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Math Help - The Cantor set and infinitesimals

  1. #1
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    The Cantor set and infinitesimals

    I am not sure into which rubric to put this, but since there is some Model Theory here, I am putting it in this one.

    First a technical question: the TEX button here doesn't give me the usual choices of what kind of symbol to use, only saying "wrap [tex] tags around selected text." Where do I differentiate between, say, subscript, superscript, Greek letters, quantifiers, and so forth?

    So, until I learn that, I ask for patience with my clumsy notation.

    First, I define the Cantor set informally:
    A(0) = [0,1]
    A(n+1) = the set of closed intervals obtained by taking out the open middle third of each interval contained in A(n), for natural numbers n.
    The Cantor set = the points not removed at any step.

    It is tempting to put
    The Cantor set = the intersection of all A(n), but this would cause problems, as follows:

    I concentrate on p = (the number corresponding to) the right end-point of the left-most interval of the Cantor set. If we stayed in the real numbers, we would have p=0, which is not what we want. So the intersection definition is not adequate. (Even if it were adequate, the following reasoning would still hold. Just covering my bases.)

    Rather, p fulfills the following description:

    for all natural numbers n, 0<p<1/3^n.

    By the Archimedean property of the real numbers, p is not a real number, but by the Compactness Theorem, we assume the existence of a model which includes both real numbers and p, and all such infinitesimals. This new model can contain all the points of the Cantor set.

    So far my reasoning. However, everywhere I look, the Cantor set is considered a subset of the real numbers. What is wrong here?
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  2. #2
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    The Cantor set = the points not removed at any step.

    It is tempting to put
    The Cantor set = the intersection of all A(n), but this would cause problems
    The set of points not removed at any step is precisely the set of points belonging to A(n) for each n. So, the Cantor set is the intersection of all A(n).

    I concentrate on p = (the number corresponding to) the right end-point of the left-most interval of the Cantor set.
    You defined not a single real number p, but a sequence p_n=1/3^n, so instead of 0<p<1/3^n, you have 0<p_n<1/3^{n-1} for all n.

    Where do I differentiate between, say, subscript, superscript, Greek letters, quantifiers, and so forth?
    Most math and other special symbols are entered using backslash-commands, such as \alpha, \sum, \int, \forall. Subscripts and superscripts are entered as follows: a_n and a^n. You can click "Reply with Quote" button under someone's response to see how they entered it. See the sticky LaTeX tutorial thread in the LaTeX help forum.
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  3. #3
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    The Cantor set and infinitesimals

    Thanks, emakarov.

    See the sticky LaTeX tutorial thread in the LaTeX help forum.
    Thanks, I went to it, and there are four links to gif files. The first one gives me a gif file, but when I click on the other three, I get directed back to the home page of this Forum. What is going on?

    ....a sequence , ....... for all n.
    That is, for n>0. Again, thanks: I think I am starting to understand, although my intuition hits the rocks when I consider that on one side the Cantor set has no intervals of non-zero length, yet on the other hand most points of the Cantor set are not endpoints of any of the intervals used to define it. Fractals are the stuff of nightmares.
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  4. #4
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    Thanks, I went to it, and there are four links to gif files. The first one gives me a gif file, but when I click on the other three, I get directed back to the home page of this Forum. What is going on?
    You are right: several of the links in the first topic are broken. Well, here are some more links.

    LaTeX Math Symbols

    Another list of LaTeX Math Symbols (PDF)

    "LaTeX" Wikibook (see the end of the "Mathematics" chapter for tables of math symbols).

    Hypertext Help with LaTeX

    LaTeX Cheat Sheet (most of it is not about math)

    "The not so Short Introduction to LaTeX" book by Tobias Oetiker (PDF; chapter 3 is about math).

    TeX Resources on the Web from the TeX Users Group
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  5. #5
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    Many thanks

    Super. Thanks a lot.
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