induction axiom
InductionAxiom
2002-08-17 23:17:17
2002-08-17 23:38:57
Definition
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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:
$$\phi(0)\wedge\forall x(\phi(x)\rightarrow\phi(x+1))\rightarrow \forall x\phi(x)\text{ where }\phi\text{ is }\Sigma^0_1$$