Cantor Confusion
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From: Dik T. Winter on 22 Oct 2006 21:05 In article <1161517636.934369.301190(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > lim{n --> oo} 1/n = 0 proclaims "omega reached". > > > > It proclaims nothing of the sort. You want to read more in notation than > > is present. > > You want to read less than present and necessary. > lim{k --> oo} a_k = pi means omega reached. Or would you state that pi > belongs to Q like every a_k for natural k? Of course not, and of course I do not state that oo is reached. Stating (for the first limit above) that omega is reached is tantamount to stating that there is an n such that 1/n = 0. There *is* no such n. And in the second statement it would mean that there is a k such that a_k = pi. Both are nonsense. > > > It is nothing but just a simplification in notation! > > > > It is not. When giving a function from R to R, f(0) might exist, but > > f(oo) *never* exists. So asking for f(0) in the first case is a > > legitimate question. > > Not more legitimate than asking for f(oo) because at noon also > infinitely many transactions must have happened. Therefore the > distinction you make is false. The distinction I make is *not* based on the number of transactions. It is simply based on the fact that when a function is given from R to R you can *not* ask for f(oo), but you *can* ask for f(0). (Whether the latter is defined is another matter.) Consider the following function from R to R: f(x) = x, you can now ask for f(0), but transform it to: g(x) = 1 / x, now you can *not* ask for g(oo) because that does not exist. The reason is that oo is not in the domain of the function. > > After the transformation you ask for f(oo), which > > is *not* legitimate. So the transformation does not simplify notation, > > it simply transforms the problem to an illegitimate problem. > > I hope you see your error. Without the actual existence of omega both > problems with omega transactions are undefined. The problem is *not* with the number of transactions. The problem is with the transformation which renders the question asked invalid, independent on the number of transactions. Consider also the following function, defined on [0,1]: f(0) = 0, f(x) = entier(-log_2(x)) when x != 0. You can not transform it to a valid function using the transformation y = 1/x. The best you can come with is: g(y) = ceiling(log_2(x)) but that does not capture the complete function, and transforming it back again leads to a function defined on (0,1], and so can never be the original function. > > > So is it not. Good heavens, that is unimportant for the present > > > argument. Every constructed number is an element of a countable set. > > > The set of all constructed numbers is countable. Every diagonal number > > > belongs to this set. > > > > Pray define what you understand under constructable number. In mathematics > > there is a precise definition. Do you think that e is constructible? > > Of course, by Sum 1/n! Ah, so you do not use standard mathematical terminology. In this case refrain from using standard theorems about constructable numbers for *your* constructable numbers. The constructable numbers according to standard definitions are countable (and that can be proven), I think that the constructable numbers according to your theory are uncountable, unless you provide us with a proof that those numbers are also countable. > > > No, I was not meaning that. I mean: constructed (by list or by > > > formula). > > > > Pray give a precise definition. > > A constructible number according to my view is a number which can be > constructed, i.e., a number of which every digit can be known either by > a formula or by a catalogue or a list or by whatever. In that case, according to your terminology, *all* real numbers are constructable. So the WM-constructable numbers are not countable. > > That deviates from the common > > mathematical meaning of constructable. > > Please leave me alone with your common definition! Darn. You did use a theorem about constructable numbers in one of your statements and applied it to WM-constructable numbers. The definition of constructable numbers is exactly that set of numbers that can be constructed using straight-edge and compass. So cuberoot(2) is not a constructable number, but is a WM-constructible number. That it is not a constructable number is because it is not possible to 'double the cube'. > The common > mathematical definition of "realism" has nothing at all to do with > reality and realism but is pure idealism. So, it seems, that common > mathematical use of words is often the opposite of the usual meaning, > but, alas, not in al cases. Then it were easy