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:
D3: An event B is said to be null if f≤g given B for any actions f and g.
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.
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.