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.

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    $\begingroup$ Let me answer the second question, the representation of a number in base $10$ has nothing to do with the number itself. Numbers are not their representations. Addition works because addition works. The methods you were taught in school were to identify a number with its decimal base, which is good for practical uses, but very bad for mathematical uses. Mathematicians don't care about the actual number (as long as it's not $0$ or $1$, and rarely $2$), which is why no one bothers with a decimal, binary, or otherwise, representation of a number. We use $n$ to represent the abstract number. $\endgroup$ – Asaf Karagila Jun 2 '13 at 23:01
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    $\begingroup$ Also relevant: math.stackexchange.com/questions/282297/… (In particular Ittay's answer) $\endgroup$ – Asaf Karagila Jun 2 '13 at 23:06
  • $\begingroup$ Thank you very much for the link. It was very useful for me. $\endgroup$ – Igor Jun 3 '13 at 11:45

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