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Thread: Does Mathematical Randomness Exist?

  1. #1 Does Mathematical Randomness Exist? 
    Forum Freshman Shubee's Avatar
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    I'm interested in the mathematical consequences of randomness if randomness is defined as follows. Let's represent an infinite sequence of numbers by the function f on the natural numbers N to the set {0,1,2,3,4,5,6,7,8,9}.

    Suppose we say that the infinite sequence f is random if no algorithm exists that generates the sequence f(n). For example, the digits of pi seem random but there are many elementary formulas that represent the numerical value of pi perfectly. Thus, in theory, pi can be computed to arbitrary accuracy.

    Question: Can it be proven that mathematically random sequences exist with this definition?

    As John von Neumann humorous said, ``Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin.''

    The method I have specified for defining non-random numbers is clearly deterministic. But how do we know that truly random numbers exist?


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  3. #2 The Indescribable Random Numbers 
    Forum Freshman Shubee's Avatar
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    When I woke up this morning it occurred to me that the set of all nonrandom numbers must be countable because the set of all algorithms that generate the nonrandom numbers can be put into a countable lexicographical order. Therefore the set of all indescribable random numbers must be uncountably infinite. But if the set of all nonrandom numbers is countable, we can use Cantor's diagonalization process, which is an algorithm, and find an unlisted nonrandom number. That's a contradiction. Did I discover a new paradox in set theory?


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  4. #3  
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    There are some truly random processes that exist (at least within our current level of knowledge) - the decay of radioactive particles is one of them. But as for mathematical randomness, can't help you there...

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  5. #4 Chaitin's constant 
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    The following links might be of interest:

    http://en.wikipedia.org/wiki/Chaitin's_constant
    http://www.expmath.org/expmath/volum...ude361_370.pdf

    I also recommend the book "Metamaths The Quest for Omega" by Gregory Chaitin
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  6. #5  
    . DrRocket's Avatar
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    Quote Originally Posted by evilwill32
    There are some truly random processes that exist (at least within our current level of knowledge) - the decay of radioactive particles is one of them. But as for mathematical randomness, can't help you there...

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    Quantum mechanics seems to be truly stochastic. To my knowledge all other applications of probability theory are in situations in which the stochastic models compensate for ignorance of initial conditions or other factors.
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  7. #6  
    Forum Radioactive Isotope MagiMaster's Avatar
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    Just to clarify that a bit, in some cases, there is no possible way to not be ignorant of the initial conditions, at least enough to prevent accurate predictions. (Weather forecasting, for example.) In such cases, there's no actual random component, but we (practically) can't tell the difference.
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