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Omega-completeness

From Encyclopedia of Mathematics
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-completeness

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

References

[1] S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951)
How to Cite This Entry:
Omega-completeness. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Omega-completeness&oldid=31445
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article