If I lived in such a universe, then it seems like I could potentially entertain uncountably many disjoint hypotheses about something, all of which I could potentially write down and potentially distinguish from one another. But I wouldn’t be able to assign more than countably many of them nonzero probability (because otherwise they couldn’t add to 1) as long as I stuck to real numbers. So it seems like I would have to revisit that particular hypothesis in Cox’s theorem…
Point.
If I lived in such a universe, then it seems like I could potentially entertain uncountably many disjoint hypotheses about something, all of which I could potentially write down and potentially distinguish from one another. But I wouldn’t be able to assign more than countably many of them nonzero probability (because otherwise they couldn’t add to 1) as long as I stuck to real numbers. So it seems like I would have to revisit that particular hypothesis in Cox’s theorem…