infinite numbers. > > > > >>There are > >>cardinalities that don't describe finite sets. And there are different > >>cardinalities which describe different infinite sets. Cantor used whole infinite numbers in order to distinguish them from fractions. Not only in his early works. In his final presentation of set theory: Beitrýge zur Begrýndung der transfiniten Mengenlehre. [Math. Annalen Bd. 46, S. 481-512 (1895); Bd. 49, S. 207-246 (1897).] Cantor wrote: Es ist zweckmýýig, sich zunýchst mit denjenigen Zahlen von Z(aeph_0) vertraut zu machen, welche ganze (rationale) Funktionen endlichen Grades von omega sind. Jede derartige Zahl lýýt sich, und dies nur auf eine Weise, in die Form bringen ... ganze Zahlen von Z(aleph_0) = whole numbers of second number class (infinite numbers) ganze rationale Funktionen von omega = whole rational function of omega One chapter is named: ý 15. Die Zahlen der zweiten Zahlenklasse Z(aleph_0). Die zweite Zahlenklasse hat eine kleinste Zahl. = The second number class has a smallest number. Small wonder that you firmly believe in Cantor's theory. You should study it! Regards, WM
From: Ross A. Finlayson on 16 Apr 2005 09:33
Die umbegrentzheit (unendliche) Zahlen, wieder die Kardinalen, Sie aber wird die Negativzahlen zuwirden. (The transfinite cardinals are actually the infinite ordinals, and the negative integers, via dual representation as the cumulative hierarchy.) Es is nicht fortunat, wo es wirde nur eins Theorem zuwirden, aber est so. Das Eins ist Vertig. Goedel sagt so! Ich sage so. With warm regards, Ross |