Saeed Salehi, University of Tabriz
ω-Consistency: Gödel’s “much weaker” notion of soundness
As the history goes, and was confirmed recently [vP20], Gödel first proved his first incompleteness theorem [G31] for sound theories (that extend Principia Mathematica). Later he weakened the soundness condition to “ℵ0-consistency”, which later evolved to “ω-consistency”. This condition was needed for irrefutability of (what is now called) Gödelian sentences; the simple consistency of a theory suffices for the unprovability of such sentences. Gödel already notes in [G31] that a necessary and sufficient condition for the independence of Gödelian sentences of T is just a bit more than the simple consistency of T: the consistency of T with ConT, the consistency statement of T.
In this talk, we ask the following questions and attempt at answering them, at least partially.
- Why on earth Gödel [G31] had to introduce this rather strange notion?
- Does it have any applications in other areas of logic, arithmetical theories, or mathematics?
- What was Gödel’s reason that ω-consistency is “much weaker” than soundness? He does prove in [G31] that consistency is weaker (if not much weaker) than ω-consistency; but never mentions a proof or even a hint as to why soundness is (much) stronger than ω-consistency!
- Other than those historical and philosophical questions, is this a useful notion worthy of further study?
- [G31] Kurt Gödel (1931); “On formally undecidable propositions of Principia Mathematica and related systems I”, in: S. Feferman, et al. (eds.), Kurt Gödel: Collected Works, Vol. I: Publications 1929–1936, Oxford University Press, 1986, pp. 135–152.
- [vP20] Jan von Plato (2020); Can Mathematics Be Proved Consistent? Gödel’s Shorthand Notes & Lectures on Incompleteness, Springer.
Reviewed in the zbMATH Open at https://zbmath.org/1466.03001