Cantor Confusion
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From: Virgil on 17 Oct 2006 15:10 In article <1161079038.832119.285630(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > imaginatorium(a)despammed.com schrieb: > > > 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? In a sense, yes! The set of numbers of balls removed from the vase, as a function of time, is a continuingly , but discontinuously, increasing function whose "limit", and therefore its value, at noon is N.
From: Virgil on 17 Oct 2006 15:13 In article <1161079173.794258.210500(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > 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? Yes! And we have done so several times.
From: Virgil on 17 Oct 2006 15:18 In article <1161079384.757420.182710(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. There are perfectly adequate definitions of what "infinite" means, and as "set" is one of the primitive undefined terms of set theory, it is only necessary to know what the axioms say about how sets behave, there is no need to know what sets "are". And while set theory is rife with references to things being finite or infinite, I do not recall a single reference to anything being infinity. Do you have such a reference?
From: Virgil on 17 Oct 2006 15:23 In article <1161079581.867611.209000(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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. Where in the construction of bijections may the members to be bijected be "subdivided" or "shared"? > > > 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? In the subdivision and sharing of members to be bijected. No axiom or definition of set theory, or of any other theory besides "Mueckenh"'s ever mentions any such thing. In all the bijecting that I have ever been exposed to, all the members of the all the sets in question must be regarded as unique and indivisible and unshared.
From: Virgil on 17 Oct 2006 15:29
In article <1161079685.233073.120000(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > There is no "equal weight" in the proof. > > > > You haven't yet noticed it? Each digit of the infinitely many digits of > the diagonal number has the same weight or importance for the proof. It is only necessary that each digit have non-zero weight in order for the difference between the diagonal and one of the list to be non-zero. > In > mathematics, the weight of the digits of reals is 10^(-n). At least in decimal notation, but as that makes all the weights non-zero, that is sufficient to distinguish the diagonal from each of the listed numbers. >Infinite > sequences of digits with equal weight are undefined and devoid of > meaning. As no such sequences are involved, the comment is irrelevant. |