# Difference between revisions of "Carnap rule"

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− | ''rule of infinite induction, | + | {{TEX|done}} |

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

− | A [[Derivation rule|derivation rule]] stating that if for an arithmetic formula | + | 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==== | ====References==== | ||

<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Carnap, "The logical syntax of language" , Kegan Paul, Trench & 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> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Carnap, "The logical syntax of language" , Kegan Paul, Trench & 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> |

## Revision as of 14:45, 1 May 2014

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

A 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 [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 [2], [3]).

#### References

[1] | R. Carnap, "The logical syntax of language" , Kegan Paul, Trench & Truber (1937) (Translated from German) |

[2] | A.V. Kuznetsov, Uspekhi Mat. Nauk , 12 : 4 (1957) pp. 218–219 |

[3] | J.R. Shoenfield, "On a restricted -rule" Bull. Acad. Polon. Sci. Cl. III , 7 (1959) pp. 405–407 |

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Carnap rule.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Carnap_rule&oldid=32040