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Thread: Rational numbers == infinitely repeating sequences of digits

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    Quote Originally Posted by lpetrich View Post
    Quote Originally Posted by SLD View Post
    One more point about Pi. If Pi has infinite digits, then it should have an infinite number combinations of digits. At some point wouldn’t it then be repeating the entire sequence up to that point? And for say two trillion times in a row? Granted that’s not infinite, but for all practical purposes if we discovered that point, we’d be mightily confused.
    Yes, it will have repeats, but it won't have form (N0).(N1)(N2)(N2)(N2)(N2)... Only rational numbers can have that form.
    That form is just a contingent happenstance that arises from arbitrarily defining things.

    You define numbers and define functions and something will come out of it.

    Even something irrational like a digit that repeats without end.

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    Quote Originally Posted by untermensche View Post
    Even something irrational like a digit that repeats without end.
    Why is that supposed to be irrational?

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    Quote Originally Posted by lpetrich View Post
    Quote Originally Posted by untermensche View Post
    Even something irrational like a digit that repeats without end.
    Why is that supposed to be irrational?
    Because it is something that both allegedly exists yet never ends.

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    Quote Originally Posted by untermensche View Post
    Quote Originally Posted by lpetrich View Post
    Quote Originally Posted by untermensche View Post
    Even something irrational like a digit that repeats without end.
    Why is that supposed to be irrational?
    Because it is something that both allegedly exists yet never ends.
    So there is no such thing as an infinitely long sequence?

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    Quote Originally Posted by lpetrich View Post
    Quote Originally Posted by untermensche View Post
    Because it is something that both allegedly exists yet never ends.
    So there is no such thing as an infinitely long sequence?
    I've never seen one.

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    Quote Originally Posted by untermensche View Post
    Quote Originally Posted by lpetrich View Post
    Quote Originally Posted by untermensche View Post
    Because it is something that both allegedly exists yet never ends.
    So there is no such thing as an infinitely long sequence?
    I've never seen one.
    Our finite minds have ways of comprehending mathematical infinities. These ways are extensions of what we do for large finite sets. We usually don't try to list every element of them, but instead find some rule which generates all of them and no others. Limited imagination is not much of an argument.

    (ETA: some of what I had posted here I've moved to the "Infinite Sets" thread, where it belonged)

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    Quote Originally Posted by lpetrich View Post
    We usually don't try to list every element of them, but instead find some rule which generates all of them and no others.
    I can see extremely finite rules but no infinities.

    To see something requires at least some time.

    To see infinite elements though can't be done even if you have infinite time to do it in.

    Infinite elements is by definition an amount of elements that can never be expressed. The end of them can never be observed.

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    Quote Originally Posted by untermensche View Post
    Quote Originally Posted by lpetrich View Post
    We usually don't try to list every element of them, but instead find some rule which generates all of them and no others.
    I can see extremely finite rules but no infinities.
    It's more like

    "The first hundred positive integers"
    vs.
    {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100}

    The first description is much shorter than the second one, and by untermensche's argument, there is no such thing as a large finite set, since he seems to consider only the second kind of description a valid description of a set. Once one accepts rules for generating set elements, like "The first hundred positive integers", it is a small step to infinite sets: "All positive integers".

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    In theory there is a large finite set.

    And in theory any finite set can be expressed.

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    Quote Originally Posted by untermensche View Post
    And in theory any finite set can be expressed.
    Even a set with more members than there are elementary particles in the observable Universe? A number which is approximately 1086.

    So I can write "the first 10^100 positive integers" without having to write them all down, because doing so is a physical impossibility. What is the fundamental difference between "the first 100 positive integers" and "the first 10100 positive integers"? Or between those two descriptions and "all positive integers"?

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