If you ever plan on talking about your hypothesis, you need to be able to describe it in a language with a finite alphabet (such as English or a programming language). There are only countably many things you can say in a language with a finite alphabet, so there are only countably many hypotheses you can even talk about (unambiguously).
Only if you live in a universe where you’re limited to writing finitely many symbols in finite space and time.
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…
Only if you live in a universe where you’re limited to writing finitely many symbols in finite space and time.
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…