## Journal of Symbolic Logic

### On the induction schema for decidable predicates

Lev D. Beklemishev

#### Abstract

We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, $I\Delta1$. We show that $I\Delta1$ is independent from the set of all true arithmetical $\Pi2$-sentences. Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of $\Delta1$-induction.

An open problem formulated by J. Paris (see \cite{CloKra,HP}) is whether $I\Delta1$ proves the corresponding least element principle for decidable predicates, $L\Delta1$ (or, equivalently, the $\Sigma1$-collection principle $B\Sigma1$). We reduce this question to a purely computation-theoretic one.

#### Article information

Source
J. Symbolic Logic, Volume 68, Issue 1 (2003), 17-34.

Dates
First available in Project Euclid: 21 February 2003

https://projecteuclid.org/euclid.jsl/1045861504

Digital Object Identifier
doi:10.2178/jsl/1045861504

Mathematical Reviews number (MathSciNet)
MR1959310

Zentralblatt MATH identifier
1041.03042

#### Citation

Beklemishev, Lev D. On the induction schema for decidable predicates. J. Symbolic Logic 68 (2003), no. 1, 17--34. doi:10.2178/jsl/1045861504. https://projecteuclid.org/euclid.jsl/1045861504

#### References

• L. D. Beklemishev Induction rules, reflection principles, and provably recursive functions, Annals of Pure and Applied Logic, vol. 85 (1997), pp. 193--242.
• P. Clote and J. Krajíček Open problems, Arithmetic, proof theory, and computational complexity (P. Clote and J. Krajíček, editors), Oxford University Press, Oxford,1993, pp. 1--19.
• P. Hájek and P. Pudlák Metamathematics of first order arithmetic, Springer-Verlag, Berlin, Heidelberg, New York,1993.
• R. Kaye, J. Paris, and C. Dimitracopoulos On parameter free induction schemas, Journal of Symbolic Logic, vol. 53 (1988), no. 4, pp. 1082--1097.
• J. Paris A hierarchy of cuts in models of arithmetic, Model theory of algebra and arithmetic, Proceedings, Karapascz, Poland, 1979, Lecture Notes in Mathematics, vol. 834, Springer-Verlag,1980, pp. 312--337.
• C. Parsons On a number-theoretic choice schema and its relation to induction, Intuitionism and proof theory (A. Kino, J. Myhill, and R. E. Vessley, editors), North-Holland, Amsterdam,1970, pp. 459--473.
• A. Rastsvetaev and L. Beklemishev On the query complexity of finding a local maximum point, Logic Group Preprint Series 206, University of Utrecht,2000.
• H. Schwichtenberg Some applications of cut-elimination, Handbook of mathematical logic (J. Barwise, editor), North-Holland, Amsterdam,1977, pp. 867--896.