4 ms·
It's unclear what precisely you mean by 'weaker', but if you mean with regard to completeness - the reals are the so-called completion of (that is, the smallest
by sverona 7y ago
It's unclear what precisely you mean by 'weaker', but if you mean with regard to completeness - the reals are the so-called completion of (that is, the smallest complete metric space containing) the rationals; so you're kind of stuck.
- liuyao 7y agoYou 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
- sverona 7y agoThis is really cool. My advisor is a strict constructivist and will tell anyone with ears about finitism after a few beers, and I've always found it a useful philosophical position.