I agree that it’s unlikely that we’d ever want infinite cardinal numbers to play a utility-like role. But it’s worth noting Conway’s “surreal numbers”, which extend the familiar real numbers with a rich variety of infinities and infinitesimals, and the infinities get (handwave handwave) “as big as the infinite ordinals”. (They are definitely more ordinal-like than cardinal-like, though they’re also definitely not the exact same thing as the ordinals either.)
So in the right framework these set-theoretic exotica do function as quantities, at least if being part of an ordered field that extends the real numbers counts as “functioning as quantities”, which I think it should.
That makes sense. I wasn’t really familiar with surreal numbers, but they indeed seem to be an ordered field extending the reals, and yet also including whatever large cardinals one cares to postulate in one’s foundations.
It looks like there’s a little bit of work already on using surreal numbers for probabilities and utility values, proving a surreal version of the vNM theorem and applying it to various Pascal’s-Wager-type scenarios. This seems like a solid direction if one really does want maximal expressive power in one’s degrees of infinitude.
I agree that it’s unlikely that we’d ever want infinite cardinal numbers to play a utility-like role. But it’s worth noting Conway’s “surreal numbers”, which extend the familiar real numbers with a rich variety of infinities and infinitesimals, and the infinities get (handwave handwave) “as big as the infinite ordinals”. (They are definitely more ordinal-like than cardinal-like, though they’re also definitely not the exact same thing as the ordinals either.)
So in the right framework these set-theoretic exotica do function as quantities, at least if being part of an ordered field that extends the real numbers counts as “functioning as quantities”, which I think it should.
That makes sense. I wasn’t really familiar with surreal numbers, but they indeed seem to be an ordered field extending the reals, and yet also including whatever large cardinals one cares to postulate in one’s foundations.
It looks like there’s a little bit of work already on using surreal numbers for probabilities and utility values, proving a surreal version of the vNM theorem and applying it to various Pascal’s-Wager-type scenarios. This seems like a solid direction if one really does want maximal expressive power in one’s degrees of infinitude.