## Notre Dame Journal of Formal Logic

### Intuitionistic Open Induction and Least Number Principle and the Buss Operator

#### Abstract

In "Intuitionistic validity in $T$-normal Kripke structures," Buss asked whether every intuitionistic theory is, for some classical theory $T$, that of all $T$-normal Kripke structures ${\cal H}(T)$ for which he gave an r.e. axiomatization. In the language of arithmetic $\mathit{Iop}$ and $\mathit{Lop}$ denote PA$^{-}$ plus Open Induction or Open LNP, $\mathit{iop}$ and $\mathit{lop}$ are their intuitionistic deductive closures. We show $\mathcal{H}\mathit{(Iop)}$ $=\mathit{lop}$ is recursively axiomatizable and $\mathit{lop}\vdash_{i\ c}\dashv \mathit{iop}$, while $i\forall_{1}\not\vdash \mathit{lop}$. If $iT$ proves PEM $_{\mathrm{atomic}}$ but not totality of a classically provably total Diophantine function of $T$, then $\mathcal{H}(T)\not\subseteq iT$ and so $iT\not\in \mathrm{range({\cal H})}$. A result due to Wehmeier then implies $i\Pi_{1}\not\in\mbox{range}({\cal H})$. We prove $\mathit{Iop}$ is not $\forall_{2}$-conservative over $i\forall_{1}$. If $\mathit{Iop}\subseteq T\subseteq I\forall_{1}$, then $iT$ is not closed under MR $_{\mathrm{open}}$ or Friedman's translation, so $iT\not\in$ range (${\cal H}$). Both $\mathit{Iop}$ and $I\forall_{1}$ are closed under the negative translation.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 39, Number 2 (1998), 212-220.

Dates
First available in Project Euclid: 7 December 2002

https://projecteuclid.org/euclid.ndjfl/1039293063

Digital Object Identifier
doi:10.1305/ndjfl/1039293063

Mathematical Reviews number (MathSciNet)
MR1714956

Zentralblatt MATH identifier
0968.03074

#### Citation

Ardeshir, Mohammad; Moniri, Mojtaba. Intuitionistic Open Induction and Least Number Principle and the Buss Operator. Notre Dame J. Formal Logic 39 (1998), no. 2, 212--220. doi:10.1305/ndjfl/1039293063. https://projecteuclid.org/euclid.ndjfl/1039293063

#### References

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• Wehmeier, K. F., “Fragments of HA based on $\Sigma_1$-induction,” Archive for Mathematical Logic, vol. 37 (1997), pp. 37–49. Zbl 0886.03040 MR 99a:03062 \AFDepartment of Mathematics Sharif University of Technology Tehran IRAN and Logic Group, IPM P.O. Box 19395-5746 Tehran IRAN email: ardeshir@karun.ipm.ac.ir Department of Mathematics Tarbiat Modarres University Tehran IRAN and Logic Group, IPM P.O. Box 19395-5746 Tehran IRAN email: mojmon@karun.ipm.ac.ir