"More interestingly, the natural first-order theory of arithmetic of real numbers (with both addition and multiplication), the so-called theory of real closed f

"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/ 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).