A non-constant sequence that converges to zero encodes an infinitesimal, and I think any infinitesimal has an encoding of that form. But a sequence that’s bounded in absolute value but doesn’t converge, e.g.
), also encodes some real plus some infinitesimal. It’s this latter kind that involves the axiom of choice, to put it in an equivalence class with some convergent sequence.
[1, 1⁄2, 1⁄3, 1⁄4, …] is the infinitesimal in the proposed definition of a uniform prior, but the hyperreal outcome of the expected utility calculation is
,%20{1\over%202}\sum_{i=1}%5E2U(o%7CH_i),%20{1\over%203}\sum_{i=1}%5E3U(o%7CH_i),%20...]) which might very well be the divergent kind.
Agreed that my first objection was more important.
A non-constant sequence that converges to zero encodes an infinitesimal, and I think any infinitesimal has an encoding of that form. But a sequence that’s bounded in absolute value but doesn’t converge, e.g.
), also encodes some real plus some infinitesimal. It’s this latter kind that involves the axiom of choice, to put it in an equivalence class with some convergent sequence.[1, 1⁄2, 1⁄3, 1⁄4, …] is the infinitesimal in the proposed definition of a uniform prior, but the hyperreal outcome of the expected utility calculation is
,%20{1\over%202}\sum_{i=1}%5E2U(o%7CH_i),%20{1\over%203}\sum_{i=1}%5E3U(o%7CH_i),%20...]) which might very well be the divergent kind.Agreed that my first objection was more important.