Cantor Confusion
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From: William Hughes on 12 Nov 2006 12:57 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > > > > > > > Lines with finitely many indexes cannot exhaust the column with its > > > > > infinitely many > > > > > > > > An infinite number can. > > > > > > A God can do everything he wants. Infinite numbers are of the same > > > power? > > > > > > > No. God can do anything. Infinite numbers cannot. However, > > Infinite numbers can do some things. > > In particular an infinite number of finite lines can > > "exhaust": an infinite number of indexes. > > > > > Can you apply your reasoning to the following matrix? > > > > > > 1 > > > 21 > > > 321 > > > 4321 > > > ... > > > > > > > Reversing the order of the lines does not change thing. > > It should show you that without an infinite line there is no infinite > column. > Well, since I already agree there is no infinite column this does not change anything. This is just more of you reading "X says there is an infinite number of columns", and concluding therefore "X must believe that there is an infinite column". This is where we started "Dik believes that 0.111... has an infinite number of places", therefore "Dik must believe that there is an infinite place". We have a and b a: Every infinite set contains an infinite element. b: There exists an infinite set that does not contain an infinite element. They cannot both be true. If you want to show that assuming b leads to a contradiction, you cannot first assume b and then say "Set A is infinite so it must contain an infinite element." > > There > > are an infinite number of lines. Each line is finite. > > There is an infinite number of initial segments of columns. Each one is > finite Which is just what I said (changing names from "line" to "initial segments of columns" changes nothing). > And nothing more is there. Nothing more needs to be there. > > There are > > are omega lines and omega columns. There is no last line. > > There is no omega th line. > > There are no omega lines. The number of lines is greater than any finite natural number. What would you call the number of lines? > > > There is no inifinite line. > > > > > What ordinal numbers have the following sequences? > > > > > > > None. These are not ordinals. > > An ordinal number must consist of an > > initial segment of ordinals. However, in > > a slightly sloppy way they can be said to > > represent ordinals (note that this > > representation is no longer uniqe). Note that you can > > represent any countable ordinal (not just omega) by > > using just finite integers, you do not need (athough > > you can use if you want) infinite ordinals. > > > > > 3,4,5,...,1,2 > > > > A infinite set consisting of finite integers. > > This represents omega+1 > > No, the sequence above represents omega + 2. > > > (For n>=3 you have put n in the spot one > > would usually have n-2, 1 is in the spot where > > you would usually have omega, and 2 is in > > the spot where you would usually have omega+1) > > > > > > > > 3,4,5,...,omega,1,2 > > > > > > > An infinite set consisting of the finite integers and > > the non-natural number omega. This represents > > omega+2 > > No, the sequence above represents omega + 3. But you see the problem. > No. There are two representations of omega + 3. Why is this a problem? - William Hughes
From: David Marcus on 12 Nov 2006 12:57 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > Okay, so for you there is no point to draw about your quote other than > > > > > > understanding Cantor's own writings. And, I'll add that in this > > > > > > particular instance, Cantor, as you translated, is not in conflict with > > > > > > the theorem of current set theory that omega is a limit ordinal and the > > > > > > first oridinal that is greater than all natural numbers. > > > > > > > > > > That is what I said. omega is a limit. In modern set theory there are > > > > > limits. > > > > > > > > Moe said that "omega is a limit ordinal". He did not say that "omega is > > > > a limit". The two statements are not the same. > > > > > > Moe said "Cantor, as you translated, is not in conflict with the > > > theorem of current set theory". Cantor said "omega can be understood as > > > a limit". > > > > Cantor said "understood". That is not the same as saying "is". > > Cantor also said: "is". Therefore it is clear that here he meant "is". Are you saying that even though Cantor wrote "understood", he really meant "is"? Since the two words don't mean the same, please give your evidence that Cantor wrote the wrong word. > > Regardless, Cantor's papers are very old and predate the modern > > formulations. > > That does not mean that the modern formulations are better. No, but the two formulations are different. If your point is that the old formulations have problems, then I doubt anyone will disagree. But, if your point is that the modern formulations have problems, then (obviously) you need to stick to the modern formulations in your discussion. > > > > The phrases "actually existing" and "cannot exist" are not defined. > > > > > Eetiquette and honesty requires you to say, "I'm sorry, but I do not > > > know what the definitions of these words are. And then you should > > > attach a list of words you know. It can't be too long. So I will look > > > whether there are words which could be used to explain "actually > > > existing" and "cannot exist". > > > > Amusing. I (and others) have repeatedly asked you what you mean. I have > > suggested that you use the terminology in Halmos's Naive Set Theory or > > in Kunen's Set Theory. If you don't like these books, then pick a > > different one. However, Cantor's papers are much too old to be used for > > a discussion today. > > No. The hidden errors can better be recognized at the roots. Perhaps, but irrelevant. If you can't find the "hidden errors" in the modern formulations, a likely explanation is that the "hidden errors" have been removed in the process of changing the formulations. Regardless, if the "hidden errors" are still there, your only hope of convincing people is to point to them in the formulations that they know. -- David Marcus
