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):

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$\forall x(x^{\prime}\neq 0)$ ( $0$ is the first number)
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$\forall x,y(x^{\prime}=y^{\prime}\rightarrow x=y)$ (the successor function is one-to-one)
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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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$\forall x(\neg(x<0))$ ( $0$ is the smallest number)
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$\forall x,y(x<y^{\prime}\leftrightarrow x<y\vee x=y)$
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$\phi(0)\wedge\forall x(\phi(x)\rightarrow\phi(x^{\prime}))\rightarrow\forall x% \phi(x))\text{where }\phi\text{ is arithmetical}$

Title	PA
Canonical name	PA
Date of creation	2013-03-22 12:56:36
Last modified on	2013-03-22 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