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, secondorder, 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):

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

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$\forall x,y({x}^{\prime}={y}^{\prime}\to x=y)$ (the successor function is onetoone)

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

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

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$\forall x(x\cdot 0=0)$

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

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$$ ($0$ is the smallest number)

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$$

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

Canonical name  PA 
Date of creation  20130322 12:56:36 
Last modified on  20130322 12:56:36 
Owner  Henry (455) 
Last modified by  Henry (455) 
Numerical id  10 
Author  Henry (455) 
Entry type  Definition 
Classification  msc 03F30 
Synonym  Peano arithmetic 
Synonym  first order Peano arithmetic 
Related topic  PeanoArithmetic 