What Gödel theorems don't say

[josejorgexl]

Gödel theorems don't say that a theory that is "complex enough", if is consistent, is incomplete.

The system of real numbers is both consistent and complete. And I think it is at least as complex as the system of natural numbers is.

[AlDante2]

"More interestingly, the natural first-order theory of arithmetic of real numbers (with both addition and multiplication), the so-called theory of real closed fields (RCF), is both complete and decidable, as was shown by Tarski (1948); he also demonstrated that the first-order theory of Euclidean geometry is complete and decidable. Thus, one should keep in mind that there are some non-trivial and interesting theories to which Gödel’s theorems do not apply."

https://plato.stanford.edu/entries/goedel-incompleteness/

The incompleteness theorems apply only to formal systems which are able to prove a sufficient collection of facts about the natural numbers. As there is no function f in RCF which can determine if a given real is also a natural number, RCF can make no statements about natural numbers. Gödel's first incompleteness theorem does not apply (and hence his second also does not apply).

[josejorgexl]

Excellent article and excellent comment as well! It's hard to find some people to talk about this subject. Thanks!

[rfergie]

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)

[josejorgexl]

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.

https://en.m.wikipedia.org/wiki/Decidability_of_first-order_...

[scott31]

This is not twitter

https://news.ycombinator.com/newsguidelines.html

[josejorgexl]

Sorry, could you be more specific? I don't understand what I did wrong. Whatever it is, it was not intentionally and I wouldn't like to do it again

[kleer001]

Do a bit more lurking and pattern recognition on highly voted HN posts, then it will become clearer.