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Andreas_Giger comments on Naturalism versus unbounded (or unmaximisable) utility options - Less Wrong
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Actually, it seems you can solve the immortality problem in ℝ after all, you just need to do it counterintuitively: 1 day is 1, 2 days is 1.5, 3 days is 1.75, etc, immortality is 2, and then you can add quality. Not very surprising in fact, considering immortality is effectively infinity and |ℕ| < |ℝ|.
But that would mean that the utility of 50% chance of 1 day and 50% chance of 3 days is
0.5*1+0.5*1.75=1.375, which is different from the utility of two days that you would expect.You can't calculate utilites anyway; there's no reason to assume that u(n days) should be 0.5 * (u(n+m days) + u(n-m days)) for any n or m. If you want to include immortality, you can't assign utilities linearly, although you can get arbitrarily close by picking a higher factor than 0.5 as long as it's < 1.