We can approach (non) collapse of PH from (non) collapse of hierarchy of Buss’s theories
(PH = Polynomial Hierarchy)
Our approach • Separate 𝑆𝑖2 by Gödel incompleteness theorem • Use analogy of separation of 𝐼Σ𝑖
Separation of 𝐼Σ𝑖
… 𝐼Σ3 ⊢ Con(IΣ2)
⊆ 𝐼Σ2 ⊢ Con IΣ2 ⊆ 𝐼Σ1
Consistency proof inside 𝑆𝑖2 • Bounded Arithmetics general y are not capable to prove consistency. – 𝑆2 does not prove consistency of Q (Paris, Wilkie) – 𝑆2 does not prove bounded consistency of 𝑆1 2(Pudlák) – 𝑆𝑖 𝑏 2 does not prove consistency the 𝐵𝑖 fragement of 𝑆−1 2 (Buss and Ignjatović)
Where… • 𝐵𝑏𝑖 − 𝐶𝐶𝐶 𝑇 – consistency of 𝐵𝑏𝑖 −proofs – 𝐵𝑏 𝑏 𝑖 −proofs : the proofs by 𝐵𝑖 -formule – 𝐵𝑏 𝑏 𝑏 𝑏 𝑖 :Σ0 (Σ𝑖 )… Formulas generated from Σ𝑖 by Boolean connectives and sharply bounded quantifiers. • 𝑆−1 2 – Induction free fragment of 𝑆𝑖2
𝑆𝑗 b −1 2 ⊢ 𝐵i − Con 𝑆2 , j > i Then, Buss’s hierarchy does not collapse.
Consistency proof of 𝑆−1 𝑖 2 inside 𝑆2
Problem • No truth definition, because • No valuation of terms, because • The values of terms increase exponentially • E.g. 2#2#2#2#2#...#2 In 𝑆𝑖2 world, terms do not have values a priori. • Thus, we must prove the existence of values in proofs. • We introduce the predicate 𝐸 which signifies existence of values.
Conjecture • 𝑆−1 2 𝐸 is weak enough – 𝑆𝑖+2 −1 2 can prove 𝑖-consistency of 𝑆2 𝐸 • While 𝑆−1 2 𝐸 is strong enough – 𝑆𝑖 𝑖 2𝐸 can interpret 𝑆2 • Conjecture 𝑆−1 𝑖 𝑖+2 2 𝐸 is a good candidate to separate 𝑆2 and 𝑆2 .