I'm still a little bit confused about your stance here. Let me clarify some things from my side first. When I say "probability theory", I am talking about the branch of mainstream measure theory that studies sets, sigma-algebras on those sets, and countably-additive, non-negative measures on those sigma-algebras whose total mass is 1. This approach to probability theory was introduced by Kolmogorov. Perhaps you could elaborate on the four theories you mentioned, with links if possible, and explain their relationship to what I am calling "probability theory".
You said,
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I agree that the envelope paradox uses standard mathematical tools to arrive at two opposite conclusions. Mathematics must be consistent, or you can prove everything, so we have to rule out one or the other chain of logic. In order to do that, we have to rule out calculations that are used every day in statistics ... My beef is with people who insist mathematics dictate the choice of assumptions. That's wrong. There is no reason outside of personal preference to prefer one approach over the other.
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I still don't understand what these two "approaches" are. Here's one approach: we assume that
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[1] if A and B are events, then P(A | B) = P(A and B)/P(B),
[2] if A and B are independent events, then P(A | B) = P(A),
[3] if A and B are mutually exclusive events, then P(A or B) = P(A) + P(B),
[4] if A is an event, then 0 <= P(A) <= 1.[/list]As far as I can tell, if we assume these four things, then it cannot be the case that
P(Y = 2X | X = k) = 0.5
for all k.
What is the other approach? Which of these assumptions does it reject? Does it involve one of these other "alternative" probability theories you mentioned? Because as far as I can tell, the work of Shannon and Arrow-Debreu was within the context of Kolmogorov's formalism, and Savage's main departure from the formalism was his use of "measures" which were only finitely additive. But the above assumptions do not require countable additivity, nor do they even require sigma-algebras, only algebras. You also suggest that this other approach, whatever it is, is used every day in statistics. So are you saying that there are mainstream statisticians who reject one or more of the above assumptions on a daily basis?