Fragments of Heyting arithmetic

Journal of Symbolic Logic 65 (3):1223-1240 (2000)
We define classes Φ n of formulae of first-order arithmetic with the following properties: (i) Every φ ∈ Φ n is classically equivalent to a Π n -formula (n ≠ 1, $\Phi_1:=\Sigma_1)$ . (ii) $\bigcup_{n\in \omega} \Phi_n = \mathscr L_A$ . (iii) IΠ n and iΦ n (i.e., Heyting arithmetic with induction schema restricted to Φ n - formulae) prove the same Π 2 -formulae. We further generalize a result by Visser and Wehmeier, namely that prenex induction within intuitionistic arithmetic is rather weak: After closing Φ n both under existential and universal quantification (we call these classes Θ n ) the corresponding theories iΘ n still prove the same Π 2 -formulae. In a second part we consider iΔ 0 plus collection-principles. We show that both the provably recursive functions and the provably total functions of iΔ 0 + {∀ x ≤ a ∃ y φ(x,y) → ∃ z ∀ x ≤ a ∃ y ≤ z φ(x,y) ∣ φ ∈ L A } are polynomially bounded. Furthermore we show that the contrapositive of the collection-schema gives rise to instances of the law of excluded middle and hence $i\Delta_0 + \{B\varphi, C\varphi \mid \varphi \in \mathscr L_A\} \vdash PA$
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DOI 10.2307/2586698
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Geoffrey E. Ostrin & Stanley S. Wainer (2005). Elementary Arithmetic. Annals of Pure and Applied Logic 133 (1):275-292.
Toshiyasu Arai (2011). Quick Cut-Elimination for Strictly Positive Cuts. Annals of Pure and Applied Logic 162 (10):807-815.

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