Cantor Confusion
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From: mueckenh on 11 Jan 2007 05:38 Dik T. Winter schrieb: > In article <1168444071.169167.287510(a)i39g2000hsf.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > The union of all finite trees (= union of all levels) of the tree > > > > automatically contains all paths, because there is no path or part of a > > > > path outside of this union. > > > > > > And again, in that case the set of paths in the union is not the union of > > > the sets of paths. I did show that with a finite graph. So I wonder > > > why you think it holds for an infinite graph (which a tree is)? > > > > We are concerned with real numbers which are represented in the tree > > just as I defined it. > > May be. > > > Every paths splits at a node into a left one and > > a right one. In the unin of trees there are all paths or pats of paths > > (in the sense defined and also the crippled ones) which exist in this > > union. Nothing can be added unless you add another edge. > > That still does *not* show that the union of the sets of paths in the > finite trees contains infinite paths. You still fail to provide a > proper definition of the union of trees. So I can only guess. Here is the formal proof: Theorem. The set of real numbers in [0, 1] is countable. Lemma. Each digit a_n of a real number r of the real interval [0, 1] in binary representation has a finite index n. r = SUM (a_n * 2^-n) with n in N and a_n in {0, 1}. Proof. A natural number n can be represented in a special unary notation: n = 0.111...1 with n digits 1 (the leading 0. playing no role). Example: 1 = 0.1, 2 = 0.11, 3 = 0.111, ... In this notation the definition of the set of natural numbers, (1, 2, 3, ...} = N, reads {0.1, 0.11, 0.111, ...} = 0.111.... (*) Note that also the union of all finite initial segments of N, {1, 2, 3, ...., n}, is N = {1, 2, 3, ...}. Therefore (*) can also be interpreted as union of initial seqments of the real number 0.111.... A real number r of the real interval [0, 1] can be represented as one (ore two) path in the infinite binary tree. The set of all real numbers r of the real interval [0, 1] is then given by the infinite binary tree: 0. / \ 0 1 / \ / \ 0 1 0 1 .................... A finite binary tree is the infinite binary tree, cut off below a level n with n in N. Here is a tree with two levels: 0. / \ 0 1 / \ / \ 0 1 0 1 namely level 1 and level 2. (The root at level 0 is conveniently not counted, because 0 is not a natural number.) The union of binary trees is defined as the union of levels. The union of two or finitely many different finite binary trees simply is the largest on. Taking the uninion of all finite binary trees, we get the complete infinite binary tree with all levels. All infinite paths representing real numbers r of the real interval [0, 1] are in this union. We can see this by the path always turning right, 0.111..., which is present in the tree, according to (*). Conclusion: Every finite binary tree contains a finite set of path. The countable union of finite sets is countable. The set of paths is countable. The set of real numbers in [0, 1] is countable. QED. Regards, WM
From: mueckenh on 11 Jan 2007 05:39 mueck...(a)rz.fh-augsburg.de schrieb: > Franziska Neugebauer schrieb: > > > >> Since there is no largest element in "potentially" infinite sets (in > > >> "actual/complete/finished", too) this sentence makes no sence at all. > > > > > > A potentially infinite quantity (set or not) is always finite. > > > > There is no time in maths. > > So you write these letters and develop new ideas in zero time? There is > no existence outside of time. > > > > > Therefore in a linearly ordered set here is a last element. Contrary > > > to the claim of set theorists, a set is not fixed in reality. > > > > When your arguments (by the way: What exactly are your arguments?) > > Here you can read it: > > Theorem. The set of real numbers in [0, 1] is countable. > > Lemma. > Each digit a_n of a real number r of the real interval [0, 1] in binary > representation has a finite index n. > r = SUM (a_n * 2^-n) with n in N and a_n in {0, 1}. > > Proof. > A natural number n can be represented in a special unary notation: n = > 0.111...1 with n digits 1 (the leading 0. playing no role). Example: 1 > = 0.1, 2 = 0.11, 3 = 0.111, ... > In this notation the definition of the set of natural numbers, (1, 2, > 3, ...} = N, reads > > {0.1, 0.11, 0.111, ...} = 0.111.... (*) > > Note that also the union of all finite initial segments of N, {1, 2, 3, > ..., n}, is N = {1, 2, 3, ...}. Therefore (*) can also be interpreted > as union of initial seqments of the real number 0.111.... > > A real number r of the real interval [0, 1] can be represented as one > (ore two) path in the infinite binary tree. The set of all real numbers > r of the real interval [0, 1] is then given by the infinite binary > tree: > > 0. > / \ > 0 1 > / \ / \ > 0 1 0 1 > ................... > > A finite binary tree is the infinite binary tree, cut off below a level > n with n in N. > Here is a tree with two levels: > > 0. > / \ > 0 1 > / \ / \ > 0 1 0 1 > > namely level 1 and level 2. (The root at level 0 is conveniently not > counted, because 0 is not a real number.) CORRECTION: 0 is not a natural number. > The union of binary trees is defined as the union of levels. > The union of two or finitely many different finite binary trees simply > is the largest on. > Taking the uninion of all finite binary trees, we get the complete > infinite binary tree with all levels. All infinite paths representing > real numbers r of the real interval [0, 1] are in this union. We can > see this by the path always turning right, 0.111..., which is present > in the tree, according to (*). > > Conclusion: Every finite binary tree contains a finite set of path. The > countable union of finite sets is countable. The set of paths is > countable. The set of real numbers in [0, 1] is countable. QED. > > Regards, WM
