Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Virgil on 21 Oct 2006 15:48 In article <1161435318.373825.152830(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > If we could not get to omega, we would not need it. > > > > If you do not need it, so be it. Do not use current set-theory and base > > your analysis on whatever is provided with a finitistic view. There is > > *no* reason to attack people for using it, except that it is possibly > > against your ethical views. > > I know that we cannot count to omega! But we use it in definitions of > limits. > > > > > And one would not > > > be able to count omega + 1 and so on. Cantor could have refrained from > > > introducing it. > > > > In order to count all finite sets you need omega. That > > > is the essence of Cantor's theory. > > > > Yes, that was Cantor's theory, but that was an error, and it was not the > > essence. See my discussion with Dave Seaman about just that point. I > > do not think he still had that position when he wrote his 'Contributions > > to set theory' some 11 years after his 'About infinite linear point-sets', > > page 213 in the 'Gesammelte Abhandlungen'. The essence of his theory was > > that there were sets that could not be put in bijection with the set of > > natural numbers. > > Yes. But this recognition is impossible or useless unless all natural > numbers do exist. > > Therefore lim [n-->oo] {1,2,3,...,n} = N. Depends on how one defines lim [n-->oo] {1,2,3,...,n}. It is certainly true if one takes the limit to be the union of all of them, as guaranteed by the axiom of union in ZF. > > Therefore lim [n-->oo] {1 + 1/2^2 + 1/3^2 + ... + 1/n^2} = (pi^2)/6. This limit is not guaranteed by the axiom of union, so that "Mueckenh"'s "Therefore" is false. > > Therefore in the vase experiment: lim [t-->oo] X(t) = X(omega). Again, "Mueckenh"'s "Therefore" is false. "Mueckenh"has three quite distinct limit situations no one of which justifies or is justified by any of the others. > > > > No. The oo in standard analysis is still potential. That is, it can not > > be attained. So the limit lim{n = 1 --> oo} 1/n = 0, does indicate a > > potential infinity, not an actual infinity. > > For which natural number n is 1/n = 0? > Potential means always finite. > lim{n = 1 --> oo} 1/n = 0 proclaims "omega reached". On the contrary, it means no such thing. One definition of limit of a sequence is that lim_{n in N and n -> oo} f(n) = L means that for every positive real epsilon, the set {n in N: | f(n) - L | > epsilon} is finite. All other definitions of the limit of a sequence of reals, however phrased, are equivalent to this. > > > But you are dishonest in transforming the vase problem (where the answer > > was asked at t = 0) to another problem (where the answer was asked at > > t = oo). > > It is nothing but just a simplification in notation! It is a complication and misrepresentation, not a simplification.
From: Virgil on 21 Oct 2006 15:57 In article <1161435575.019298.164830(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1161378187.155995.290420(a)f16g2000cwb.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > > In article <1161276574.792436.186750(a)i3g2000cwc.googlegroups.com> > > > > mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > > Yes, but it is infinitely many times not continuous in each > > > > neighbourhood > > > > of the limit point. > > > > > > Let t be the ordinal number of transactions. > > > > > > It *is* continuous like a staircase. > > > X(t) = 9 for t = 1 until t = 2 where X(t) switches to 18. That is > > > enough to excflude X(omega) = 0. > > > > How do you *define* X(omega). As far as I know X is only defined for real > > numbers, and omega is not one of them. And I see no reason to exclude > > X(omega) = 0, = 1, = -1 at all from this reasoning. > > lim {t --> oo} X(t) = X(omega) Then X(omega) = 0. > > > > > > That is still not a mathematical formulation. More is required. You > > > > need to actually state what you mean with 'number of balls in the vase > > > > at noon' or 'natural numbers in the set at noon'. > > > > > > The number of transactions t is then t = omega at noon. > > > > Is *that* a mathematical definition? Pray provide a real mathematical > > definition. > > lim {t --> omega} t = omega Define "lim {t --> omega} t". Note that all usages of "lim" in mathematics must have precise definitions before they are acceptable. But "lim {t --> omega} t" does not have any such definition. > > > > > > The limit can in mathematics *always* be determined by the terms (if > > > > there > > > > is a limit). But the limit in no case defines the function value at > > > > the > > > > limit point. > > > > > > Then the irrational numbers as limit points are undefined. > > > > You think so. The irrational numbers are defined to be the limits of some > > particular sequences (or rather as equivalence classes of sequences). I > > Equivalence classes of sequences with same limit like lim {t --> oo} > a_t. The meaning of such a limit statement as "lim {t --> oo} a_t = L" is precisely defined. Yours is not, so is meaningless.
