PlanetMath: induction axiom

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induction axiom

(Definition)

An induction axiom specifies that a theory includes induction, possibly restricted to specific formulas. IND is the general axiom of induction:

$\displaystyle \phi(0)\wedge\forall x(\phi(x)\rightarrow\phi(x+1))\rightarrow \forall x\phi(x)$ for any formula $\displaystyle \phi$

If $\phi$ is restricted to some family of formulas then the axiom is called F-IND, or F induction. For example the axiom $\Sigma^0_1$ -IND is:

$\displaystyle \phi(0)\wedge\forall x(\phi(x)\rightarrow\phi(x+1))\rightarrow \forall x\phi(x)$ where $\displaystyle \phi$ is $\displaystyle \Sigma^0_1$

"induction axiom" is owned by Henry.

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Other names:

IND, -IND, axiom of induction

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Cross-references: axiom, formulas, restricted, induction, theory
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This is version 4 of induction axiom, born on 2002-08-17, modified 2002-08-17.
Object id is 3306, canonical name is InductionAxiom.
Accessed 5932 times total.

Classification:

AMS MSC:

03F35 (Mathematical logic and foundations :: Proof theory and constructive mathematics :: Second- and higher-order arithmetic and fragments)

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