Cantor Confusion
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From: David Marcus on 7 Oct 2006 18:37 mueckenh(a)rz.fh-augsburg.de wrote: > You got it several times already. According to the axiom of infinity > the set of all natural numbers does exist. And with it the following > statements are true: > > 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? -- David Marcus
From: Virgil on 7 Oct 2006 19:49 In article <MPG.1f91f63c6c27b1d0989699(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > You got it several times already. According to the axiom of infinity > > the set of all natural numbers does exist. And with it the following > > statements are true: > > > > 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? The number of balls in the urn is a is function of time whose derivative is zero almost everywhere, and is right-continuous everywhere.
From: David Marcus on 7 Oct 2006 20:58 David Marcus wrote: > Tonico wrote: > > Omega has NEVER, ever being a cardinal, as far as I know (perhaps I > > missed something, true), but an ORDINAL!! > > And since when cardinals are...sets?!?! Talking of mixing up terms...!! > > Actually, this is standard. For example, the book "Set Theory" by Kunen > has the following definition. > > Definition. If A can be well-ordered, then the cardinality of A (denoted > |A|) is the least ordinal alpha such that there is a bijection from > alpha to A. > > Kunen then remarks that with AC, |A| is defined for every A. Next comes the defintion of "cardinal": Definition. An ordinal alpha is a "cardinal" iff alpha = |alpha|. After a couple of lemmas, comes Corollary. omega is a cardinal and each n in omega is a cardinal. Then he gives a definition: Definition. A is "finite" iff |A| < omega. A is "countable" iff |A| <= omega. "Infinite" means not finite. "Uncountable" means not countable. The definition of cardinal multiplication and addition comes next. -- David Marcus
From: David Marcus on 7 Oct 2006 21:05 Virgil wrote: > In article <MPG.1f91f63c6c27b1d0989699(a)news.rcn.com>, > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > You got it several times already. According to the axiom of infinity > > > the set of all natural numbers does exist. And with it the following > > > statements are true: > > > > > > 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? > > The number of balls in the urn is a is function of time whose derivative > is zero almost everywhere, and is right-continuous everywhere. I was asking mueckenh to translate the statement of the problem into mathematics. You've stated some properties of the "number of balls" function that may be true of the translation, but the first step is to do the translation. -- David Marcus
From: Dik T. Winter on 7 Oct 2006 21:55
In article <1160252933.461908.146830(a)m7g2000cwm.googlegroups.com> Han.deBruijn(a)DTO.TUDelft.NL writes: > Tonico wrote: > > Han.deBruijn(a)DTO.TUDelft.NL wrote: .... > > > And therefore there is NO guarantee that it "isn't grounded on dogma". > > > > > Wrong: whether maths is a science or not, it is NOT grounded on dogma > > There are several forms of mathematics. Some of them are NOT grounded > on dogma (presumably the forms you are aware of), but others certainly > are. I would say that all forms of mathematics are grounded on axioms (or dogmas as you prefer to say). But contrary to dogmas, axioms can be negated to get another form of mathematics. Dogmas are absolute truths, axioms are only absolute truths within some realm of discourse. In the same way in most countries it is an axiom that you should drive on the right. But an Englishman would state, rightly, the right side is not the right side to ride. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |