Why is multiplication axiomatized? When we learned exponentiation, we first introduced it as an iterated multiplication (for integer powers), then using nth root we defined it for rational powers, and finally we took the limit to define it for arbitrary real power.

My question: why can't we do the same with multiplication? Why is multiplication defined with axioms, rather than as an iterated addition?
My understanding is that standard propositional logic, which has axioms for multiplication and addition, is complicated enough to be undecidable etc. (Gödel's theorems). However, an axiom set that uses only addition is decidable and is thus much less powerful. I don't understand why using multiplication is sufficient to get the full power of mathematics while addition is not sufficient, but that is the case logically.

Perhaps I should state explicitly that I am talking about the natural numbers, not the real numbers, but I think that the point of the power of multiplication (excuse the pun) is relevant.
My understanding is that standard propositional logic, which has axioms for multiplication and addition, is complicated enough to be undecidable etc. (Gödel's theorems). However, an axiom set that uses only addition is decidable and is thus much less powerful. I don't understand why using multiplication is sufficient to get the full power of mathematics while addition is not sufficient, but that is the case logically.

Perhaps I should state explicitly that I am talking about the natural numbers, not the real numbers, but I think that the point of the power of multiplication (excuse the pun) is relevant.