(Disclaimer: I’m no mathematician, but I think I’m sufficiently familiar with the math in this post and comment.)
I’m quite confused about why you would use extended reals when you could use hyperreals (or surreals, which are just an extension of hyperreals). For example, my intuitions about infinite utility say we should have
∞+1 > ∞
2*∞ > ∞
∞*∞ > ∞
where “∞” is the amount of utility in some canonical infinite-utility universe and “>” means “is better than”.
Each of these works if “∞” represents an infinite hyperreal, but not if it’s just the infinite extended-real, right? There’s not just one positive and one negative possible-universe-value off the real line; it feels much more like there’s such a value for each hyperreal.
(Disclaimer: I’m no mathematician, but I think I’m sufficiently familiar with the math in this post and comment.)
I’m quite confused about why you would use extended reals when you could use hyperreals (or surreals, which are just an extension of hyperreals). For example, my intuitions about infinite utility say we should have
∞+1 > ∞
2*∞ > ∞
∞*∞ > ∞
where “∞” is the amount of utility in some canonical infinite-utility universe and “>” means “is better than”.
Each of these works if “∞” represents an infinite hyperreal, but not if it’s just the infinite extended-real, right? There’s not just one positive and one negative possible-universe-value off the real line; it feels much more like there’s such a value for each hyperreal.