 On the other hand, not all theories of arithmetic are incomplete. The theory of only addition of natural numbers but without multiplication (often called “Presburger arithmetic”), for example, is complete (and decidable) (Presburger 1929), as is the theory of multiplication of the positive integers (Skolem 1930). These theories are, though, very weak. But in any case, at least a theory which deals with both addition and multiplication is needed. 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 \((\mathbf{RCF})\), 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.