Journal of Symbolic Logic

Fragments of Heyting Arithmetic

Wolfgang Burr

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Abstract

We define classes $\Phi_n$ of formulae of first-order arithmetic with the following properties: (i) Every $\varphi \in \Phi_n$ is classically equivalent to a $\Pi_n$-formula (n $\neq$ 1, $\Phi_1 :=\Sigma_1)$. (ii) $\bigcup_{n\in \omega} \Phi_n = \mathscr L_A$. (iii) $I\Pi_n$ and $i\Phi_n$ (i.e., Heyting arithmetic with induction schema restricted to $\Phi_n$- formulae) prove the same $\Pi_2$-formulae. We further generalize a result by Visser and Wehmeier, namely that prenex induction within intuitionistic arithmetic is rather weak: After closing $\Phi_n$ both under existential and universal quantification (we call these classes $\Theta_n$) the corresponding theories i$\Theta_n$ still prove the same $\Pi_2$-formulae. In a second part we consider i$\Delta_0$ plus collection-principles. We show that both the provably recursive functions and the provably total functions of $i\Delta_0 + \{\forall x \leq a \exists y \varphi(x,y) \rightarrow \exists z \forall x \leq a \exists y \leq z \varphi(x,y) \mid \varphi \in \mathscr L_A\}$ 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 $i\Delta_0 + \{B\varphi, C\varphi \mid \varphi \in \mathscr L_A\} \vdash PA$.

Article information

Source
J. Symbolic Logic, Volume 65, Issue 3 (2000), 1223-1240.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183746179

Mathematical Reviews number (MathSciNet)
MR1791374

Zentralblatt MATH identifier
0966.03052

JSTOR
links.jstor.org

Citation

Burr, Wolfgang. Fragments of Heyting Arithmetic. J. Symbolic Logic 65 (2000), no. 3, 1223--1240. https://projecteuclid.org/euclid.jsl/1183746179


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