View source for Omega-completeness

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068150/o0681502.png" />-completeness''

The property of formal systems of arithmetic in which, for any formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068150/o0681503.png" />, from a deduction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068150/o0681504.png" /> it follows that one can infer the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068150/o0681505.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068150/o0681506.png" /> is a constant signifying the natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068150/o0681507.png" />. If this is not true, the system is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068150/o0681509.png" />-incomplete. K. Gödel in his incompleteness theorem (cf. [[Gödel incompleteness theorem|Gödel incompleteness theorem]]) actually established the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068150/o06815010.png" />-incompleteness of formal arithmetic. If all formulas which are true in the standard model of arithmetic are taken as axioms, then an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068150/o06815011.png" />-complete axiom system is obtained. On the other hand, in every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068150/o06815012.png" />-complete extension of Peano arithmetic, every formula which is true in the standard model can be deduced.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.C. Kleene,   "Introduction to metamathematics" , North-Holland  (1951)</TD></TR></table>