I read that Gödel's incompleteness theorems don't effect Peano Arithmetic that doesn't include multiplication sign. This confuses me, since multiplication can be defined through addition. So, even if "PA without multiplication" doesn't have multiplication sign in itself, it provides everything that's needed for defining mul. sign. So what's the difference? That is, why is "PA without multiplication" (but that contains everything needed for defining mul.) different from PA (that already has multiplication defined)?
From "Gödel without tears":
"The formalized interpreted language L contains the language of basic arithmetic if L has at least the standard rst-order logical apparatus (including identity), has a term '0' which denotes zero and function symbols for the successor, addition and multiplication functions defined over numbers - either built-in as primitives or introduced by definition - and has a predicate whose extension is the natural numbers."
Is there any difference between having those symbols defined as primitives (through axioms) and introducing them by definitions? I guess there is a difference, since PA is incomplete and PA without X isn't (and mul. definition is the only difference). If definitions (even if innocent-looking and seemengly talk only about what's already in system) are so powerful, then shouldn't people be more careful, because of it, when defining stuff? I noticed Gödel makes some other (primitive recursive) definitions in his proof (for example, function that returns n-th prime number). Could it be, in theory, that some of these definitions added certain functionality to system P that wasn't there before in P, and that without this new functionality P would be too weak for Gödel's theorems to affect it?