*Originally posted by Palynka*

**[b]but my point is that as the set of possible values in the envelope grows, P(N<2n) shrinks.**

The set of possible values is not bounded, so I don't know what you mean by that. My prior already includes all possible values of N up to infinity. And I'm integrating over that support (up to infinity). Think about it. If the integral of n up to 2n is 1/2, n interval size. So I'm now thinking how to work it out in a more general setting...[/b]

ok now i see where we separated in terms of my understanding of your argument. now it seems ok, except i will ask you to consider this:

1) wouldn't setting the problem over the reals contain an infinitude of irrational numbers that, in the spirit of the problem, wouldn't really work as a "dollar amount" in the envelope? since they are uniformly distributed throughout the real line, this wouldn't hurt our ability to uniformly distribute the RATIONALS, and so i can see an argument for using this as a setting for the problem. however,

2) using the rationals requires an infinitesimally accurate system of "value," that is to say the number you see in the envelope could be as tiny as .0000....0001, which i think again is not within the spirit of the problem. also the rationals include things like 5/7 dollars and 41/3 dollars, which it seems to me goes against the spirit of the problem as a dollar amount. if it were posed as "you open an envelope and there is a number in it... and are told it is either exactly half or exactly double the number in the other envelope" etc. then i could see this as plausible. but i think these restrictions are not flippant, but really within the spirit of the problem, and as such

3) i believe the most "natural" setting (har har) for the problem is over the natural numbers. but then since we get into the issue that any non-power of 2 would give you too much information about the other envelope, so we restrict our scope to {1, 2, 4, 8, 16, ...}

perhaps these restrictions are too strong, but i personally think it is more in line with the spirit of the problem. that being said, infinitesimally accurate decimals and a placement over the rationals is entirely arguable, and i think that all the maths you've done so far would work over the rationals...i just think it doesn't seem right to tell the player "in your envelope is 147 and 93756/142019274 dollars... do you think there is double that or half that in the other envelope?" ðŸ™‚ just being cheeky.