From: cbrown on 11 Jan 2007 06:51 Andy Smith wrote: > > > > Are you trolling? > > > I'm not sure what trolling is, but I infer from context > that it is undesirable. > http://en.wikipedia.org/wiki/Troll_%28Internet%29 Not generally a compliment! > > Anyway, the thread is called Cantor confusion, and I am > definitely confused. > > You mathematicians have had 100 years of intense nit-picking > over infinite sets, and I am sure that everything that > you say is 100% self-consistent. This is prima facie evidence that you are not trolling, but are instead curious why your (admittedly, possibly naive or hell, even dumb-dumb) argument does not follow. > But I still think that your response to Zeno's paradox > (there is no last jump) even if it is consistent > does violence at the least does violence to "common sense" ...if that term can ever be applied in the context of infinite sets. > But no one (but you, hypothetically) is claiming that your flea could "really" exist. Mathematics is a fantasy land, replete with elves and dwarves and trolls (the latter being the magical kind). But under certain conditions, if something is true in this fantasy land, it is assuredly true in reality. For example, if we can prove that something is true for all sequences {a_n} of the naturals (each of whose very existence is a physical impossibility: what process generates an infinite number of sequential elements?), then we have also proven that it is true for all /finite/ sequences of naturals. > No I don't believe any of the guff that I wrote a few posts > ago, as I thought was plain from context. But you lot > take Peano as sacred writ - maybe one might get something > a lot less tricky with some slightly different axioms? > If you think about it, Peano really zoomed in on exactly the "non-tricky" parts of it. I start with either nothing or something. Perhaps controversial, but certainly easily understood. This gives "nothing" a non-ambiguous meaning (often formalized by equating it with the symbol for, e.g., the empty set, or 0). Since I accept that there is also "something", I'll go further in that there are other things: one thing, and then another thing. and so on. Peano is essentially trying to capture: do you agree that "the naturals", whatever you may think that may /mean/, at the very least obeys these very simple rules: we start with 0, we have a rule that says that given some number of things, we can always add 1 to that number and come up with different number, etc. Now, "your" naturals may be different than "my" naturals because "my" naturals obey certain additional idiosyncratic (from "your", sadly, limited viewpoint) properties. E.g., that primes of the form 6*n + 1 are red, and primes of the form 6*n - 1 are green; and that therefore I have a theorem that no yellow primes exist. On the other hand, we /can/ agree that, for example, 24 is the unique product of powers of primes 2^3*3. The "pathetically limited" Peano definition of the naturals (which after all, makes /no/ mention of color /whatsoever/!) allows us to agree on facts such as the above. And that is the usefulness of mathematical abstraction. Metaphorically: Not everything that can be true in some part of fairyland can be true in reality; but everything that must be false in all parts of fairyland must be false in reality. And often, it is easier to show that something is false in all fairyland than it is to show that that same thing is false in reality. Most mathematical proofs can be read (by a physicist) as asserting "in any magical fairyland such that (some very particular definitions) holds true, it follows that (some admittedly /intuitively/ impossible thing) holds true". Whatever attitude one may have regarding the /reality/ of one's own vs. other's particular mathematical fairyland, we can then agree that certain things are true, and others are false. > Anyway as i said earlier, I am giving up and finding a book. "Oh, no! Please don't throw me in the briar patch!" Cheers - Chas
From: Franziska Neugebauer on 11 Jan 2007 07:16 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> "Different line ends are always >> >> elements of one single line" is not true for the potentially >> >> infinite set of all "Different line ends". >> > >> > Of course it is. The potentially infinite set is always a finite >> > set. >> >> First of all "potentially infinite" is an attribute not a noun. > > The "potentially infinite" can be used as a noun. But I spoke of the > "potentially infinite set" where "potentially infinite" is used as an > attribute. My remark went into the reply of a different posting. Let me refer to ,----[ <1168511487.581261.39710(a)77g2000hsv.googlegroups.com> ] | He misundertands the meaning of potential infinity. `---- >> Since maths (with or without "potential infinite sets") does not >> entail a concept of time your last sentence does not make sence. As >> far as I can see William has proved that a "potential infinite" set >> is _not_ finite. > > He misundertands the meaning of potential infinity. 1. Please explain what exactly he misunderstands. 