PlanetMath: PA

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(Definition)

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:

$\displaystyle \phi(0)\wedge \forall x(\phi(x)\rightarrow\phi(x'))\rightarrow \forall x\phi(x))$ where $\displaystyle \phi$ 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):

$\forall x (x'\neq 0)$ (0 is the first number)
$\forall x,y (x'=y'\rightarrow x=y)$ (the successor function is one-to-one)
$\forall x (x+0=x)$ (0 is the additive identity)
$\forall x,y(x+y'=(x+y)')$ (addition is the repeated application of the successor function)
$\forall x(x\cdot 0=0)$
$\forall x,y(x\cdot(y')=x\cdot y+x)$ (multiplication is repeated addition)
$\forall x(\neg (x<0))$ (0 is the smallest number)
$\forall x,y(x<y'\leftrightarrow x<y\vee x=y)$
$\phi(0)\wedge \forall x(\phi(x)\rightarrow\phi(x'))\rightarrow \forall x\phi(x))$ where $\phi$ is arithmetical

"PA" is owned by Henry.

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