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).
This means that if there are constants floating around which can have arbitrary real values, then you can’t talk about all but countably many of those values. (What you can do instead is, for example, specify them to arbitrary but finite precision.)
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…
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).
This means that if there are constants floating around which can have arbitrary real values, then you can’t talk about all but countably many of those values. (What you can do instead is, for example, specify them to arbitrary but finite precision.)
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…