Cantor Confusion
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From: David Marcus on 11 Jan 2007 01:50 Andy Smith wrote: > Or more easily just extend Peano to the negative numbers. What does that mean? > Then I think you can show that -1 = infinity > -2 = infinity -1 > etc. > > (which solves my problem of a label for the last element > in an infinite sequence). I fail to see how you can label something that doesn't exist. > The infinite integers implied by negative values do not > however let one index the irrational numbers ... Are you trolling? -- David Marcus
From: Andy Smith on 10 Jan 2007 18:11 > > Are you trolling? > I'm not sure what trolling is, but I infer from context that it is undesirable. 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. 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. 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? Anyway as i said earlier, I am giving up and finding a book. Cheers
From: Andy Smith on 10 Jan 2007 19:24 > > Andy Smith wrote: > > Or more easily just extend Peano to the negative > numbers. > > > > Then I think you can show that -1 = infinity > > -2 = infinity -1 > > etc. > > > > (which solves my problem of a label for the last > element > > in an infinite sequence). > > Hmm. Of course, you could define anything you like, > using any names you > like, as long as other people can follow your > writing. (If you want to > do sollipsistic mathematics, no need to bother with > Usenet) > > But I wonder why you use the name "infinite sequence" > here? I would > think of this expression - more or less by definition > - as meaning "a > sequence with no end, no last element". In this case > there is hardly a > need for a label for something that doesn't exist. > I was trying to get my head around Zeno's paradox - where there is an infinite series, with no last element, which sums to a finite amount (no problem so far for me), but the paradox is made graphic by enumerating successive terms in amounts of time that decrease pro-rata with the size of the terms. Then the finite sum of the infinite series is achieved by a process in a finite amount of time, also made up as an infinite series in direct correspondence with the original one. As I understand the professional answer kindly provided here (several times) it is that we can't identify a last instant of time, so there isn't a corresponding last jump. So, no problem there then, other than in my head.... And again, by setting out an infinite series with a finite sum twice, back to back, then it feels like one can have a correspondence with a corresponding infinite sequence, which definitely has a finite first and last element, but an infinite number of terms in between. But that was sat on hard too. On the Peano thing, what happens if you remove the axiom that says 0 is not the successor of any natural number? Then you get negative numbers included as natural numbers, other properties unchanged for the positive numbers. What does the e.g. binary representation of -1, -2 then look like ? A horror, with ...1111 as -1. Do -inf and +inf converge on the same number? Can one then just regard -1 as the point that one counts round to eventually? Then you could have a start and an end with an infinite number of points in between, was how my uneducated thought processes went, and then I wouldn't have to fret about Zeno. I am not qualified to comment on that, don't really believe it, and am liable to be called a troll (whatever that is) for suggesting even the possibility of thinking of it....
From: mueckenh on 11 Jan 2007 05:31 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. > > 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. > > > 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. > > >> > > 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". I merely emphasize the complete existence by adding the "actually". Potential infinite sets do not exist (or actually exist), but sets are always finite, though there is no upper bound. That is the definition. > > > 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. It is an intermingling of different notions. > > Potential infinity is changing (growing) finity. > > Says who? The pope? Cantor and Hilbert (and everybody else who knows about it). Regards, WM
From: mueckenh on 11 Jan 2007 05:35
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.) 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 |