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The property of formal systems of arithmetic in which, for any formula $A(x)$, from a deduction of $A(\bar0),\ldots,A(\bar n),\ldots,$ it follows that one can infer the formula $\forall xA(x)$, where $\bar n$ is a constant signifying the natural number $n$. If this is not true, the system is called $\omega$-incomplete. K. Gödel in his incompleteness theorem (cf. Gödel incompleteness theorem) actually established the $\omega$-incompleteness of formal arithmetic. If all formulas which are true in the standard model of arithmetic are taken as axioms, then an $\omega$-complete axiom system is obtained. On the other hand, in every $\omega$-complete extension of Peano arithmetic, every formula which is true in the standard model can be deduced.


[1] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)
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Omega-completeness. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article