# induction axiom

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

 $\phi(0)\wedge\forall x(\phi(x)\rightarrow\phi(x+1))\rightarrow\forall x\phi(x)% \text{ for any formula }\phi$

If $\phi$ is restricted to some family of formulas $F$ then the axiom is called F-IND, or F induction. For example the axiom $\Sigma^{0}_{1}$-IND is:

 $\phi(0)\wedge\forall x(\phi(x)\rightarrow\phi(x+1))\rightarrow\forall x\phi(x)% \text{ where }\phi\text{ is }\Sigma^{0}_{1}$
Title induction axiom InductionAxiom 2013-03-22 12:56:51 2013-03-22 12:56:51 Henry (455) Henry (455) 7 Henry (455) Definition msc 03F35 IND -IND axiom of induction