View source for Carnap rule

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''rule of infinite induction, $\omega$-rule''

A [[Derivation rule|derivation rule]] stating that if for an arithmetic formula $\phi(x)$ the propositions $\phi(0),\phi(1),\ldots,$ have been proved, then the proposition $\forall x\phi(x)$ can be regarded as being proved. This rule was first brought into consideration by R. Carnap [[#References|[1]]]. Carnap's rule uses an infinite set of premises and is therefore inadmissible within the structure of the formal theories of D. Hilbert. The concept of a derivation in a system with the Carnap rule is undecidable. In mathematical logic one uses, for the study of formal arithmetic, the constructive Carnap rule: If there is an algorithm which for a natural number $n$ provides a derivation of the formula $\phi(n)$, then the proposition $\forall x\phi(x)$ can be regarded as being proved (the restricted $\omega$-rule, the rule of constructive infinite induction). Classical arithmetic calculus, which by Gödel's theorem is incomplete, becomes complete on adding the constructive Carnap rule (see [[#References|[2]]], [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Carnap,   "The logical syntax of language" , Kegan Paul, Trench &amp; Truber  (1937)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Kuznetsov,   ''Uspekhi Mat. Nauk'' , '''12''' :  4  (1957)  pp. 218–219</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.R. Shoenfield,   "On a restricted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020480/c02048011.png" />-rule"  ''Bull. Acad. Polon. Sci. Cl. III'' , '''7'''  (1959)  pp. 405–407</TD></TR></table>