Infinity minus one isn’t smaller than infinity. That’s not useful in that way.
The thing being added or subtracted is not the mere number of hypotheses, but a measure of the likelihood of those hypotheses. We might suppose an infinitude of mutually exclusive theories of the world, but most of them are extremely unlikely—for any degree of unlikeliness, there are an infinity of theories less likely than that! A randomly-chosen theory is so unlikely to be true, that if you add up the likelihoods of every single theory, they add up to a number less than infinity.
It is for this reason that it is important when we divide our hypotheses between something likely, and everything else. “Everything else” contains infinite possibilities, but only finite likelihood.
Well, no—it’s a set of explanations. A very large set, consisting of every explanation other than ‘the sun is powered by nuclear fusion’, but smaller than T | ~T, and therefore somewhat useful, however slightly.
This was talking about set sizes, which is what I replied about.
You can’t quantify your fallibility in the sense of knowing how likely you are to be mistaken in an unexpected way. That’s not possible.
The thing being added or subtracted is not the mere number of hypotheses, but a measure of the likelihood of those hypotheses. We might suppose an infinitude of mutually exclusive theories of the world, but most of them are extremely unlikely—for any degree of unlikeliness, there are an infinity of theories less likely than that! A randomly-chosen theory is so unlikely to be true, that if you add up the likelihoods of every single theory, they add up to a number less than infinity.
It is for this reason that it is important when we divide our hypotheses between something likely, and everything else. “Everything else” contains infinite possibilities, but only finite likelihood.
This was talking about set sizes, which is what I replied about.
You can’t quantify your fallibility in the sense of knowing how likely you are to be mistaken in an unexpected way. That’s not possible.