EDIT: I guess you’d need to make more distributional assumptions, since there’s no uniform distribution over infinitely many distinct elements, or you’d draw infinitely many individuals with duplicates from a finite set, and your observations wouldn’t distinguish you from your duplicates.
Adding to this, I think it would follow from your argument that the credence you must assign to the universe having infinitely many individuals must be 0 or 1, which seems to prove too much. You could repeat your argument, but this time with any fixed finite number of individuals instead of 1 for solipsism, and infinitely many individuals as the alternative, and your argument would show that you must assign credence 0 to the first option and so 1 to the infinite.
For each natural number k∈N, and N representing the number of actual people, you could show that
(This is assuming the number of individuals must be countable. I wouldn’t be surprised if the SIA has larger cardinals always dominate in the same way infinity does over each finite number. But there’s no largest cardinal, although maybe we can use the class of all cardinal numbers? What does this even mean anymore? Or maybe we just need to prove that the number of individuals can’t be larger than some particular cardinal.)
EDIT: I guess you’d need to make more distributional assumptions, since there’s no uniform distribution over infinitely many distinct elements, or you’d draw infinitely many individuals with duplicates from a finite set, and your observations wouldn’t distinguish you from your duplicates.
Adding to this, I think it would follow from your argument that the credence you must assign to the universe having infinitely many individuals must be 0 or 1, which seems to prove too much. You could repeat your argument, but this time with any fixed finite number of individuals instead of 1 for solipsism, and infinitely many individuals as the alternative, and your argument would show that you must assign credence 0 to the first option and so 1 to the infinite.
P(N=k|I exist and (N=k or N=∞))=0For each natural numberk∈N, andNrepresenting the number of actual people, you could show that
P(N<∞|I exist)=∞∑k=1P(N=k|I exist)≤∞∑k=1P(N=k|I exist and (N=k or N=∞))=0and soAnd henceP(N=∞|I exist)=1.(This is assuming the number of individuals must be countable. I wouldn’t be surprised if the SIA has larger cardinals always dominate in the same way infinity does over each finite number. But there’s no largest cardinal, although maybe we can use the class of all cardinal numbers? What does this even mean anymore? Or maybe we just need to prove that the number of individuals can’t be larger than some particular cardinal.)