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Blueberry comments on Normal Cryonics - Less Wrong
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That wouldn't be mapping into the real numbers, of course, since infinity isn't a real number.
As I understand it, utility functions are supposed to be equivalence classes of mappings into the real numbers, where two such mappings are said to be equivalent if they are related by a (positive) affine transformation (x -> ax + b where a>0).
Right, which is why I was saying that some ethical theories can't be expressed by a utility function. And there could be many such incomparable qualities: even adding in infinity and negative infinity may not be enough (though the transfinite ordinals, or the surreal numbers, might be).
I'm surprised at that +b, because that doesn't preserve utility ratios.
Ah, I see. But I'm still not actually sure that's true, though...see below.
Indeed not; utilities are measured on an interval scale, not a ratio scale. There's no "absolute zero". (I believe Eliezer made a youthful mistake along these lines, IIRC.) This expresses the fact that utility functions are just (scaled) preference orderings.