to talk this language. I > mean that a constructible number can be constructed. With straight-edge and compass. Or else, please solve the problems of 'doubling the cube', 'squaring the circle' and 'trisecting an angle', using straight-edge and compass. Apparently you are back to 15-th century mathematics. > Of couse 0.110001000... and all its sums with rational numbers are > constructible numbers in my sense. But the constructable numbers in your sense are not countable. > > Which "real" number is not constructable in your view? > > The great majority of real numbers does not belong to the set of > constructible numbers because this set is countable. This is extreme nonsense. Every real is constructable with your meaning of constructable, but you have *not* provided a proof that WM-constructable numbers are countable. -- 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 22 Oct 2006 21:51 In article <1161518008.776999.238550(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1161435575.019298.164830(a)e3g2000cwe.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > > > 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) > > > > Yes, if you define X(omega) like that your conclusion is obvious. > > Not I define omega as the limit ordinal number. That is a matter of set > theory. You use oo, which is *not* the same as omega. But when we ignore that difference, more problems prop up. X(t) was a function from R to R, I think, not a function from ordinals to something else. But you stated that X(t) was 9.t. Again lacking precision. On the one hand, if X(t) = 9.t, we get: lim{t -> omega} X(t) = X(omega) but with X(t) = t.9, we get: lim{t -> omega} X(t) != X(omega). Yes, in the ordinals multiplication is not commutative... > Not I define omega as the limit ordinal number. That is a matter of set > theory. I only use the continuity of a function which is already > required to find lim 1/n = 0. There is no continuity required to find such a limit. Limits are independent on continuity. The function: f(x) = 1 when x = 0 and = 0 when x != 0 is certainly not continuous at x = 0, nevertheless the limit for x -> 0 exists, and is 0. What continuity do you use when you calculate: lim{x -> 0} f(x) ? > > > lim {t --> omega} t = omega > > > > Oh. Provide a mathematical definition of that limit, please. In standard > > mathematics that limit is undefined. > > Cantor used omega with two slightly different meanings. omega is the > set N and omega is the first infinite ordinal number, i.e., the > smallest number larger than any natural. These two definitions yield: > lim {t --> oo} t = omega > and > lim {t --> oo} {1,2,3,...,t} = omega. Yes, so what? I asked for a mathematical definition, not for handwaving. And none of the usages of Cantor do in any way define the limit. What *is* your mathematical definition of that limit? > > > > 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. > > > > Wrong. > > Wrong is wrong. The limit *is* the irrational number. You can use > these and only these numbers in a Cantor list, not the equivalence > classes of sequences. You really do not understand how the reals are defined. The limit is *not* the irrational number. The limit does not even exist. > > > > > (7) assume sequences of rationals. Create equivalence classes amongst > > > > those sequences (a_n ~ b_n if |a_n - b_n| goes to 0; but this is > > > > losely speaking and quite a few other methods are known, all > > > > equivalent). > > ... > > > > So, at what stage in this process is the limit of a function used to > > > > define the irrationals? > > > > > > At (7). The equivalence classes of sequences of rationals with same > > > limit. > > > > Wrong. At that point you can not talk about sequences of rationals with > > the same limit, because many of such sequences do not have a limit in the > > rationals. So (7) is formulated as I wrote it (in one of the forms to > > define the reals from the rationals). It is not the *limit* that is the > > irrational, it is the equivalence class of sequences. > > The sequences belong to Q. So your irrational numbers belong to Q? That > is nonsense. In Q we have sequences with Cauchy-convergence and, > therefore, perhaps without a limit in Q. But the irrational numbers are > definitely *not* in Q. Pray re-read what I wrote. The real numbers are defined as equivalence classes of sequences of rational numbers. The sequences do not belong to Q. They are sequences of elements of Q. So when defining reals (according to this methodology) we start with sequences of rationals. We call two sequences equivalent if their difference goes to 0, the concept of limit has not yet even been defined. It is easily shown that that is an equivalence relation, so we can