From: William Hughes on 12 Nov 2006 13:18 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > omega 1,2,3,4... > > > > omega+1 2,3,4 ...,1 > > > > omega+2 3,4,5 ...,1,2 > > > > 2*omega 1,3,5... 2,4,6... > > omega = 1,3,5,... > omega = 2,4,6,... > omega, omega = 2omega > 1,3,5,..., omega, 2,4,6,...,omega = 2omega + 1 > > > > It can however be a term in another sequence representing > > omega+1. There is no single sequence which represents > > omega+1. > > > > > but omega is the first part of it, omega = 2,3,4,... > > > > > > This dilemma is what I have been trying to explain for years now. > > > > > > > The explanation is that if we do not insist that an ordinal be > > represented by an initial seqment of ordinals, then there > > is no unique representation of an ordinal. > > I we do not insist, then by definition omega = 1,2,3,... = n, n+1, n+2, > ... > omega + 1 = 1,2,3,..., omega. Yes. So what? You are not going to do any unjustified formal manipulations are you? > > Cardinal numbers like one, tweo, ... and ordinal numbers like first, > second, ... are closely connected. So every iinitial segment of natural > numbers is counted by a natural number, namely |{1,2,3,...,n}| = n. > Therefore no initial segment of natural numbers can be counted by an > unnatural number like omega. Only if we say that {1,2,3,...} is not an initial segment of natural numbers. No set of the form {1,2,3..,n} can be counted by an unnatural number like omega. The set {1,2,3,...} is not a set of the form {1,2,3,...,n} > This leads to the problem |{1,2,3,...}| = > omega and |{1,2,3,...,omega}| = omega + 1. So we have |{1,2,3,...,a}| = > a or a + 1, corresponding to the kind of a. Yes. This is true. It is not however a problem. [Sometimes this is used to argue that starting at 0 is more elegent. Then the ordinal represented by {0,1,2,3,...a} is always a+1 (no matter what type of ordinal a is). While I would agree, I would also say that it is not a major issue.] - William Hughes
From: Virgil on 12 Nov 2006 14:06 In article <1163351315.961078.253970(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > > > > > > > Lines with finitely many indexes cannot exhaust the column with its > > > > > infinitely many > > > > > > > > An infinite number can. > > > > > > A God can do everything he wants. Infinite numbers are of the same > > > power? > > > > > > > No. God can do anything. Infinite numbers cannot. However, > > Infinite numbers can do some things. > > In particular an infinite number of finite lines can > > "exhaust": an infinite number of indexes. > > > > > Can you apply your reasoning to the following matrix? > > > > > > 1 > > > 21 > > > 321 > > > 4321 > > > ... > > > > > > > Reversing the order of the lines does not change thing. > > It should show you that without an infinite line there is no infinite > column. Why should something true show something false? > > > There > > are an infinite number of lines. Each line is finite. > > There is an infinite number of initial segments of columns. Each one is > finite. And nothing more is there. How is it that there are more-than-any-finite-number of lines but not more-than-any-finite-number of first elements of lines? At least according to WM. > > > There are > > are omega lines and omega columns. There is no last line. > > There is no omega th line. > > There are no omega lines. There is no omega-th line, but there are omega lines
From: Virgil on 12 Nov 2006 14:11
In article <1163351435.092455.162270(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > David Marcus schrieb: > > > > > > > If you think that your "mathematical proof" is valid and you prove > > > > something that is absurd, this only shows that your mathematical > > > > reasoning leads to absurdities. Everyone who uses mathematical > > > > reasoning > > > > consistent with ZFC cannot prove your absurdity. So, if you learn ZFC, > > > > you will stop proving absurdities. > > > > > > Only then the absurdities would begin. But I would not notice. > > > > > > > > > We need no experts in trees. This tree does nothing else but > > > > > represent > > > > > the real numbers of an interval. Here it is in a form which can be > > > > > understod by ever mathamatician. > > > > > > > > > > Consider a binary tree which has (no finite paths but only) infinite > > > > > paths representing the real numbers between 0 and 1 as binary > > > > > strings. > > > > > The edges (like a, b, and c below) connect the nodes, i.e., the > > > > > binary > > > > > digits 0 or 1. > > > > > > > > > > > > > > > > > > > > 0. > > > > > > > > > > /a \ > > > > > > > > > > 0 1 > > > > > > > > > > /b \c / \ > > > > > > > > > > 0 1 0 1 > > > > > > > > > > .......................... > > > > > > > > > > > > > > > > > > > > The set of edges is countable, because we can enumerate them. 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. > > > > > > > > Half of the paths include edge a. One quarter of the paths include edge > > > > b. > > > > > > > > Suppose we have a finite tree of height n. Then there are 2^n paths. > > > > Consider a path (e1,e2,...,en). Edge e^j is contained in 2^-j of the > > > > paths. We can define a real-value function on paths by g(e1,e2,...,en) > > > > = > > > > sum_{j=1}^n 2^-j = 1 - 2^-n. Not sure what to do next... > > > > > > > > > Continuing in this > > > > > manner in infinity, we see by the infinite recursion > > > > > > > > > > f(n+1) = 1 + f(n)/2 > > > > > > > > What is f(n)? > > > > > > f(n) is the number of edges related to the initial segment of one path > > > which has passed through the first n edges. > > > > Are you saying that f(1) is the number of edges "related" to the paths > > that contain edge a? What is the value of f(1)? > > f(1) = 1 is the number of edges related to initial segment (of the > path) that contains edge a. > > We start with the first edge a of a path. Then we see the path splits > into two paths. So half of edge a is related to each one and so on. But then when those paths split only 1/4 of edge 'a' is related to any of those, and for the next split only 1/8 of 'a' is related, so that for only 1/ (lim-{n -> oo} 2^n) of edge 'a' is related to any infinite path. |