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 “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.

  1. Why on earth Gödel [G31] had to introduce this rather strange notion?
  2. Does it have any applications in other areas of logic, arithmetical theories, or mathematics?
  3. 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!
  4. Other than those historical and philosophical questions, is this a useful notion worthy of further study?
We will also review some properties of ω-consistency in the talk.