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| Content Provider | Springer Nature Link |
|---|---|
| Author | Wehmeier, Kai F. |
| Copyright Year | 1997 |
| 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 We then sharpen an unpublished theorem of Visser's to the effect that quantifier alternation alone is much less powerful intuitionistically than classically: $iI\!\Sigma_1$ together with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 $\mbox{MR}_{PR}$ . |
| Starting Page | 37 |
| Ending Page | 49 |
| Page Count | 13 |
| File Format | |
| ISSN | 09335846 |
| Journal | Archive for Mathematical Logic |
| Volume Number | 37 |
| Issue Number | 1 |
| e-ISSN | 14320665 |
| Language | English |
| Publisher | Springer-Verlag |
| Publisher Date | 1997-12-01 |
| Publisher Place | Berlin Heidelberg |
| Access Restriction | One Nation One Subscription (ONOS) |
| Content Type | Text |
| Resource Type | Article |
| Subject | Philosophy Logic |
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