Omega-completeness

-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)