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Phrak
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What's the value of a random variable?
Do you mean: "What are random variables good for" ?
Or do you mean the "value" in the sense [itex]X(\omega) [/itex] where [itex] X[/itex] is a random variable and [itex] \omega[/itex] is a sample point?
Probability theory was given an axiomatic setting by Kolmogoroff in the 1930's. It looks in some ways like measure theory.
The experimental trial can be conceptual, like analysis of what happens after a series of coin flips by an ideal coin.
I don't know what you mean by separated.So I am curious as to the answer to the question: Can probability theory be separated from physics (in this case classical stochastic processes), so that axiomatic quantum mechanics is, in part, defined upon classical uncertainty?
I don't know what you mean by separated.
Probability theory is a branch of mathematics. Physicists use it just like they use many other branches such as calculus, vector analysis, differential equations, etc.This is the pivotal question, isn't it? I hadn't been sure at the time. Now, however:-
If the axioms and theorems of probability theory are so abstract as to be free of any reference to any physical system, then probability theory is separate from physics. If there is an explicit reliance they are not.
Does this sound meaningful to you?
Probability theory is a branch of mathematics. Physicists use it just like they use many other branches such as calculus, vector analysis, differential equations, etc.
Other fields where probability is used include statistics, finance, biology, etc.
I guess you don't understand ordinary American jargon. Besides the point is another way of saying irrelevant.
In any case the axioms for probability theory are essentially the same as those for measure theory, with the additional condition that the total measure (probability) is one.
It is not irrelevent. If an axiomatic system cannot be encoded in a symbolic language, it is flawed.
It is not irrelevent. If an axiomatic system cannot be encoded in a symbolic language, it is flawed.
By encoding, do you mean the automation of the logic of probability theory (PT) to prove theorems? If so, the answer is yes. There is no flaw in PT if you accept the axioms and definitions. The definition of a random variable in particular, is that it is simply a mapping from a probability space to an event space with no guidance at all as to what value the function assigns to an event provided it is in the interval [0,1]. The following link is book sized, but the preface should at least shed some light on your question. The basic thrust is that the idea of "randomness" should be replaced by "incomplete information". There is also a good article (IMHO) on Quantum Logic and Probability Theory in the Stanford Encyclopedia of Philosophy.
http://bayes.wustl.edu/etj/prob/book.pdf
I'm sorry I can't reply intelligently to your post. I'll have to satisfy for myself that PT is consistent, unflawed or otherwise by putting in the effort.
That's an unrealistic goal. No one has proved the axioms of ZFC are self consistent or "unflawed".