You wrote the above in regards to hyperreal infinities, “the hyperreal encoded by a given divergent sequence”. I’m under the impression that hyperreal infinitesimals are encoded by convergent sequences: specifically, sequences that converge to zero. The hyperreal [1, 1⁄2, 1⁄3, 1⁄4, …] is the one that corresponds to the limit you gave. Does that adequately dispel the computability issue you raised?
In any case, non-computability isn’t a major defect of the utilitarian prior vis-a-vis the also non-computable Solomonoff prior. It is an important caution, however.
Your first objection seems much more damaging to the idea of a utilitarian prior. Indeed, there seems little reason to expect max(U(o|Hi)) to vary in a systematic way with a useful enumeration of the hypotheses.
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.
Thanks!
You wrote the above in regards to hyperreal infinities, “the hyperreal encoded by a given divergent sequence”. I’m under the impression that hyperreal infinitesimals are encoded by convergent sequences: specifically, sequences that converge to zero. The hyperreal [1, 1⁄2, 1⁄3, 1⁄4, …] is the one that corresponds to the limit you gave. Does that adequately dispel the computability issue you raised?
In any case, non-computability isn’t a major defect of the utilitarian prior vis-a-vis the also non-computable Solomonoff prior. It is an important caution, however.
Your first objection seems much more damaging to the idea of a utilitarian prior. Indeed, there seems little reason to expect max(U(o|Hi)) to vary in a systematic way with a useful enumeration of the hypotheses.
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.
[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
Agreed that my first objection was more important.