Namespaces
Variants
Actions

Omega-consistency

From Encyclopedia of Mathematics
Revision as of 17:06, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

-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)
How to Cite This Entry:
Omega-consistency. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Omega-consistency&oldid=31621
This article was adapted from an original article by V.N. Grishin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article