From: David Marcus on 21 Oct 2006 19:39 cbrown(a)cbrownsystems.com wrote: > David Marcus wrote: > > cbrown(a)cbrownsystems.com wrote: > > > I took HdB's statement as "it is not possible to have a theory that is > > > empirically supported and states 'A and not A, simultaneously' ". > > > > > > The fact that there are empirically supported theories which state "it > > > is not possible for A and ~A to be true simultaneously" doesn't negate > > > the fact that there are equally empirically supported theories that > > > state the opposite. > > > > Perhaps. Although, I suppose it depends on whether you think the > > empirical evidence really supports the theory or the people claiming it > > does are just confused. > > Until an experiment which distingushes between two similar but distinct > theories is proposed and perfomed, there's no scientific way of knowing > whther adherents to either theory are confused or correct. But, if one theory is vague or ill defined, then it can be hard to say whether an experiment really supports it. > > However, many > > physicists seem to prefer the illogical "Copenhagen" explanations of > > Bohr. > > I would claim that the Copenhagen interpretation is not illogical /on > its own terms/ - it's simply inconsistent with other interpretations > (including "common sense"). The "terms" of the Copenhagen interpretation seem to be that anytime you try to ask a logical question, you are told you can't. > > > The main limitation I can see in his theory (from my exhaustive 30 > > > minute study :-) ) is that it seems to rely on the assumption of > > > non-locality, in a theory that isn't relativistic. > > > > > That seems a /lot/ easier to swallow than it would be in a relativistic > > > theory. > > > > I'm not quite sure I follow what you mean. > > Non-locality isn't a particularly vexing issue in a non-relativistic > setting, because the question of what we mean by "event A is > simultaneous with event B" is perfectly clear. > > In a relativistic setting "event A is simultaneous with event B" is > much more complicated; so when we say "non-locality implies that some > event A simultaneously affects event B", this is much more complicated. > I might be wrong, but isn't that the point of the EPR experiment? It wasn't the point Einstein, Podolsky, and Rosen were trying to make. Their point was that if we take it for granted that nature is local, then the EPR experiment shows that the wave function can't be a complete description of nature. This is because the wave function at the two separated detectors is the same, so if the detectors are far enough apart that a measurement at one can't influence the other, then you can't explain the observed correlations. I think this also explains why Einstein never embraced Bohmian Mechanics. It was always clear that Bohmian Mechanics was non-local. Einstein was looking for a local theory, so he wouldn't be interested in Bohmian Mechanics. This was quite reasonable on Einstein's part, since until John Bell came along and proved his inequality, there was no reason to think that you couldn't come up with a local theory that agreed with all the experiments. Once you realize you have to give up locality, then Bohmian Mechanics looks very attractive. > Not that I claim to fully understand (after my 30 minutes of study) > Bohm's theory, but I can imagine that the idea that changes in the > pilot wave's state propagate "at an infinite speed" (i.e., is > non-local) is going to be a real issue here when we consider > relativistic frames of reference. I think you have this backwards. It is the Copenhagen interpretation that suffers from this problem. In orthodox Quantum Mechanics, in addition to the Schroedinger Equation, you need rules that tell you how the wave function collapses when a measurement is made. (Without these rules, it isn't clear that the theory actually predicts anything.) And, this collapse must happen instantaneously. This is why the EPR authors weren't satisfied with the Copenhagen interpretion. Unfortunately, Einstein's catchy "God does not play dice" has given people the mistaken impression that Einstein objected to a non-deterministic theory; this isn't true. In Bohmian Mechanics, the wave function always evolves according to the Schroedinger Equation. So, whatever problem the Schroedinger Equation has with Lorentz invariance is shared by both Bohmian mechanics and orthodox quantum mechanics. -- David Marcus
From: David Marcus on 21 Oct 2006 19:43 Han de Bruijn wrote: > Bob Kolker wrote: > > > The are mathematical systems we know for sure are consistent since they > > have finite models. For example Just Plain Old Group Theory. No > > contradictions can be inferred from the group postulates simpliciter. > > True. And that is because the axioms of group theory actually serve as > a _definition_ of what a group _is_. Right? Just as the axioms of set theory define what a set is, the axioms of the natural numbers define what a natural number is, and the axioms of the real numbers define what a real number is. -- David Marcus
From: David Marcus on 21 Oct 2006 19:51
mueckenh(a)rz.fh-augsburg.de wrote: > I am not willing to waste my time by considering such a mess. Considering that all of your posts are such a mess that no one can understand what you are saying (assuming for the moment that you actually have something to say), please do stop wasting your (and our) time. -- David Marcus |