I really tried to find similar questions but didn't manage to find them. Please, forgive me if this question is a duplicate. I also apologize for my English.
So. The question.
We're given five Peano axioms. I consider the case when the set of natural numbers is considered to contain 0.
Suppose we defined numbers 1,2,...,9 as successor of 0, successor of 1, ..., successor of 8 or somehow else.
1) How can we derive from Peano axioms that all natural numbers can be uniquely represented in decimal system?
2) And how can we show that addition (and multiplication) of natural numbers in decimal representation (which we all learned at school) follows from Peano axioms? For example how do we know that in Peano arithmetic $132 + 223 = 355$.
I hope this question is not to strange and i'm grateful in advance for any answers.