Cantor Confusion
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From: mueckenh on 31 Oct 2006 08:55 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Every mathematician whose mind has not yet suffered the drill of set > > theory you probably were exposed to knows that my argument is correct. > > How could they know it is correct regarding set theory if they don't > know set theory? It is enough to know that it is correct. Every path is related to two edges which are not related to any other path. And that is easy enough to see. If set theory denies that, then set theory is wrong. Regards, WM
From: William Hughes on 31 Oct 2006 08:58 Albrecht wrote: > Sebastian Holzmann schrieb: > > > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > > "dirty" is a property which without the axiom of dirt is as well > > > defined as "infinite" without the axiom of infinity. > > > > No. A set x (which is here to denote an element of a model M of ZF-INF) > > is called "finite" if x satisfies one of the following conditions: > > > > 0: x does not have an element > > 1: x has exactly one element > > 2: x has exactly two elements > > and so on > > > > otherwise, x is called "infinite". Where do I need the axiom of infinity > > to do this? > > Since there is no x which don't follow one of your conditions 0, 1, 2, > ... you may call the otherwise x as you want. You can call them > "muggles" without an axiom of muggles if you want. > You miss the point. The point is not whether an infinite set exists or not. The point is that it is possible to define the properties that an infinite set must satisfy, without answering the questions "does an infinite set exist". Similarly, I can say "A unicorn is a horse like creature with a white horn in its forehead", without having to decide whether or not unicorns exist. We can say that the definition of unicorn exists, even if no unicorn exists, and thus it makes sense to talk about the definition. (An interesting question is: "Under the assumption that unicorns do not exist, does the question "Does a unicorn have a white horn?" make sense?" (I would say yes, and that furthermore, one correct answer to the question is no.)) - William Hughes
From: stephen on 31 Oct 2006 09:01 Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > stephen(a)nomail.com wrote: >> Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >> >>>stephen(a)nomail.com wrote: >> >>>>Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: >>>> >>>>>I'm DOING mathematics. Mathematics is NOT independent of Physics. >>>> >>>>5*10^8 m/s + 5*10^8 m/s = 10*10^8 m/s >> >>>There are 10 men in a room. Each has a body temperature of 37 Celcius. >>>This means that the temperature in the room is: 10 x 37 = 370 Celcius. >>>Satisfied? Or do you rather prefer it in Kelvin? >> >> Satisfied? I have no idea what you mean. Either you are agreeing with >> me or you totally missed my point. [ ... snip ... ] > Who is missing who's point? As I said, I have no idea what your point was supposed to be. Your example seemed to support my claim that mathematics is independent of physics. Of course you apparently have your own personal definition of both "mathematics" and "physics" and perhaps "indendent" as well, so who know what you are trying to say. Stephen
From: mueckenh on 31 Oct 2006 09:14 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > 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. > > The word 'accomplished' is not in set theory. E. ZERMELO: Beweis, daß jede Menge wohlgeordnet werden kann, Math. Ann. 59 (1904) 514-516. Proof that evey set can be well ordered. That is the same as well-ordering be accomplished. E. ZERMELO: Neuer Beweis für die Möglichkeit einer Wohlordnung, Math. Ann. 65 (1908) 107-128. New proof for the possibility of a well ordering. >ZFC proves the sentence > 'AxEr r well orders x'. But there is no assertion about 'accomplished' > since 'accomplished' is not in the language of the theory. ZFC proves > that any set has a well ordering, but ZFC doesn't show us how to > describe anything about that well ordering. ZFC just tells us that such > a well ordering exists. Period. Where does it exist? In our minds it does not exist. As a finite formula it does not exist. As a ist it cannot exist, because R is uncountable. >That is exists. Indeed, that is quite > understandably unsatisfying for many people. But it is not in itself a > contradiction. The assertion that something does exist, but that it never can be found, is nonsense. >Yes, the axiom of choice may be regarded as unintuitive; > but being unintuitive does not in itself entail contradiction. On the > other hand, it may be unintuitive NOT to adopt the axiom of choice. For > example, let me ask you this: > > Suppose we have a function f from x onto y. So f "covers" every member > of y with a member of x. So, intuitively, there must be at LEAST as > many members of x as there are of y, since every member of y gets > mapped to (is "covered") by at LEAST (maybe more than one) member of x. > So we would think that there must be an injection from y into x. That > is, we would think there must be a function that maps each member of y > to one of the members of x that y is mapped to by the function f. At > least, at first blush, that seems intuitive, right? Okay, but try to > prove it without the axiom of choice. It is impossible to consistently do mathematics with infinite sets. See the vase or the tree. Therefore we need not ponder whether a well order does exist or not. The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma? (Jerry Bona) visit:http://www.fh-augsburg.de/~mueckenh/MR/Zitate.mht Regards, WM
From: imaginatorium on 31 Oct 2006 09:42
David Marcus wrote: <snip> > I wonder: do Lester, Ross, Han, Tony, and WM all agree that noon doesn't > exist? It is such an odd thing to say. Imagine walking up to someone in > the street and trying to convince them that noon doesn't exist. > > It is so hard to keep the nonsense straight. It all seems to run > together--although there are stylistic differences. More than stylistic, I think. For example, don't offhand recall Lester ever saying anything mathematical that was even wrong, whereas our Tony is reponding in another thread, saying something that makes sense, seems to be correct, and (gasp!) includes use of the f-word. http://groups.google.com/group/sci.math/browse_frm/thread/ea85cc0bd40c0ca8/43d661d74974c55d#43d661d74974c55d Question: If a_n is such that sum(a_n) converges is it true that sum((a_n)^3) converges? Tony's answer: It would seem so, at least assuming all positive terms. The number of a_n greater than 1 must be finite if sum(a_n) converges, so the number of a_n^3 greater than 1 would be finite as well, and that portion converge. For all a_n less than or equal to 1, a_n^3 is less than or equal to a_n, and so that portion would converge as a series of lesser terms. I think he may have missed the point slightly, since I suppose the nontrivial question is about the case where all terms are not positive, but let's not be negative. Brian Chandler http://imaginatorium.org |