Cantor Confusion
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From: mueckenh on 30 Sep 2006 06:03 William Hughes schrieb: > For every real number x, there exists a list, L_x such > that x is a member of L_x Where do they exist? It is impossible for most real numbers even to name them. You cannot name or construct more than aleph_0 real numbers. Nevertheless you insist, that also the other ones, which have no names and no other identification properties, should have complete lists? > > There exists a list L, such that every real number x is > a member of L. > > The first is true, the second is false. There is no way to put > all the L_x together to get a "countable set of entries" > (the list L). But if they existed, then we could put them together. Why can't we connect in our thoughts all these thought lists such that there is only one thought list, i.e., the thought list of all thought lists? At least a square of all thought lists should be possible. But if you like, you can schematically consider all real numbers by the infinite binary tree which contains them all represented by a countable set of nodes and edges. I have shown by a rational relation that the set of branches (corresponding to real numbers) is not larger than the set of edges. Regards, WM
From: mueckenh on 30 Sep 2006 06:09 Dave L. Renfro schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > But the set of all lists is countable (as is any quantized > > or discontinuous set), so is the set of all list entries. > > Nevertheless, there is no real number missing in every > > list? So every real number is in at least one of the > > list? So every real number is one element of a countable > > set of entries? And there is nothing real really outside > > of this countable set? > > The set of all lists of _what_? Also, "lists" is a term > that for some people means one thing, and for other people > it can mean something else. For me, a list is the physical representation of an injective sequence. > If you're going to argue about > details, rather than give a general overview of what's > going on, you should use the correct mathematical terms, > such as one-to-one correspondences (i.e. bijective functions). > Otherwise, the discussion is going to degenerate into > disagreements over word usage. In fact, you're doing it > yourself by bringing up "quantized" and "discontinuous", > two words that, to most people in math, have no direct > relevance to the discussion at hand. A list as a physical representation requires some space and is a quantized object with a finite volume dV. It can be shown that everything which is not a spaceless entity like a point or a line or an area belongs to a countable set. > > It never ceases to amaze me that people can have so much > trouble with the main gist (not the details) of the > argument. But there are opposite conclusions which can be derived by mathematical deduction. One of them is the theorem that there is a countable set which contains more elements than its power set. 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) I find it puzzling that people have so much trouble with the main gist (not the details) of the argument. Regards, WM
From: mueckenh on 30 Sep 2006 06:11 cbrown(a)cbrownsystems.com schrieb: > Therefore, the assertion "M is a complete list of reals" is only true > if the assertion "M is complete, and M is not complete" is true. > > (A and ~A) = false. A system has the property W, if it can be proved that the reals can be well-ordered. A system has the property ~W if it can be proved that the reals cannot be well-ordered. A system is self-contradictive, if W and ~W can be proved. Therefore the system does not exist. Regards, WM
From: mueckenh on 30 Sep 2006 06:13 Tonico schrieb: > The argument seems extremely simple to me...so simple that some seem to > feel it MUST be wrong, for some reason (perhaps the reason is that if > something ain't complex then it can't be maths...): take ANY list of > real numbers (or take ANY injective mapping > f:N --> R). Then it can be shown with the genial and simple diagonal > argument that Cantor came up, Unfortunately, his method fails in some cases which is fatal for an impossibility proof. The list, for instance, 0.0 0.1 0.11 0.111 .... with the prescription that the diagonal digit 0 is replaced by 1, delivers a number which is not different from any list number, except in the last digit, which, however, does not exist. Regards, WM
From: mueckenh on 30 Sep 2006 08:40
Virgil schrieb: > In article <451dddd9(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > > > > > > > The original proof was regarding a complete language using > > at least two symbols, m and w, no? > > Not quite. Two disjoint sets of synbols. > > > That was later conflated to a proof about the reals. > > It was later shown that it could be modified to form a proof that the > set of all reals is uncountable. This was *not* "later shown", but at the very time of publishing in 1890/91 Cantor considered this very proof as the proof of the uncountability of he reals. Cantor, in the first paragraph: " Es läßt sich aber von jenem Satze [uncountability of the reals] ein viel einfacherer Beweis liefern, der unabhängig von der Betrachtung der Irrationalzahlen ist." My translation: "Here is a much simpler proof of the theorem [uncountability of the reals] which is independent of the reference to irrational numbers" Regards, WM |