Examples of when $0\cdot x\neq 0$ or $x+0\neq x$.

Question

I'm trying to prove that the arithmetic axioms are independent by constructing a model in which all bar one of the axioms are satisfied, for each of the axioms below.

A first order theory with arithmetic has equality, one constant ($\mathbf{0}$), one unary function ('successor') and two binary functions ($+,\cdot$).

1. $(\forall x)( x^+\neq0)$
2. $(\forall x)(\forall y)(x^+=y^+\implies x=y)$
3. $(\forall x)(x+0=x)$
4. $(\forall x)(\forall y)(x+y^+=(x+y)^+)$
5. $(\forall x)(x\cdot 0=0)$
6. $(\forall x)(\forall y)(x\cdot (y^+)=(x\cdot y)+x)$
7. Induction

I would appreciate some help constructing models in which the 3rd and 5th axioms are false in particular.

Answer 1

Consider $\{0,1,\ldots\}$ with the same $+$, ${}^+$ and $0$, but a different multiplication: $a \cdot b = 1 + ab$ (where $ab$ is the usual product). This satisfies all but (5).

Answer 2

In the same fashion of Robert's answer, redefine $a+b=a^+ + b$. This should violate the third axiom.

Answer 3

All but 3: On $\mathbb N_0$, define $x+ y=(x+_{\text{standard}}y)^+$. Then use 5 and 6 to define multiplication recursively. Actually, one verifies that this results in $x\cdot y=x^+\cdot_{\text{standard}}y$