Such a preference ordering has nothing to do with real numbers or expected values, though. Orderings don't by themselves lead to real numbers without some sort

Such a preference ordering has nothing to do with real numbers or expected values, though. Orderings don't by themselves lead to real numbers without some sort of countability or Archimedean requirement, and they certainly don't lead to expected values without quite a bit more. (IMO Savage's Theorem[0] provides a better foundation than VNM, but that's nitpicking.) [0]See my article on it, if you're unfamiliar: https://www.lesswrong.com/posts/5J34FAKyEmqKaT7jt/a-summary-of-savage-s-foundations-for-probability-and https://www.lesswrong.com/posts/5J34FAKyEmqKaT7jt/a-summary-...