2. Does he use a wrong definition of "potentially infinite set"? If so, please give the "right" one, you want to base your game on. >> > Don't you know that? Have you another definition of potential >> > infinity? Do you think it is actual infinity? >> >> The question is whether you accept (for the sake of reasoning) that >> there are "potentially infinite" sets. I. e. sets which are not >> finite but also not provably-infinite in the sense of "actual >> inifinite". > > i.e., they are always finite but have no upper bound and, therefore, > no fixed last line. This makes no sense. For the following reasons: 1. Sets in general have no lines at all. 2. "Always finite" makes no sense or is contradictory. "Always" refers to a temporal aspect which is not present in mathematics due to the lack of time. Finite in size implies having a last element. 3. As has been pointed out, if a set has "no last element" (not line) it is not finite in size. It seems to me that there is a misunderstanding on your side, which may be due your process-like imagination of what sets "really are". >> >> > > L_D does not exist. >> >> > >> >> > Of course does it not exist, because the diagonal does not >> >> > exist. >> >> > >> >> >> >> Make up your mind. Your repeated claim is that L_D does exist. >> >> >> > My claim was that L_D exists as a fixed line IF the complete >> > diagonal exists actually. (This was argued in order to disprove >> > actual infinity.) >> >> In the last say one hundered posts your have been discussing with >> William Hughes the issue under the presumption that the diagonal >> exists precluding the provable-existence of every of its elements. >> >> Since you now want to discuss the existence of L_D under the >> presumption that the diagonal "actually" exists, you are now >> discussing a different issue. > > Wrong understanding of pot and act. "To exist" means "to exist > actually". Usually "an entity x exists" simply means there is an axiom or a (proven) theorem which states "x exists". The game we played so far was: To not use the "each element of the diagonal exists" and of course to assume "the diagonal exists". This makes the diagonal "potentially infinite" as far as I can see. Would we have assumed that "each element of the diagonal exists" the diagonal would be "actually infinite". These are obviously two different assumptions in both of which you get a contradiction after you introduce the existence of a last line L_D. Hence L_D does not exist. This is what WH proved and what remains valid. > I merely emphasize the complete existence by adding the > "actually". 1. Adding to what? 2. Means what? > Potential infinite sets do not exist (or actually exist), If we did not (at least for the sake of reasoning, "as-if") assume that potentially infinite sets do exist, it would be meaningless to talk about them. So far we have been assuming the diagonal to (at least "potentially infinitely") exist. > but sets are always finite, though there is no upper bound. That is > the definition. I can't spot what is to be defined here. >> > For the potentially infinite diagonal, a last line also exists, but >> > not as a fixed line. >> >> There is no time in maths. Hence your sentence is meaningless. Is it >> possible that you confuse a "potentially infinite" set with a >> computer program which step-by-step generates the set members? If so, >> than your arguing is obviously driven by wrong imaginations. >> >> William's proof that no last L_D exists even if the diagonal exists >> only "potentially" remains valid. >> >> If you want to attack his proof you must show an error _therin_. > > I do not "attack" this "proof" because it is not a proof. WH posed some presumptions and applied valid reasoning to yield the statement "there is no last line L_D". This is considered to be a proof of "there is no last line L_D". If you want to play a different game you should state its rules. F. N. -- xyz
From: Franziska Neugebauer on 11 Jan 2007 07:26
mueckenh(a)rz.fh-augsburg.de wrote: ,----[ <45a3ae6b$0$97254$892e7fe2(a)authen.yellow.readfreenews.net> ] | No. I do refer to "complete" in the sense of "finished" in contrast to | "potential" inifite. Since neither the list and hence nor the diagonal | is finite there are no largest elements. OTOH since we do not assume | every member of the list or the diagonal to provably exist, the list | and the diagonal are "potential" infinite. | | This said William Hughes has shown that the assumption of the | exist[e]nce of a "potentially infinite" "last line" L_D leads to a | contradiction. Hence "L_D exists" is wrong. `---- So you agree to this part you have cut? Fine! > Franziska Neugebauer schrieb: >> >> Since there is no largest element in "potentially" infinite sets >> >> (in "actual/complete/finished", too) this sentence makes no sence >> >> at all. >> > >> > A potentially infinite quantity (set or not) is always finite. >> >> There is no time in maths. > > So you write these letters and develop new ideas in zero time? My posts and my ideas do not take place *in* maths. They take place in the real _physical_ world. > There is no existence outside of time. _Mathematical_ existence is not to be confused with physical existence. A mathematical entity x exists if there is a proof of "x exists". As I have pointed out many times before: If and if so how the physical world determines our reasoning is off topic in sci.math. Please consult the neuro sciences groups for that complex of issues. This said your statement "A potentially infinite quantity (set or not) is always finite" makes no sense. >> >> > Therefore in a linearly ordered set here is a last element. >> > Contrary to the claim of set theorists, a set is not fixed in >> > reality. >> >> When your arguments (by the way: What exactly are your arguments?) > > Here you can read it: > > Theorem. The set of real numbers in [0, 1] is countable. Please create a new thread with an appropriate topic. If you won't do, I will help you. F. N. -- xyz |