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induction axiom
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(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: $$\phi(0)\wedge\forall x(\phi(x)\rightarrow\phi(x+1))\rightarrow \forall x\phi(x)\text{ where }\phi\text{ is }\Sigma^0_1$$
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"induction axiom" is owned by Henry.
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(view preamble | get metadata)
| Other names: |
IND, -IND, axiom of induction |
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Cross-references: axiom, formulas, induction, theory
There are 7 references to this entry.
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 7422 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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Pending Errata and Addenda
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