Well, I appear to be somewhat confused. Here is the logic that I’m using so far:
If:
1: A hypothesis space can contain mathematical constants,
2: Those mathematical constants can be irrational numbers,
3: The hypothesis space allows those mathematical constants to set to any irrational number,
4: And the set of irrational numbers cannot be ennumerated.
Then:
5: A list of hypothesis spaces is impossible to enumerate.
So If I assume 5 is incorrect (and that it is possible to enumerate the list) I seem to either have put together something logically invalid or one of my premises is wrong. I would suspect it is premise 3 because it seems to be a bit less justifiable then the others.
On the other hand, it’s possible premise 3 is correct, my logic is valid, and this is a rhetorical question where the answer is intended to be “That’s impossible to enumerate.”
I think the reason that I am confused is likely because I’m having a hard time figuring out where to proceed from here.
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…
It looks like you’re right, but let’s not give up there. How could we parametrize the hypothesis space, given that the parameters may be real numbers (or maybe even higher precision than that).
Well, I appear to be somewhat confused. Here is the logic that I’m using so far:
If:
1: A hypothesis space can contain mathematical constants,
2: Those mathematical constants can be irrational numbers,
3: The hypothesis space allows those mathematical constants to set to any irrational number,
4: And the set of irrational numbers cannot be ennumerated.
Then:
5: A list of hypothesis spaces is impossible to enumerate.
So If I assume 5 is incorrect (and that it is possible to enumerate the list) I seem to either have put together something logically invalid or one of my premises is wrong. I would suspect it is premise 3 because it seems to be a bit less justifiable then the others.
On the other hand, it’s possible premise 3 is correct, my logic is valid, and this is a rhetorical question where the answer is intended to be “That’s impossible to enumerate.”
I think the reason that I am confused is likely because I’m having a hard time figuring out where to proceed from here.
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…
It looks like you’re right, but let’s not give up there. How could we parametrize the hypothesis space, given that the parameters may be real numbers (or maybe even higher precision than that).