## Abstract

A major theme in the study of degree structures of all types has been the question of the decidability or undecidability of their first order theories. This is a natural and fundamental question that is an important goal in the analysis of these structures. In this paper, we study decidability for theories of upper semilattices that arise from the theory of numberings. We use the following approach: given a level of complexity, say Σα0, we consider the upper semilattice RΣα0 of all Σα0-computable numberings of all Σα0-computable families of subsets of N. We prove that the theory of the semilattice of all computable numberings is computably isomorphic to first order arithmetic. We show that the theory of the semilattice of all numberings is computably isomorphic to second order arithmetic. We also obtain a lower bound for the 1-degree of the theory of the semilattice of all Σα0-computable numberings, where α≥ 2 is a computable successor ordinal. Furthermore, it is shown that for any of the theories T mentioned above, the Π_{5}-fragment of T is hereditarily undecidable. Similar results are obtained for the structure of all computably enumerable equivalence relations on N, equipped with composition.

Original language | English |
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Pages (from-to) | 485-500 |

Number of pages | 16 |

Journal | Archive for Mathematical Logic |

Volume | 58 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 9 May 2019 |

## Keywords

- Computability theory
- Computably enumerable equivalence relation
- Elementary definability
- First order arithmetic
- Hereditary undecidability
- Numbering theory
- Rogers semilattice
- Second order arithmetic
- Upper semilattice

## OECD FOS+WOS

- 6.03.UA PHILOSOPHY
- 1.01.QL LOGIC