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Jonathan_Graehl comments on A summary of Savage's foundations for probability and utility. - Less Wrong
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You mean P7 is implied already by P1-6 for finite B, I assume.
No, I meant that P1-P6 imply the expected utility hypothesis for finite gambles, i.e., if f and g each only take on finitely many values (outside a set of probability 0). They therefore also imply P7 for finite gambles, and hence in particular for finite B, but "finite B" is a very strict condition - under P1-P6, any finite B will always be null, so P7 will be true for them trivially!
Okay. I was considering finite gambles backed by a finite S, although of course that need not be the case. Do these axioms only apply to infinite S? If so, I didn't notice where that was stated - is it a consequence I missed? I'm also curious why P1-P6 imply that any finite B must be null:
A finite B necessarily has only finitely many subsets, while any nonnull B necessarily has at least continuum-many subsets, since there is always a subset of any given probability at most P(B).
Basically one of the effects of P6 is to ensure we're not in a "small world". See all that stuff about uniform partitions into arbitrarily many parts, etc.
Yes, P6 very clearly says that. Somehow I skipped it on first reading. So when you add P6, S is provably infinite. Thanks.