Cantor Confusion
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From: Dik T. Winter on 9 Oct 2006 18:40 In article <1160404949.384518.260480(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1160308871.194701.44520(a)c28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > > > 1) Before noon every ball comes out of the vase. At noon the vase is > > > > > empty. > > > > > 2) Before and at noon there are more balls in the vase than have come > > > > > out. .... > > > > How do you translate the words of the problem into mathematics? > > > > > > 0) There is a bijection between the set of balls entering the vase and > > > |N. > > > 1) There is a bijection between the set of escaped balls and |N. > > > 2) There is a bijection between (the cardinal numbers of the sets of > > > balls remaining in the vase after an escape)/9 and |N. > > > > How do you *define* division between cardinal numbers? > > "/9" in (2) is completely negligible. That is irrelevant. I ask you how you define definition between cardinals. > Therefore no division need be > defined. But, in principle, division by a finite cardinal number is > defined. Also it is easy to estimate: X/9 = X*(1/9) =< X. More is not > required here. How do you define division in cardinals? How do you define (1/9) in cardinals? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 9 Oct 2006 18:42 In article <1160405521.639470.295030(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <virgil-F7008E.15353808102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > > ... > > > If one chooses to work within ZFC or NBG, the vase is empty at noon. > > > > I doubt this. The problem is not defined with enough precision to state > > that. It has not been defined by most what is meant with "the number of > > balls in the vase at noon". Of course, you can use that the infinite > > intersection of sets does exist (and that is what you are using), and > > so get at the result. > > Then the infinite intersection of the cardinal numbers A(t) with t = 1, > 2, 3, ... of the set in the vase after completing action t does also > exist. It is 9. Impossible. The infinite intersection is a set. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 9 Oct 2006 18:43 In article <1160407082.513322.311190(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > William Hughes schrieb: > > > > My conclusion is: > > > Either > > > (S is covered up to every position <==> S is completely covered by at > > > least one element of the infinite set of finite unary numbers > > > > Straight quatifier dyslexia. The fact that "for every x there exists > > a y such that" does not imply "there exists a y such that for every x" > > A nonsense argument. Your assertion is wrong in a linear set. Give an > example where the linear set covers a number which is not covered by > one member of the linear set. The set of natural numbers. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 9 Oct 2006 18:52 In article <1160407188.555466.59730(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > The limit {1,...,n} for n-->oo is N, if N does exist. > > > > By what definitions? > > By the definition of the limit ordinal omega (= N). Pray, explain. The ordinal omega is the smallest ordinal larger than each finite ordinal. That is the definition. How do you come at the above limit from that? What *might* be a sensible definition of a limit for a sequence of sets of naturals is, that (given each A_n is a set of naturals), the limit lim{n = 1 ... oo} A_n = A exists if and only if for every p in n, there is an n0, such that either (1) p in A_n for n > n0 or (2) p !in A_n for n > n0. In the first case p is in A, in the second case p !in A. With that definition, indeed, lim{n = 1 ... oo} A_n = N, but also lim{n = 1 ... oo} {n + 1, ..., 10n} = 0. I do not think you are meaning that definition. So what *is* your definition? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 9 Oct 2006 19:18
In article <b7f51$452a1029$82a1e228$25909(a)news2.tudelft.nl> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > Dik T. Winter wrote: > > > So, if you come here, and simply state: "the axiom of infinity is false, > > because no infinite sets do exist", that is just a bald statement of > > opinion. > > Mathematicians have found another name for scientitic facts. What facts? In my opinion (and the opinion of many others) infinite sets do exist "in the mathematical sense". That they are not physically realisable is irrelevant. > They call > them "just an opinion". No! The burden is yours. _You_ have to provide > arguments why it is admissible to allow for infinite sets. Because we can think about them. > While _all_ > eyes and all instrumentation in the cosmos can only make observations > of things that are _finite_. Read "The Physics of Infinity" at: What is the relevance of that for mathematics? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |