thinks—Okay, so if I understand you correctly now, the essential thing I was missing that you meant to imply was that the utility of living forever must necessarily be equal to (cannot be larger than) the limit of the utilities of living a finite number of years. Then, if u(live forever) is finite, p times the difference between u(live forever) and u(live n years) must become arbitrarily small, and thus, eventually smaller than q times the difference between u(live n years) and u(live googolplex years). You then arrive at a contradiction, from which you conclude that u(live forever) = the limit of u(live n years) cannot be finite. Okay. Without the qualification I was missing, the condition wouldn’t be inconsistent with a bounded utility function, since the difference wouldn’t have to get arbitrarily small, but the qualification certainly seems reasonable.
(I would still prefer for all possibilities considered to have defined utilities, which would mean extending the range of the utility function beyond the real numbers, which would mean that u(live forever) would, technically, be an upper bound for {u(live n years) | n in N} -- that’s what I had in mind in my last paragraph above. But you’re not required to share my preferences on framing the issue, of course :-))
thinks—Okay, so if I understand you correctly now, the essential thing I was missing that you meant to imply was that the utility of living forever must necessarily be equal to (cannot be larger than) the limit of the utilities of living a finite number of years. Then, if u(live forever) is finite, p times the difference between u(live forever) and u(live n years) must become arbitrarily small, and thus, eventually smaller than q times the difference between u(live n years) and u(live googolplex years). You then arrive at a contradiction, from which you conclude that u(live forever) = the limit of u(live n years) cannot be finite. Okay. Without the qualification I was missing, the condition wouldn’t be inconsistent with a bounded utility function, since the difference wouldn’t have to get arbitrarily small, but the qualification certainly seems reasonable.
(I would still prefer for all possibilities considered to have defined utilities, which would mean extending the range of the utility function beyond the real numbers, which would mean that u(live forever) would, technically, be an upper bound for {u(live n years) | n in N} -- that’s what I had in mind in my last paragraph above. But you’re not required to share my preferences on framing the issue, of course :-))