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Sniffnoy comments on A summary of Savage's foundations for probability and utility. - Less Wrong
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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.