Cantor Confusion
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From: Lester Zick on 17 Nov 2006 14:37 On 16 Nov 2006 13:09:32 -0800, "david petry" <david_lawrence_petry(a)yahoo.com> wrote: > >Dik T. Winter wrote: > >> If you want to find absolute truth you should not look at mathematics. Correction. One should not look at mathematikers who proclaim the irrelevance of truth to mathematics. >Perhaps we should replace "absolute truth" with "culturally neutral >truth", or in other words, truth without any cultural, religious, or >philosophical bias. Sure that'll work. If truth is not absolute, mathematical, physical, cultural, religious, or philosophical it's not really clear what the content of "truth" might be or what could be true. > We can reason about this concept by asking the >question: what will the mathematics of advanced alien civilizations >(i.e. from other planets) look like? Thinking about this question >leads most of us to believe that there is a core of mathematics which >every such civilization will accept. Obviously there can only be such a core if it's true. ~v~~
From: Virgil on 17 Nov 2006 14:50 In article <455de26d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Franziska Neugebauer wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >> David Marcus schrieb: > >> > >>>>>> I see. But recently you used the word "completed infinity". > >>>>> I don't think I ever said that. Do you have a quote? > >>>> Here it is: > >>> In the below post, I was just trying to paraphrase what you are > >>> saying. > >> Please don't try to paraphrase what I said, because I don't believe > >> that you understand it sufficiently. > > > > This is due to your conceptional weakness. > > > > [...] > > > >>> I didn't say I would say that or that I understood what you were > >>> trying to say. > >> The question was whether you "ever said that" it. I hope this > >> question as been settled now. > >> > >>> In fact, I don't know what you you mean by the phrase. Did you > >>> really misunderstand what I wrote? > >> There is no misunderstanding possible. You refuted Lester's > >> interpretation, by proposing to have a better one: > >> > >> "That doesn't seem to be what WM is saying. He seems to be saying that > >> the notion of a completed infinity leads to either absurdities or > >> contradictions. Perhaps he thinks the way to avoid these absurdities > >> is to only consider things that can be physically produced." > >> > >> I [k]now that cranks never admit having made an error. > > > > Introspection? > > > >> But do you think that obvious lies like this are a way to reach your > >> aim?> > > > > I cannot spot any lie. > > > >>> I don't know what you mean by "completed anywhere". > >> The completed initial segment contains every natural number. > > > > http://mathworld.wolfram.com/InitialSegment.html > > > > A "completed initial segment" is not an *initial* segment. > > > >> Another segment contains not every natural number. > > > > No *initial* segment contains every element of its underlying set. > > > > F. N. > > How exactly d you define "initial segment"? It would seem to me to be a > subset of an ordered set which, if it contains element x, also contains > every element with an index in the ordered set which is less than x's. > If this is the definition, then the complete set IS an initial segment > of itself. If you disagree with thi definition, can you provide a better > one? An equally valid definition of "initial segment" of an ordered set S, which specifically excludes the set itself, is a set of all elements of S preceding (strictly less than) a given element of S. There will then be a natural bijection between any ordered set S and the set of its initial segments: s <--> { x in S: x < s}
From: Virgil on 17 Nov 2006 15:10 In article <1163782414.605780.279260(a)j44g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > The length of the diagonal cannot surpass the length of any line. (This > > > is easy to show for every matrix.) > > > > it has never been shown because it is false for a list in which the nth > > member is of length n. > > > > In such a list, for every n, the diagonal has "length" >= n, and in > > fact, has "length">= n+1 (as here is always a n+1'st line) so: > > > > For every n in N, length(diagonal) > n. > > Yes. But for every digit of the diagonal there is a line supplying it. > Therefore the diagonal cannot be longer than every natural number. Most natural numbers have "lengths" much shorter than their values, in any base other than unary. For every natural number n, the diagonal that we have been discussing has a position n+1, so is of length > n. Therefore there is no n for which the length of the diagonal is equal to n and there is no n for which the lenght of the diagonal is less than n and for every n the length of the diagonal is greater than that n. If WM wisheds to claim that there is some n for which the length of the diagonal is NOT greater than n, let him produce it now, or forever hold his peace. > > ============= > > > Then, according to WM, one can have a set containing NO infinite > > element, such as the rationals, and a proper subset of it which MUST > > contain an infinite element, such as the integral rationals. > > > I find such self-contradictory systems quite unsatisfactory. > > > As do, I suspect, all those who are mathematically at least minimally > > competent. > > Please check your own competence. Done, and while not perfect, it is leagues better than yours. > As usual you misunderstood and drew > the wrong conclusion. In principle it was possible that the set 1/n had > infinitely many elements. WM specified that the rationals, because dense, could be infinite without containing any infinite elements, but that any ordered set in which there was "a positive minimum distance between members", or equivalent, would have to contain an infinite member to be an infinite set. Since there are proper subsets of the rationals having "a positive minimum distance between members" but being equally numerous (bijectable) with the rationals, WM is claiming an obvious self-contradiction. > > A sequence, being a function, a sort of set, is of fixed "size", in the > > sense that its membership does not vary with time or with any other > > "variable". > > O, yeah, you are the guy who has imagined all the natural numbers and > all the terms of all sequences. That is nowhere near so difficult as imagining that a proper subset of a set having to contain a member not in the superset, as WM's essay on infinite sets requires.
From: Virgil on 17 Nov 2006 15:11 In article <1163782520.578365.191290(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > david petry schrieb: > > > Dik T. Winter wrote: > > > > > If you want to find absolute truth you should not look at mathematics. > > > > Perhaps we should replace "absolute truth" with "culturally neutral > > truth", or in other words, truth without any cultural, religious, or > > philosophical bias. We can reason about this concept by asking the > > question: what will the mathematics of advanced alien civilizations > > (i.e. from other planets) look like? Thinking about this question > > leads most of us to believe that there is a core of mathematics which > > every such civilization will accept. > > without axioms, yes. For instance: I + I = II (after translating "+" > and "="). Therefore I call this an absolute truth. I think it loses too much in translation.
From: Virgil on 17 Nov 2006 15:26
In article <1163782643.189768.296460(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > > > > http://mathworld.wolfram.com/InitialSegment.html > > > > What you call "complete initial segment" is not an *initial* segment > > but the whole set of lines. > > What is a name? > > What *counts* is this: It is asserted that the number of natural > numbers is omega It is asserted that the /number/ of naturals is aleph_0, in the sense of cardinality. This is not quite the same as saying that the ordinality of the set of all finite ordinals is omega. (the elements of the first column = number of lines). One can say that the /number/ of elements in the first column is equal to some number, but what you said is nonsense. > By adding 1 element to the first column we see that, if this assertion > is correct, the number increases to omega + 1. If you are talking numbers (cardinals), that is false. if you are taling ordinals, say so, and also say where in order the new element is to be added, as that makes considerable difference. Note that when one appends an object to the "end" of a "list" of the sort we are discussing, the result is no longer a list. > But are there enough > lines? Every element must be in some column and in some row (line) if it is to be anywhere. When WM "adds" an element to that first column, in a place where no line previously existed, he must also add a line for it to be put into or he does not have any place anywhere to put it. |