We investigate the “mathematical” strength of the theory I*2. In particular we prove the quadratic reciprocity law and Bertrand's postulate, using fragments of I*2 which employ some well-known number-theoretic functions.

We study the strength (over bounded induction) of axioms expressing particular cases of the Chinese Remainder Theorem with respect to the axiom ∀ x, y∃ z (z = xy).

. We provide an alternative proof of a theorem of P. Clote concerning end extensions of models of $\Sigma_n$ -induction, for $n \geq 2$.

We study problems of Clote and Paris, concerning the existence of end extensions of models of Σ n -collection. We continue the study of the notion of ‘Γ-fullness’, begun by Wilkie and Paris (Logic, Methodology and Philosophy of Science VIII (Moscow, 1987). Stud. Logic Found. Math., vol. 126, pp. 143–161. North- Holland, Amsterdam, 1989) and introduce and study a generalization of it, to be used in connection with the existence of Σ n -elementary end extensions (instead of plain end extensions). (...) We obtain (a) alternative proofs of results (Adamowicz in Fund. Math. 136, 133–145, 1990) and (Wilkie and Paris in Logic, Methodology and Philosophy of Science VIII (Moscow, 1987). Stud. Logic Found. Math., vol. 126, pp. 143–161. North-Holland, Amsterdam, 1989) related to the problem of Paris and (b) a partial solution to the problem of Clote. (shrink)

Over 25 years ago, the first author conjectured in [15] that the existence of arbitrarily large primes is provable from the axioms I Δ₀(π) + def(π), where π(x) is the number of primes not exceeding x, IΔ₀(π) denotes the theory of Δ₀ induction for the language of arithmetic including the new function symbol π, and de f(π) is an axiom expressing the usual recursive definition of π. We prove a modified version in which π is replaced by a more general (...) function ξ that counts some of the primes below x (which primes depends on the values of parameters in ξ), and has the property that π is provably Δ₀(ξ) definable. (shrink)