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Difference between revisions of "Omega-completeness"

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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068150/o0681502.png" />-completeness''
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''$\omega$-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.
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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 <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 $\omega$-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.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.C. Kleene,  "Introduction to metamathematics" , North-Holland  (1951)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.C. Kleene,  "Introduction to metamathematics" , North-Holland  (1951)</TD></TR></table>

Revision as of 13:40, 9 April 2014

$\omega$-completeness

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 -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.

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