You can weaken "completeness" though. In each of its many manifestations, completeness asks for the existence of something for all subsets (sequences, functions

You can weaken "completeness" though. In each of its many manifestations, completeness asks for the existence of something for all subsets (sequences, functions, ...) of reals. We can instead ask for the ones that are definable by finite means (aka computable, constructive). To find (=construct) the extreme value, for example, you probably won't be able to find a general procedure; it has to be treated case by case. Now you'll appreciate how easy real numbers are. It gives the assurance that the extreme values exist. I would not have thought it would interest Hacker News readers, but here's more about computable numbers: https://observablehq.com/@neobourbaki/computable-numbers https://observablehq.com/@neobourbaki/computable-numbers