Omega-consistency

$\omega$-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)