Cantor Confusion
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From: David Bernier on 30 Oct 2006 04:49 The Ghost In The Machine wrote: > 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. [...] I was trying to find out what the part of philosophy that studies being and existence was called; I found out it's named ontology: cf: http://en.wikipedia.org/wiki/Ontology There are a number of books either authored or edited by Stewart Shapiro, cf.: <http://www.amazon.com/s/ref=sr_st/104-6965854-8209521?page=1&rh=n%3A1000%2Cp_55%3AStewart+Shapiro&sort=reviewrank&x=14&y=11> Rarely have I seen discussions of the different philosophies of mathematics that were as easy to follow. Concerning sqrt(2), algebraists might be quite happy with Q[x]/(x^2 -2) , but analysts would want to show that f: R->R x |-> x^2 -2 has precisely one zero in the interval [0,2]. David Bernier
From: Sebastian Holzmann on 30 Oct 2006 06:24 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > Sebastian Holzmann schrieb: >> mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: >> > Sebastian Holzmann schrieb: >> >> It is, more than that, quite true that ZF without INF _is_ infinite >> > Are you really sure? >> >> I am. >> > So you also must be sure that ZF is yellow. > >> >> (the axiom schema of separation alone provides infinitely many axioms). >> > Are you really sure? >> >> I am. > > But most of them are very dirty? Oh, they are all very similar to each other (but distinct non the less). >> >> The point is: ZF without INF does not prohibit the existence of infinite >> >> sets, nor does it force them to exist. >> > >> > It prohibits to speak of infinite sets and to recognize such sets. So >> > one cannot be sure, but you are? >> >> Given any model of ZF-INF, one cannot be sure if there are infinite >> sets. > > In particular the sets consisting of infinitely many rabbits would be > of great interest. Learn about logic, model theory and set theory (preferably in that order). You might be surprised.
From: stephen on 30 Oct 2006 07:20 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > David Marcus wrote: >> Han.deBruijn(a)DTO.TUDelft.NL wrote: >> >>>mueckenh(a)rz.fh-augsburg.de wrote: >>> >>>>stephen(a)nomail.com schrieb: >>>> >>>>>Can you describe a continuous version of the problem where each >>>>>"unit" of water has a well defined exit time? A key part of >>>>>the original problem is that the time at which each ball is >>>>>removed is defined and reached. This is crucial to the problem. It is >>>>>not just a matter of rates. If you added balls 1-10, then 2-20, >>>>>3-30, ... but you removed balls 2,4,6,8, ... then the vase is >>>>>not empty at noon, even though the rates of insertions and removals >>>>>are the same as in the original problem. So you cannot just >>>>>say the rate is 10 in and 1 out and base an answer on that. >>>> >>>>The answer for any time t *before noon* is independent of the chosen >>>>enumeration of the balls. Doesn't that fact make you think a bit >>>>deeper? >>> >>>And add this to the fact that noon and beyond cannot exist >>>in this problem. >> >> Are you still doing physics, water, and a finite number of molecules? >> Let us know when you switch to mathematics. > I'm DOING mathematics. Mathematics is NOT independent of Physics. > Han de Bruijn 5*10^8 m/s + 5*10^8 m/s = 10*10^8 m/s Stephen
From: mueckenh on 30 Oct 2006 08:49 David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > stephen(a)nomail.com schrieb: > > > > > Can you describe a continuous version of the problem where each > > > "unit" of water has a well defined exit time? A key part of > > > the original problem is that the time at which each ball is > > > removed is defined and reached. This is crucial to the problem. It is > > > not just a matter of rates. If you added balls 1-10, then 2-20, > > > 3-30, ... but you removed balls 2,4,6,8, ... then the vase is > > > not empty at noon, even though the rates of insertions and removals > > > are the same as in the original problem. So you cannot just > > > say the rate is 10 in and 1 out and base an answer on that. > > > > The answer for any time t *before noon* is independent of the chosen > > enumeration of the balls. Doesn't that fact make you think a bit > > deeper? > > In other words, we have two (or more) situations (depending on how we > decide when the balls are removed). For these different situations, the > number of balls in the vase before noon are the same. Are you saying > that this implies that the number of balls in the vase at noon are the > same for the different situations? > > Suppose we define two functions by > > f(x) = 1 if x < 0, > 0 if x >= 0, > > g(x) = 1 if x <= 0, > 0 if x > 0. > > Then for x < 0, f(x) = g(x). Are you saying that this implies that f(0) > = g(0)? If you think that you can arbitrarily define the value at t = 0, how can you be sure that in the vase problem this value is "empty"? Das Operieren mit dem Unendlichen kann nur durch das Endliche gesichert werden. (David Hilbert) Operating in the infinite (t = 0) can only be established by the finite (t < 0). Regards, WM
From: mueckenh on 30 Oct 2006 08:56
David Marcus schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > David Marcus wrote: > > > > Not sure if he ever said precisely "within Z set theory", but he > > > > certainly said things very similar. Below are a few messages that I > > > > found. There are probably others. > > > > > > > > In the first, he says that "standard mathematics contains a > > > > contradiction". > > > > as ar as standard mathematics is derived from set theory an comes to > > the conclusion of an empty vase at noon or an uncountable set of reals. > > Classical mathematics is free of such contradictions. > > > > > >In the next two, he states there are "internal > > > > contradictions of set theory". In the next, I say that he says that > > > > "standard mathematics contains a contradiction", and he does not > > > > dispute this. > > > > > > Thanks. And at least a couple of times I said that I was reading his > > > argument to see whether it does sustain his claim about set theory, as > > > I mentioned specifically Z set theory. > > > > My proof of the binary tree covers all possible theories. And it should > > not cost you too much time to see that it is true. > > Unfortunately, I don't know what "covers all possible theories" means. > (You seem to use the word "cover" a lot.) Here it means, the result of my proof is valid for all possible theories which allow to define the infinite binary tree. > > Are you saying that your proof works *within* all possible theories, > i.e., all possible theories contain your proof? For example, does your > proof work within ZFC (i.e., are the axioms and rules of inference of > ZFC all that you need for your proof)? You just said: "The tree is that > mathematics which deserves this name. It is outside of your model, > independent of ZFC". > Either you have a proof that can be given in ZFC or you don't. Which is > it? This shouldn't be a difficult question to answer. But it is not an interesting question. Regards, WM |