It has been a long time since I've studied this; can you link me to something more about the reals being consistent and complete? Or write a quick explanation in a comment?
I thought there were undecidable propositions in the reals (e.g. cardinality)
I can't find right now a link to some good sources in internet about the subject but you can start with a Wikipedia entry. I also recommend "Gödel's theorems: an incomplete guide to its use and abuse" by Torkel Franzel.
You can see the other comment that was made which have some explanation about the reason that Gödel theorems don't hold on real numbers. But the intuitive idea is that Gödel's proposition needs some natural numbers axioms to be built. The real numbers system do not "explain" natural numbers in the way Peano's axioms do. So the axioms aren't valid and hence the Gödel proof isn't valid either.
josejorgexl|5 years ago
You can see the other comment that was made which have some explanation about the reason that Gödel theorems don't hold on real numbers. But the intuitive idea is that Gödel's proposition needs some natural numbers axioms to be built. The real numbers system do not "explain" natural numbers in the way Peano's axioms do. So the axioms aren't valid and hence the Gödel proof isn't valid either.
https://en.m.wikipedia.org/wiki/Decidability_of_first-order_...