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Re: Proportions of Infinity

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  • Barry Brent
    Unless he denies the Continuum Hypothesis, kevinsworkingemail was already saying that there is a 1-1 function between the sets, when he said they were both
    Message 1 of 53 , Sep 1, 2003
      Unless he denies the Continuum Hypothesis, "kevinsworkingemail" was already
      saying that there is a 1-1 function between the sets, when he said they
      were both uncountable.

      His point, it seems to me, is only that cardinality doesn't equate
      unerringly to measure (size), which mathematicians use to handle
      probabilities in this context. That's true.

      Devising and then analyzing any particular procedure for choosing a number,
      to get to the probabilities in this question, is at least hard. For
      example,kevinsworkingemail's procedure doesn't terminate, so it doesn't
      terminate in a choice. Not sure how serious an objection that is. But,
      also, as K. Kahn seems to be suggesting, a reasonable but different
      procedure can give a different result. I think he's saying this: if we
      choose a "preliminary number" using the original coin flip procedure, and
      then [if we can imagine a "then" after a non-terminating procedure] take
      its reciprocal, it looks like this slightly different procedure exactly
      reverses the probabilities by the same analysis. So finding and analyzing
      a choice-procedure to compute this kind of probability is subtle or
      impossible. I don't know which.

      But the probabilities in this question would not usually be computed like
      that. People use measure. Say, Lebesgue measure. (It's fairly standard.)
      The ratio

      (measure of (0,1)/(measure of (1,infinity))

      is 1/infinity

      = 0, more or less.

      That's the usual reason math people, I think, evaluate this particular
      probability as zero.

      On the other hand, some sets of real numbers are not measurable in the
      Lebesgue theory, so there is room for another approach. But my guess is,
      people would just look for a more expansive theory of measure to handle
      those sets.

      Barry

      >Sorry about the 3 month delay but I am confused about the following.
      >
      >--- In Fabric-of-Reality@yahoogroups.com, PaintedDevil@a... wrote:
      >> In a message dated 5/6/2003 12:46:47 PM GMT Daylight Time,
      >> kevinsworkingemail@y... writes:
      >>
      >>
      >> For example, the irrational numbers between 0 and 1 and between 1
      >and
      >> infinity both form uncountably infinite sets, yet if you choose an
      >irrational
      >> number at random (e.g. with a (countably?) infinite series of coin
      >tosses to
      >> generate the digits before the decimal point (in binary) and
      >another infinite
      >> series to get the digits after the decimal point) your chances of
      >getting a
      >> value between 0 and 1 is zero compared to your chances of getting
      >one between
      >> 1 and infinity.
      >>
      >
      >This example makes sense until I think about the fact that the
      >reciprocal of all those irrational number between 1 and infinity are
      >between 0 and 1. There is a one-to-one function between the two
      >sets. So it must depend upon how you generate the irrational number
      >as to whether you're likely to find it between 0 and 1 or greater
      >than 1. Why is one way of generating numbers better than another?
      >
      >-ken kahn
      >
      >
      >
      >
      >
      >
      >
      >Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
    • Barry Brent
      Unless he denies the Continuum Hypothesis, kevinsworkingemail was already saying that there is a 1-1 function between the sets, when he said they were both
      Message 53 of 53 , Sep 1, 2003
        Unless he denies the Continuum Hypothesis, "kevinsworkingemail" was already
        saying that there is a 1-1 function between the sets, when he said they
        were both uncountable.

        His point, it seems to me, is only that cardinality doesn't equate
        unerringly to measure (size), which mathematicians use to handle
        probabilities in this context. That's true.

        Devising and then analyzing any particular procedure for choosing a number,
        to get to the probabilities in this question, is at least hard. For
        example,kevinsworkingemail's procedure doesn't terminate, so it doesn't
        terminate in a choice. Not sure how serious an objection that is. But,
        also, as K. Kahn seems to be suggesting, a reasonable but different
        procedure can give a different result. I think he's saying this: if we
        choose a "preliminary number" using the original coin flip procedure, and
        then [if we can imagine a "then" after a non-terminating procedure] take
        its reciprocal, it looks like this slightly different procedure exactly
        reverses the probabilities by the same analysis. So finding and analyzing
        a choice-procedure to compute this kind of probability is subtle or
        impossible. I don't know which.

        But the probabilities in this question would not usually be computed like
        that. People use measure. Say, Lebesgue measure. (It's fairly standard.)
        The ratio

        (measure of (0,1)/(measure of (1,infinity))

        is 1/infinity

        = 0, more or less.

        That's the usual reason math people, I think, evaluate this particular
        probability as zero.

        On the other hand, some sets of real numbers are not measurable in the
        Lebesgue theory, so there is room for another approach. But my guess is,
        people would just look for a more expansive theory of measure to handle
        those sets.

        Barry

        >Sorry about the 3 month delay but I am confused about the following.
        >
        >--- In Fabric-of-Reality@yahoogroups.com, PaintedDevil@a... wrote:
        >> In a message dated 5/6/2003 12:46:47 PM GMT Daylight Time,
        >> kevinsworkingemail@y... writes:
        >>
        >>
        >> For example, the irrational numbers between 0 and 1 and between 1
        >and
        >> infinity both form uncountably infinite sets, yet if you choose an
        >irrational
        >> number at random (e.g. with a (countably?) infinite series of coin
        >tosses to
        >> generate the digits before the decimal point (in binary) and
        >another infinite
        >> series to get the digits after the decimal point) your chances of
        >getting a
        >> value between 0 and 1 is zero compared to your chances of getting
        >one between
        >> 1 and infinity.
        >>
        >
        >This example makes sense until I think about the fact that the
        >reciprocal of all those irrational number between 1 and infinity are
        >between 0 and 1. There is a one-to-one function between the two
        >sets. So it must depend upon how you generate the irrational number
        >as to whether you're likely to find it between 0 and 1 or greater
        >than 1. Why is one way of generating numbers better than another?
        >
        >-ken kahn
        >
        >
        >
        >
        >
        >
        >
        >Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
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