Cantor Confusion
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From: imaginatorium on 17 Oct 2006 00:58 Ross A. Finlayson wrote: > Virgil wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > I warned you that this point is new: The edges are split in shares of > > > 1/2, 1/4, and so on. But when fractions were introduced in mathematics, > > > most people may have had the same problems as you today. I am sure you > > > can understand it from the written text above. (Many others have > > > already understood it.) > > > > Name one. > > Oh, don't worry, that's a problem everywhere. Ah, nice of you to step in to clear up the linguistic confusion, Ross. Uh, is the "everywhere" in the last sentence the same as that in "There are everywhere reals between 0 and 1"? What does it mean? That some of the reals between 0 and 1 are actually on platform 4 of Waterloo Station? But let's not quibble about this problem. I like salad a lot. I've recently discovered that pumpkin seeds are the best way of adding Real Meatiness (tm) to the usual mix of lettuce, mizuna, and onion. > Consider the differences between for any, for each, for all, and for > every. They're normally lumped under interchangeable representations > of the universal quantifier. The transfer principle doesn't always > hold. There is also the problem of the interpretation of 'every' in the following, which is my attempt to piece together Wolfgang Muechenheim's answer to the original question above - who is this that understands his proof? "One cannot know him, cannot call him by his name, but he is provably present. You should be familiar with this from of existence. It is like the well-order of the reals: present but very." > Yeah, like some years ago there weren't methods to solve the wave > equation. Right, but these have surely been developed from of existence. The problem with salad is not that every day is lettuce. I could, and perhaps will, eat lettuce every day until I die. But I want new lettuce, succulent and tasty, not just the same pieces of lettuce as in every previous post. > Steve: It is. Hard to argue with that. > Alan: deperdit numerus. Consider the chicken farm. Uh, not sure of the connection here? > It's in the null axiom theory. Right - well-order the reals!! Brian Chandler http://imaginatorium.org
From: mueckenh on 17 Oct 2006 05:57 imaginatorium(a)despammed.com schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Randy Poe schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Randy Poe schrieb: > > > > > > > > > Han de Bruijn wrote: > > > > > > > I merely note that there is no requirement in the problem that > > > > > > > the limit be the value at noon. > > > > > > > > > > > > The limit at noon - iff it existed - would be the value at noon. > > > > > > > > > > Wrong. That is a flat out incorrect statement showing a > > > > > fundamental misunderstanding about what limits mean. > > > > > > > > > > A CONTINUOUS function at x0 has the property that the > > > > > limit of f(x) as x->x0 is f(x0). But not all functions are > > > > > continuous. > > > > > > > > And you are in charge of determining which functions are continuous and > > > > which are not? > > > > > > Where do you get this stuff from? > > > > > > How do you translate a statement that some functions are not > > > continuous into "I am in charge of determining if some functions > > > are continuous"? > > > > Because it seems a bit mysterious, how you know or can define that the > > function of balls in the vase was not continuous. Or, may be, because > > you will accept that the function is continuous if the balls are taken > > out in the sequence 1, 11, 21, .... > > > > > > No, I am not in charge. Non-continuous functions are non-continuous > > > now and forever. They were non-continuous before I existed, they will > > > remain non-continuous after I'm gone. > > > > > > The number of balls in the vase is such a function. > > > > How do you acquire that knowledge? > > By the application of a little elementary mathematical knowledge, I > should think. > > (1) For any positive value nu (integer, but actually a real will do > too), for a sufficiently small value of tau, the number of balls in the > vase at time noon-tau is greater than nu. This can be derived > tediously, but obviously, from the set of step functions, one for each > natural n, representing the state of the nth ball. Therefore the limit > to the number of balls as time approaches noon from the "left" > diverges. > > (2) For the value t=noon, there does not exist any n for which the step > function representing the state of ball n has the value IN. Therefore > the set of balls with the corresponding value IN is empty, and the > number of balls is zero. > > Therefore the limit as t->noon does not equal the value at noon. This > simply follows from the statements in the problem. But the function of balls/numbers removed from the vase is a continuously (stepwise) increasing one, containing all natural numbers at noon? Regards, WM
