David Hopkins wrote:
Obi-Wan Kenobi wrote:
From the math point of view, the risk/reward calculation is infinity/infinity, which is an undefined and therefore meaningless term.
I don't see that
Today, there is a 50% chance, tomorrow it is 66%, the next day is 75% and so on. Mathmatically, your odds improve.
All the normal understandings go out the window when infinite values enter the calculation.
Assume that the value of staying in Hell forever is infinitely negative while the value of entering Heaven is infinitely positive (surely there's no argument with either of those). Then the expected value of playing the game on day
i is (i / (i + 1)) * [positive infinity] + (1 / (i + 1)) * [negative infinity]. But the value of any positive number times positive infinity is still positive infinity, and the value of any positive number times negative infinity is still negative infinity. The equation therefore reduces to [positive infinity] + [negative infinity] = [positive infinity] - [positive infinity], and (unlike with "normal" numbers) infinity - infinity doesn't equal 0, but is indeterminate.
Why?
The set of positive integers (1, 2, 3, 4, 5 . . . . ) is of infinite magnitude.
* If I subtract from that the equally infinite set of prime numbers, I still have an infinite number of elements left, so in this case, infinity - infinity = infinity.
* If I subtract the set from itself, then there are no elements left, and infinity - infinity = 0.
* If I subtract the set (2, 3, 4, 5 . . . . ) from it, then infinity - infinity = 1, and more generally if I omit the first n positive integers, then infinity - infinity = n.
So there is no way to evaluate the original equation because there is no point at which I can say that a 6% (or whatever) chance of being in Hell forever is an acceptable risk in light of a 94% (or whatever_ chance of being in Heaven forever.