Is there a problem if I don't use $0$ in Peano arithmetic?

Question

Peano arithmetic is the following list of axioms (along with the usual axioms of equality) plus induction schema.

1. $\forall x \ (0 \neq S ( x ))$
2. $\forall x, y \ (S( x ) = S( y ) \Rightarrow x = y)$
3. $\forall x \ (x + 0 = x )$
4. $\forall x, y \ (x + S( y ) = S( x + y ))$
5. $\forall x \ (x \cdot 0 = 0)$
6. $\forall x, y \ (x \cdot S ( y ) = x \cdot y + x )$

In my country's general education, the natural number does not include $0$.

Is there a problem if I write this as follows?

1. $\forall x \ (1 \neq S ( x ))$
2. $\forall x, y \ (S( x ) = S( y ) \Rightarrow x = y)$
3. $\forall x \ (x + 1 = S(x) )$
4. $\forall x, y \ (x + S( y ) = S( x + y ))$
5. $\forall x \ (x \cdot 1 = x)$
6. $\forall x, y \ (x \cdot S ( y ) = x \cdot y + x )$

Answer 1

If you want a Peano-like axiomatization of the positive integers, then that's possible, but what you've written is not it. Axiom 5 should say $\forall x\,(x \cdot 1 = x)$, instead. Otherwise, you get an operation $\cdot$ that acts a bit like multiplication, but is not commutative (we can prove that $S(1) \cdot 1 = 1$ but $1 \cdot S(1) = S(1)$, for example), instead of having the axioms describe what we think of as "normal" arithmetic.

After you have corrected axiom 5, there is no problem with those axioms. (Essentially, all you've done is provided a different base case for how multiplication works.)

...but there is a problem calling them "the Peano axioms", because they are not. For the purposes of speaking with the greater mathematical community, the Peano axioms are a specific thing, and you will get confused. If there is some nationwide standardized exam on which you have to repeat that $0$ is not a natural number, then of course you do that - but you don't have to contort yourself into thinking a specific way just to pass that exam.

(The actual, correct thing to say about the natural numbers is that whether they include $0$ or not is a matter of convention, and mathematicians tend to switch the convention to whatever makes life easier. I think that, at the level when you are learning about the Peano axioms, you should be able to handle this idea, and trying to convert everything everyone does to the One True Convention is a worse idea you should abandon.)