Cantor Confusion
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From: Dik T. Winter on 29 Oct 2006 22:07 In article <1162157745.995131.292260(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > "positive numbers" with "numbers larger than 0". Because I understand the > > main domain is Anglo-Saxon mathematics. This is in contradiction to what > > I did learn at university (0 is both positive and negative). > > Really? I never heard of that. Is here anybody who learned that too? It > would interest me. No polemic intended. It is common in Bourbaki influenced mathematics, which is, I think, still prevalent in France. In that realm there is a disctinction between positive numbers (>= 0) and strictly positive numbers (> 0). The Dutch universities were largely influenced by both French mathematics and German mathematics, so I learned pretty early that there could be more than one notion about a concept. -- 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 29 Oct 2006 22:02 In article <1162157584.352015.276510(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > Euclides was apparently smarter than Cantor in this. He used the > > parallel axiom (postulate) for something he could not prove from the > > other axioms. > > But which is obviously possible and correct under special > circumstances, while Zermelo's AC is obviously false under any > circumstances. You just state so. This is nothing more than opinion. > > But when mathematicians are bothered about the parallel axiom have > > problems with the justification of that axiom and reject it, you > > aparently have no problems. But when mathematicians do the same with > > the (implicit) well-ordering axiom from Cantor you appear to have > > problems. Why? > > Because I can construct parallels (or see their absence) but I cannot > see a well-order which is not definable. Indeed, what you do not see does not exist, and when others see it they are talking nonsense. How more opinionated you can get? > > With the axiom of choice, it is possible to well-order the reals. > > But it can also be shown that such a well-ordering can not be defined > > by a formula in ZFC. > > How then can the well-ordering be accomplished? Zermelo proved tat it > could be accomplished. Oh, well, just in another thread there is a discussion about the existence proof and showing a method to calculate such a thing. An existence proof does not necessarily imply a method to actually provide such a thing. There is a mathematical proof that li(x) - pi(x) changes sign infinitely often. In your finite world you will never see such a sign change. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 29 Oct 2006 23:02 In article <MPG.1faf1538d19002279897b9(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > Dik T. Winter schrieb: > > > > > Euclides was apparently smarter than Cantor in this. He used the > > > parallel axiom (postulate) for something he could not prove from the > > > other axioms. > > > > But which is obviously possible and correct under special > > circumstances, while Zermelo's AC is obviously false under any > > circumstances. > > What do you mean an axiom is "false"? Are you discussing philosophy or > mathematics? Good point! Axioms cannot be "false" in any real sense.
From: The Ghost In The Machine on 29 Oct 2006 23:20 In sci.math, Dik T. Winter <Dik.Winter(a)cwi.nl> wrote on Mon, 30 Oct 2006 02:21:06 GMT <J7xFv6.ECF(a)cwi.nl>: > In article <1162139168.281239.183260(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > You need not tell that to me. You should tell that to Wolfgang > > > Mueckenheim who insists that sqrt(2) does not exist because it is > > > impossible to know all the decimals in its decimal expansion. > > > > It does not exist as a number. It has no b-adic representation. It > > exists as a geometric entity. It is an idea. > > It exists as the continued fraction [1,2,2,2,2,...]. It also exists > as 1 in the base sqrt(2) notation. I am extremely puzzled as to why anyone would say that sqrt(2) doesn't exist, given that numbers exist at all, which is an abstract concept in its own right; however, if 1 exists then N exists therefore J (or Z) exists therefore Q exists therefore R exists (take one's pick: Cauchy or Dedekind) therefore C exists. So...sqrt(2) is, for instance, that number r such that r^2 = 2. Since r is not in Q (you probably already know the proof for this :-) ), we enter Dedekind cut territory: S0 = {q in Q: q < 0} union {0} union (q in Q: q > 0 and q^2 < 2} S1 = {q in Q: q^2 > 2 and q > 0} It is clear that S0 union S1 covers Q and every element in S0 is less than any element in S1; therefore this is a valid cut and defines a number r not in Q. Therefore, r exists. A nice approximation sequence -- which might be related to the above continued fraction; I'm not that energetic tonight :-) -- yields the following: q_0 = 1 is in S0. q_1 = (1/2) * (1 + 2/q_0) = 3/2 is in S1. q_2 = (1/2) * (3/2 + 4/3) = 17/12 is also in S1. q_3 = (1/2) * (17/12 + 24/17) = 577/408 is in S1. q_4 = (1/2) * (577/408 + 816/577) = 665857/470832 is in S1. I'm sensing a definite pattern here. If q_i is in S1 (i.e., q_i^2 > 2) then q_{i+1} = (1/2) *( q_i + 2/q_i) is such that q_{i+1}^2 = (1/4) * (q_i^2 + 4/q_i^2 + 4) = 1 + (1/4) * (q_i^2 + 4/q_i^2) The question is whether q_i^2 = q^2 > 2 implies q^2 + 4/q^2 > 4, or, equivalently f(q^2) = q^2 + 4/q^2 - 4 is ever negative when q^2 > 2. If q^2 = a/b then f(a/b) = a/b + 4*b/a - 4 = (a^2 + 4*b^2 - 4*a*b)/(a*b) = (a - 2*b)^2 / (a*b) Since a/b > 2, (a - 2*b)^2 > 0 and therefore all q_i > sqrt(2) for i > 0, since q_1 > sqrt(2). However, if q = q_i > sqrt(2) then 2/q^2 < 1 and (1/2)*(1+2/q^2) < (1/2)*(1+1) = 1 therefore q_{i+1} = (1/2)*(q_i+2/q_i) < q_i and we get a nice Cauchy sequence here, whose limit gets arbitrarily close to sqrt(2). In any event, 1/3, 1/7, 1/17, and such don't exist either according to the above logic, since we can't write all of their decimals, although we know what they look like. -- #191, ewill3(a)earthlink.net If your CPU can't stand the heat, get another fan. -- Posted via a free Usenet account from http://www.teranews.com
From: MoeBlee on 30 Oct 2006 00:13
Lester Zick wrote: > On 27 Oct 2006 16:56:30 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > >Lester Zick wrote: > >> >> You mean that mathematical definitions can't have different "domains > >> >> of discourse" and mathematical definitions in different domains of > >> >> discourse can't borrow from one another? > >> > > >> >No, that's not what I said. > >> > >> Then why exactly are you complaining about what I said? Frankly, Moe, > >> you don't seem to have said much of anything that I can make out. If > >> mathematical definitions can have different domains of discourse then > >> what I wrote should be perfectly acceptable according to your own > >> definition of mathematical definitions and domains of discourse.. > > > >Since I never said anything that can be paraphrased as the jumble of > >nonsense you just mentioned, nothing I did write entails that the > >jumble of nonsense you wrote needs to be acceptable to me. > > What kind of jumble of nonsense do you prefer then, Moe? > > >> >No, you posted utter nonsense ("Cardinality(x)=least ordinal(y) with > >> >equinumerosity(z)") as if it is something that I had said. > >> > >> I never said you had said that. > > > >You're absurd. You quoted me asking you what I said that justified a > >certain statement you made. > > Gee it's sure too bad the relevant citations appears to have gone with > the wind. No doubt my fault as well. Gone in your one post memory span. But the post in which you posted "Cardinality(x)=least ordinal(y) with equinumerosity(z)", in reply to my question, hasn't been swept by any winds. MoeBlee |