**May 10**

**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 “ℵ_{0}-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 Con_{T}, 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?

*ω*-consistency in the talk.

**References**:

- [G31] Kurt Gödel (1931); “On formally undecidable propositions of
*Principia Mathematica*and related systems I”, in: S. Feferman, et al. (eds.),, Oxford University Press, 1986, pp. 135–152.*Kurt Gödel: Collected Works, Vol. I: Publications 1929–1936* - [vP20] Jan von Plato (2020);
, Springer.*Can Mathematics Be Proved Consistent? Gödel’s Shorthand Notes & Lectures on Incompleteness*

Reviewed in the**zbMATH**Open at`https://zbmath.org/1466.03001`