Abstract.
In the first part of this paper we investigate the intuitionistic version \(iI\!\Sigma_1\) of \(I\!\Sigma_1\) (in the language of \(PRA\)), using Kleene's recursive realizability techniques. Our treatment closely parallels the usual one for \(HA\) and establishes a number of nice properties for \(iI\!\Sigma_1\), e.g. existence of primitive recursive choice functions (this is established by different means also in [D94]). We then sharpen an unpublished theorem of Visser's to the effect that quantifier alternation alone is much less powerful intuitionistically than classically: \(iI\!\Sigma_1\) together with induction over arbitrary prenex formulas is \(\Pi_2\)-conservative over \(iI\!\Pi_2\). In the second part of the article we study the relation of \(iI\!\Sigma_1\) to \(iI\!\Pi_1\) (in the usual arithmetical language). The situation here is markedly different from the classical case in that \(iI\!\Pi_1\) and \(iI\!\Sigma_1\) are mutually incomparable, while \(iI\!\Sigma_1\) is significantly stronger than \(iI\!\Pi_1\) as far as provably recursive functions are concerned: All primitive recursive functions can be proved total in \(iI\!\Sigma_1\) whereas the provably recursive functions of \(iI\!\Pi_1\) are all majorized by polynomials over \({\Bbb N}\). 0 \(iI\!\Pi_1\) is unusual also in that it lacks closure under Markov's Rule \(\mbox{MR}_{PR}\).
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Received April 4, 1996
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Wehmeier, K. Fragments of \(HA\) based on \(\Sigma_1\)-induction. Arch Math Logic 37, 37–49 (1997). https://doi.org/10.1007/s001530050081
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DOI: https://doi.org/10.1007/s001530050081