The ability to pick a truly random integer from a uniform distribution is just a practical one. Consider: How many digits does a "typical" integer have? The answer is that it is unbounded. For any value you choose, it is easy to show that there are far more integers with more digits than with less. So indeed, even writing down your choice of a random integer would take a long time.
But I don't think the two envelope paradox has anything to do with that practical impossibility. It is about the rules of symbol manipulation and modeling.
But I don't think the two envelope paradox has anything to do with that practical impossibility. It is about the rules of symbol manipulation and modeling.
Exactly this. The two envelopes paradox is a paradox only because using a fallacious argument leads to the wrong result, and the Wiki page proposes numerous resolutions to the problem as well. I like this way of stating it (from http://en.wikipedia.org/wiki/Two_envelopes_problem#Non-proba...):
1. Let the amount in the envelope chosen by the player be A. By swapping, the player may gain A or lose A/2. So the potential gain is strictly greater than the potential loss.
2. Let the amounts in the envelopes be X and 2X. Now by swapping, the player may gain X or lose X. So the potential gain is equal to the potential loss.
In the first case, A and A/2 are actually referring to the same amount of money. You have to condition on what A is: If A is the larger value, the second envelope cannot contain 2A; A/2 is equal to the smaller amount. If A is the smaller amount, then the second envelope cannot contain A/2; A is equal to the smaller amount. So your potential gain and loss are both equal to the smaller amount of money, and you have no logical reason to swap.
But I don't think the two envelope paradox has anything to do with that practical impossibility. It is about the rules of symbol manipulation and modeling.