Title: Herbrand Consistency in Arithmetics with Bounded Induction

The language of arithmetical theories considered here is L = +, ×, ≤, 0, 1 in which the symbols are interpreted as usual in elementary mathematics. Robinson's arithmetic is denoted by Q; it is a finitely axiomatized basic theory of the function and predicate symbols in L. Peano's arithmetic PA is the first-order theory that extends Q by the induction schema for any L-formula ϕ(x): ϕ(0) & ∀x(ϕ(x) → ϕ(x + 1)) → ∀xϕ(x). Fragments of PA are extensions of Q with the induction schema restricted to a class of formulas. The most studied hierarchy of formulas is defined as follows: let ∆ 0 be the class of bounded formulas. A formula is called bounded if its every quantifier is bounded, i.e., is either of the form ∀x ≤ t(.. .) or ∃ x ≤ t(.. .) where t is a term; they are read as ∀x(x ≤ t →. . .) and ∃x(x ≤ t ∧. . .) respectively. It is easy to see that bounded formulas are decidable. The theory I∆ 0 , also called bounded arithmetic, is axiomatized by Q plus the induction schema for bounded formulas. The next level in the hierarchy are the classes of Σ 1 and Π 1 formulas which constitute bounded formulas prefixed with, respectively, a block of existential, and universal quantifiers. So, for example the formula ∃ x∀y ≤ x(y = x ∧ ∃ z≤ x[y × z = x] → y = 2) is a Σ 1-formula, and its negation ∀x∃y ≤ x(y = x ∧ ∃ z ≤ x[y × z = x] ∧ y = 2) is a Π 1-formula. We note that Σ 1-definable properties are exactly the computationally verifiable ones, and Π 1-definable properties are exactly the computationally refutable ones. The classes Σ m and Π m are defined inductively: Σ n+1-formulas are obtained from Π n-formulas by putting a block of existential quantifiers behind them, and Π n+1-formulas are Σ n-formulas prefixed with a block of universal quantifiers. The theory IΣ n is the extension of Q by the induction schema for Σ n-formulas. Note that PA = n≥0 IΣ n .