PA

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

$$\varphi (0)\wedge \forall x(\varphi (x)\to \varphi (x^{\prime }))\to \forall x\varphi (x))\text{where }\varphi \text{ is arithmetical}$$

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):

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

• $\forall x,y(x^{\prime }=y^{\prime }\to x=y)$ (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)

• ($0$ is the smallest number)

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