From: mueckenh on 17 Oct 2006 05:59 Randy Poe schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Virgil schrieb: > > > > > > According to the ZFC system: The vase is empty at noon, because all > > > > natural numbers left it before noon. > > > > By means of the ZFC system we can formulate sequences and their limits > > > > in mathematical language. From this it follows that lim {n-->oo} n > 1. > > > > And from this it follows that the vase is not empty at noon. > > > > > > By what axiom do you conclude that the limit as t increases towards noon > > > of any function and the value of that function at noon must be the same? > > > > By that or those axiom(s) which lead(s) to the result lim {t-->oo} 1/t > > = 0. > > Here is your theorem: Let f(x) be any function f:R->R. Then > lim(x->0-) f(x) = f(0). That is, the limit of f(x) as x approaches > 0 from the left is f(0). > > Can you show me how the axiom(s) you describe prove > that theorem? > > Can you then show me how the theorem applies to this > function? f(x) = 1 if x<0, f(x) = -1 if x>=0. If there is no stepwise continuity in f(t) = n, can you show me why the set of balls/numbers removed from the vase is containing all natural numbers at noon after the number of transactions t --> oo? Regards, WM
From: mueckenh on 17 Oct 2006 06:03 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > A good, if no the best source to learn about the different meanings of > > infinity would be Cantor's collected works. > > Set theory has advanced since Cantor. The best source to learn about > current set theoretic definitions of 'infinite' is not Cantor. There is no definition what infinity is and there is no definition what a set is. Current set theory has forgotten about theoretic definitions of the infinite at all and uses the notion "infinity" just as seems necessary to avoid too obvious contradictions. Regards, WM
From: mueckenh on 17 Oct 2006 06:06
MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > Han de Bruijn wrote: > > > > Set Theory is simply not very useful. The main problem being that finite > > > > sets in your axiom system are STATIC. They can not grow. > > > > > > Set theory provides for capturing the notion of mathematical growth. > > > Sets don't grow, but growth is expressible in set theory. If there is a > > > mathematical notion that set theory cannot express, then please say > > > what it is. > > > > Obviously the notion of "rational relation" as used in the binary tree > > cannot be expressed by mathematical notion: > > Consider the binary tree which has (no finite paths but only) infinite > > paths representing the real numbers between 0 and 1. The edges (like a, > > b, and c below) connect the nodes, i.e., the binary digits. The set of > > edges is countable, because we can enumerate them > > > > 0. > > /a \ > > 0 1 > > /b \c / \ > > 0 1 0 1 > > ............. > > > > Now we set up a relation between paths and edges. Relate edge a to all > > paths which begin with 0.0. Relate edge b to all paths which begin with > > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > > a is inherited by all paths which begin with 0.00, the other half of > > edge a is inherited by all paths which begin with 0.01. Continuing in > > this manner in infinity, we see that every single infinite path is > > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related to any > > other path. The set of paths is uncountable, but as we have seen, it > > contains less elements than the set of edges. Cantor's diagonal > > argument does not apply in this case, because the tree contains all > > representations of real numbers of [0, 1], some of them even twice, > > like 1.000... and 0.111... . Therefore we have a contradiction: > > > > Card(R) >> Card(N) > > || || > > Card(paths) =< Card(edges) > > What I see above is a lot of mathematical terms that are used in set > theory with precise definitions, but for which I do not know your own > personal definitions. There are no personal definitions. There is only one extension of current state, which, however, is not in contradiction with any axioms, namely that edges can be subdivided and the shares can be counted. > You may not agree with the axioms of set theory, Which part of my proof is not in agreement with current axioms and definitions of set theory? Regards, WM |