# 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:

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

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):

• $\forall x(x^{\prime}\neq 0)$ ($0$ 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

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

Title PA PA 2013-03-22 12:56:36 2013-03-22 12:56:36 Henry (455) Henry (455) 10 Henry (455) Definition msc 03F30 Peano arithmetic first order Peano arithmetic PeanoArithmetic