divide the sequences in equivalence classes. Each of those equivalence classes is a real number. So a real number is an equivalence class of rationals. It is easy to embed the rational numbers isomorphically in the set of real numbers (when arithmetic on the real numbers is properly defined). > > And (again, > > losely speaking) the equivalence classes are built in such a way that > > all members of the classe *ought* to have the same limit in the extended > > system. > > Exactly. And that is the irrational number, *not* loosely speaking. It is *extremely* losely speaking. The real (not irrational) number is an equivalence class, it is not a limit. So initially, 1/2 is *not* an element of this new system. An element of this new system is an equivalence class of which the sequence: 1/2, 1/2, 1/2, ... is a representative. > You > can use this and only this number in a Cantor list, not the equivalence > class of sequences (because the due terms are not uniquely defined). Oh, perhaps, I do not understand at all. I do not see the relation. > Summarizing the original question: You need the limit omega to > construct the irrational numbers. Where in the construction above did I use the limit omega? -- 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 22 Oct 2006 22:00 In article <1161518242.756958.103660(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <virgil-E8EF11.13483421102006(a)comcast.dca.giganews.com> Virgil <virgil(a)comcast.net> writes: > > > In article <1161435318.373825.152830(a)e3g2000cwe.googlegroups.com>, > > > mueckenh(a)rz.fh-augsburg.de wrote: > > ... > > > > 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. > > > > I would not like that as a definition. That would make: > > lim{n -> oo} {n, n + 1, n + 2, ...} = N > > Obviously wrong, because 1, 2, 3, .., n-1 all belong to N Eh? Using the definition Virgil gave leads to my conclusion without further ado. And I just stated that it was not desirable. Obviously wrong is something else. If Virgils definition is the definition, my conclusion is obviously right. So there should be a search for a definition that allows what is wanted. Virgils definition, although not wrong, definitions can not be wrong, does not lead to desirable results. > lim [n-->oo] {-1,0,1,2,3,...,n} = N > > is obviuously wrong too. Depends on how you define the limit. > Therefore lim [n-->oo] {1,2,3,...,n} = N. Depends on how you define the limit. > > I gave sometime ago a definition of the limit of sets that is (in my > > opinion) workable, but Mueckenheim did not allow that definition. The > > reason being that under some formulations of the vase problem that > > definition would make the vase empty at noon. > > So it must be wrong and needs no further attention. Why? But you never give a definition of the limit of sets. You only state that definitions are *wrong* (although I fail to see why a definition can be wrong). But you never state a proper definition. > Because the contents of the vase increases on and on. Such a process > cannot lead to emptiness in any consistent system - independent of any > "intuition". That requires proof. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: David Marcus on 22 Oct 2006 22:01 cbrown(a)cbrownsystems.com wrote: > David Marcus wrote: > > cbrown(a)cbrownsystems.com wrote: > > I should state up front that I don't actually claim that "A and not A" > is a good ontological description of entanglement; I was mostly just > tweaking HdB. A worthy endeavor. > > But, if one theory is vague or ill defined, then it can be hard to say > > whether an experiment really supports it. > > I donlt quite get what you are implying here. I assume you have some > particular physical experiment in mind where this is the case as > regards QM. Could you elaborate? Not a particular experiment, but the Copenhagen interpretation itself. See "What is the meaning of the wave function" by Jean Bricmont. Google turns up several copies on the Web. It seems that Bohr intentionally wanted to make things vague. There was an interesting article in Physics Today a few years ago that analyzed Bohr's political/philosophical views and argued that they influenced his physics: "The Sokal Hoax: At Whom Are We Laughing?" by Mara Beller, Physics Today, September 1998, pages 29-34. > If I understand correctly, very loosely, Bohm sees the probablistic > results of QM as being a result of the fact that the hidden variable > regarding position, by virtue of its being inherently unmeasurable > beyond some arbitrary degree of accuracy, introduces errors in our > computations which are essentially chaotic (in the mathematical sense). > So we see results that "look" probabilistic, when in fact they are > fully determined. I don't