Why is it impossible to define multiplication in Presburger arithmetic yet possible to define exponentiation in Peano Arithmethic?

Question

Hello my question is related to Why is it impossible to define multiplication in Presburger arithmetic? and to How is exponentiation defined in Peano arithmetic?. I would have preferred to add it as a comment to one of the above discussions but I don't have commenting powers yet :-( Anyway, when I look at the answer to how exponentiation is defined, using sequences and the Chinese remainder theorem, I assume that Presburger is simply not powerful enough to play the same trick to define $\times$ in terms of +?

thanks

Answer 1

Yes, Presburger arithmetic is not strong enough to quantify over sequences, which is the reason it can't define multiplication.