Omega-consistency
-consistency
The property of formal systems of arithmetic signifying the impossibility of obtaining -inconsistency.
-inconsistency is a situation in which, for some formula
, each formula of the infinite sequence
and the formula
are provable, where
is a constant of the formal system signifying the number 0, while the constants
are defined recursively in terms of
, signifying the number following directly after
:
.
The concept of -consistency appeared in conjunction with the Gödel incompleteness theorem of arithmetic. Assuming the
-consistency of formal arithmetic, K. Gödel proved its incompleteness. The property of
-consistency is stronger than the property of simple consistency. Simple consistency occurs if a formula not involving
is taken as
. It follows from Gödel's incompleteness theorem that there exist systems which are consistent but also
-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=31621