Peano Arithmetic (PA) is the restriction of Peano’s axioms to a first order theory of arithmetic. The only change is that the induction axiom is replaced by induction restricted to arithmetic formulas:

Note that this replaces the single, second-order, axiom of induction with a countably infinite schema of axioms.

Appropriate axioms defining $+$, $\cdot$, and $<$ are included. A full list of the axioms of PA looks like this (although the exact list of axioms varies somewhat from source to source):

• $\forall x(x^{{\prime}}\neq 0)$ ($0$ is the first number)

• $\forall x,y(x^{{\prime}}=y^{{\prime}}\rightarrow x=y)$ (the successor function is one-to-one)

• $\forall x(x+0=x)$ ($0$ is the additive identity)

• $\forall x,y(x+y^{{\prime}}=(x+y)^{{\prime}})$ (addition is the repeated application of the successor function)

• $\forall x(x\cdot 0=0)$

• $\forall x,y(x\cdot(y^{{\prime}})=x\cdot y+x)$ (multiplication is repeated addition)

• $\forall x(\neg(x<0))$ ($0$ is the smallest number)

• $\forall x,y(x<y^{{\prime}}\leftrightarrow x<y\vee x=y)$

• $\phi(0)\wedge\forall x(\phi(x)\rightarrow\phi(x^{{\prime}}))\rightarrow\forall x\phi(x))\text{where }\phi\text{ is arithmetical}$