think so. Bohmian Mechanics is 100% deterministic. All of the uncertainty in the results of an experiment is due to uncertainty in setting up the initial conditions of the experiment. The following papers (available at http://www.math.rutgers.edu/ ~oldstein/) are relevant (and discuss this much better than I can). Quantum Chaos, Classical Randomness, and Bohmian Mechanics, with D. D?rr and N. Zangh?, Journal of Statistical Physics 68, 259-270 (1992) Quantum Equilibrium and the Origin of Absolute Uncertainty, with D. D?rr and N. Zangh?, Journal of Statistical Physics 67, 843-907 (1992) > In my admittedly uninformed opinion. the Copenhagen interpretation > takes probalistic results it to be evidence that, for classical > quantities such as position, it is the case that (like my hometown > Oakland, CA) "there is no there there". Our assumption that there is a > specific "element of reality" called "the object's position" is simply > a classical intuition, like the intuition that it shoudl be possible to > state unambiguously that "A and B occur simultaneously". Instead, it > posits as its "elements of reality" more complex mathematical objects > (self adjoint operators). I don't think the operators are elements of reality in the Copenhagen interpretation. If anything at all is supposed to be real, it is the wave function. The operators are just how you describe a "measurement" (or experiment). Although, I think the Copenhagen interpretation says that nothing is real and the mathematics is just a way of predicting the results of experiments. > I understand that there are philosophical attractions to both > explanations; but I don't see why you claim that there are certain > 'logical' questions which cannot be asked in the latter interpretation. I don't think there are any philosophical attractions to the Copenhagen interpretation (except perhaps to Bohr). It is basically just a way to avoid having to admit that QM still needed some more work. Have you read Bell's book? You really should. > > > 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 your analysis is essentially correct. But if I understand > correctly, this relies on a particular definition of "element of > reality"; Not according to Einstein, Podolsky, Rosen, and Bell. I've read what they've written, and I agree with them. I suggest you read "Bertlmann's socks and the nature of reality" by Bell. It is reprinted in the book "Speakable and Unspeakable in Quantum Mechanics" by J.S. Bell. > which in the Copenhagen interpretation is the probability > distribution, not some real number. > Again, it's been a while since I've looked at this stuff, but > considering the comments found at: > > http://www.mth.kcl.ac.uk/~streater/EPR.html > http://www.mth.kcl.ac.uk/~streater/lostcauses.html#I > and > http://www.mth.kcl.ac.uk/~streater/lostcauses.html#XI > > it seems that while EPR + Bell inequalities implies there are no local > hidden variable descriptions; it doesn't imply that therefore QM need > be a non-local theory. If I read the comments above correctly, it > remains local in spite of Bell, because it identifies the "elements of > reality" as probability distributions (self adjoint operators). I'm pretty sure the self-adjoint operator describes the experiment and you combine it with the wave function to calculate the probability of the experimental outcomes. But, this doesn't get around Bell's inequality. In deriving his inequality, Bell only assumes locality. Everything else he deduces from the locality. Bell himself remarked that it was very hard to get this point across. However, if you read what he wrote, you can easily confirm that Bell is 100% correct. > Despite his obviously unsympathetic view of Bohm mechanics as a "Lost > Cause", his arguments seem appealing as at least a logical account of > EPR and Bell in the C
From: David Marcus on 22 Oct 2006 22:08
cbrown(a)cbrownsystems.com wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > cbrown(a)cbrownsystems.com schrieb: > > > To clarify: suppose we agree that "since there is a set having the > > > properties of omega, therefore ..." is not a valid for a correct > > > argument. Do you have complaints about the remaining assumptions and > > > axioms of ZFC? Or do you find them (and the logical conclusions that > > > follow from them) to be acceptable as being true, obvious, logical, > > > correct, etc. in mathematical discourse? > > > > Finite set theory is fine and true and useful . > > I wouldn't neccessarily call ZFC - AoI "finite set theory". It seems clear that Mueckenheim has no idea what ZFC is. > We cannot deduce "there does not exist a Dedekind-infinite set" from > ZFC - AoI; that too would need to be asserted as an axiom. -- David Marcus |