Omega-consistency
-consistency
The property of formal systems of arithmetic signifying the impossibility of obtaining \omega-inconsistency. \omega-inconsistency is a situation in which, for some formula A(x), each formula of the infinite sequence A(\bar0),\ldots,A(\bar n),\ldots, and the formula \neg\forall x\ A(x) are provable, where \bar 0 is a constant of the formal system signifying the number 0, while the constants \bar n are defined recursively in terms of (x)', signifying the number following directly after x: \overline{n+1}=(\bar n)'.
The concept of \omega-consistency appeared in conjunction with the Gödel incompleteness theorem of arithmetic. Assuming the \omega-consistency of formal arithmetic, K. Gödel proved its incompleteness. The property of \omega-consistency is stronger than the property of simple consistency. Simple consistency occurs if a formula not involving x is taken as A(x). It follows from Gödel's incompleteness theorem that there exist systems which are consistent but also \omega-inconsistent.
References
[1] | S.C. Kleene, "Introduction to metamathematics" , North-Holland (1951) |
Omega-consistency. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Omega-consistency&oldid=43594