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December 8th, 2015, 08:11 AM   #11
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Thanks for the discussion.

For generating random digits, I wasn't thinking of using a computer or anything else, I was just thinking conceptually of a generator for a random sequence of digits.

I understand that it would take an infinite amount of time to generate a number this way. That seems to me to be part of what it means for the number to be unspecifiable. In my mind you can at least conceptually generate random 'points' on the real number line, even if they can't be specified.

I'm not sure if it would be valid to think of these conceptually generated points as numbers. If you could think of them like that, the set of all such numbers seem to in a sense 'fill up' the segment, and make a continuum in a way that seems more natural to me than most descriptions of the real numbers. Even though it seems it would be missing all the 'normal' real numbers.

At the base of it, though, this may be just building a segment of the real number line using smaller segments, since any one of these conceptual randomly generated reals can only be found to fall into a smaller and smaller segment of the real line (depending on how much time you take to read off its digits).

Regarding if these randomly generated reals can be thought of as numbers, assuming that the digits generated for two different numbers a and b are independent, then the two numbers can not be equal to each other, and given enough time, you will be able to determine which one is larger than the other. (This may take an arbitrarily long amount of time, but I would argue it can't take forever).

This is just a train of thought - not sure how much is reasonable.
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December 8th, 2015, 09:51 AM   #12
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It seems to me that if you want to do mathematics with these points, you are going to want to find the distance between the points or between them and some reference point on the line. Either way, you'll need to measure.

Otherwise you just have a mess on the floor.
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December 12th, 2015, 01:29 AM   #13
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Here is the conclusion I draw about this:
  • you can pick a random point on the real line
  • the point can be considered a real number in every sense
  • the point will not be a pre-specified real number
  • it would take an infinite amount of time to describe exactly where on the real line the point is
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December 18th, 2015, 07:16 PM   #14
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Quote:
Originally Posted by Country Boy View Post
No. In a continuous probability distribution, we can talk about the probability that x is in a given interval but the probability that x is a specific number is always 0.
I'm not saying you're wrong, but the sum of the probabilities must be 1, and there are infinitely many numbers, so if the probability that x is a specific number is always 0 doesn't that make the equation 0*infinity = 1?
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