Abstract
We extend a classical result in ordinary recursion theory to higher recursion theory, namely that every recursively enumerable set can be represented in any model \(\mathfrak {A}\) by some Horn theory, where \(\mathfrak {A}\) can be any model of a higher recursion theory, like primitive set recursion, \(\alpha \)-recursion, or \(\beta \)-recursion. We also prove that, under suitable conditions, a set defined through a Horn theory in a set \(\mathfrak {A}\) is recursively enumerable in models of the above mentioned recursion theories.
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Acknowledgments
This research was partially supported by CONACYT grant 400200-5-32267-E. This work started during the third author’s sabbatical leave at Institut für Mathematik, Humboldt Universität, 10099 Berlin, Germany. We would like to thank the Referee for his/her interest in our work and for his/her helpful comments that will greatly improve the manuscript.
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Nido Valencia, J.A., Solís Daun, J.E. & Villegas Silva, L.M. A representation of recursively enumerable sets through Horn formulas in higher recursion theory. Period Math Hung 73, 1–15 (2016). https://doi.org/10.1007/s10998-016-0148-x
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DOI: https://doi.org/10.1007/s10998-016-0148-x
Keywords
- Horn theory
- Primitive recursive set functions
- Recursively enumerable set
- \(\alpha \)-Recursion theory
- Primitive recursively closed ordinals
- Admissible recursion
- \(\beta \)-Recursion