### Abstract

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 $\Sigma^0_{\alpha}$,

we consider the upper semilattice $R_{\Sigma^0_{\alpha}}$ of all $\Sigma^0_{\alpha}$-computable

numberings of all $\Sigma^0_{\alpha}$-computable families of subsets of $\mathbb{N}$. 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 $\Sigma^0_{\alpha}$-computable numberings, where $\alpha \geq 2$ is a computable successor ordinal.

Furthermore, it is shown that for any of the theories $T$ mentioned above, the $\Pi_5$-fragment of $T$ is hereditarily undecidable. Similar results are obtained for the structure of all computably enumerable equivalence relations on $\mathbb{N}$, equipped with composition.

Original language | English |
---|---|

Article number | 1 |

Number of pages | 15 |

Journal | Archive for Mathematical Logic |

DOIs | |

Publication status | Accepted/In press - 2018 |

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### Cite this

*Archive for Mathematical Logic*, [1]. https://doi.org/DOI: 10.1007/s00153-018-0647-y

**ELEMENTARY THEORIES AND HEREDITARY UNDECIDABILITY FOR SEMILATTICES OF NUMBERINGS.** / Mustafa, Manat; Bazhenov, Nikolay; Mars, Yamaleev.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - ELEMENTARY THEORIES AND HEREDITARY UNDECIDABILITY FOR SEMILATTICES OF NUMBERINGS

AU - Mustafa, Manat

AU - Bazhenov, Nikolay

AU - Mars, Yamaleev

PY - 2018

Y1 - 2018

N2 - A major theme in the study of degree structures of all types has been the questionof the decidability or undecidability of their first order theories. This is a natural andfundamental 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 $\Sigma^0_{\alpha}$,we consider the upper semilattice $R_{\Sigma^0_{\alpha}}$ of all $\Sigma^0_{\alpha}$-computablenumberings of all $\Sigma^0_{\alpha}$-computable families of subsets of $\mathbb{N}$. We prove thatthe theory of the semilattice of all computable numberings is computably isomorphic to first order arithmetic. We show thatthe 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 $\Sigma^0_{\alpha}$-computable numberings, where $\alpha \geq 2$ is a computable successor ordinal.Furthermore, it is shown that for any of the theories $T$ mentioned above, the $\Pi_5$-fragment of $T$ is hereditarily undecidable. Similar results are obtained for the structure of all computably enumerable equivalence relations on $\mathbb{N}$, equipped with composition.

AB - A major theme in the study of degree structures of all types has been the questionof the decidability or undecidability of their first order theories. This is a natural andfundamental 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 $\Sigma^0_{\alpha}$,we consider the upper semilattice $R_{\Sigma^0_{\alpha}}$ of all $\Sigma^0_{\alpha}$-computablenumberings of all $\Sigma^0_{\alpha}$-computable families of subsets of $\mathbb{N}$. We prove thatthe theory of the semilattice of all computable numberings is computably isomorphic to first order arithmetic. We show thatthe 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 $\Sigma^0_{\alpha}$-computable numberings, where $\alpha \geq 2$ is a computable successor ordinal.Furthermore, it is shown that for any of the theories $T$ mentioned above, the $\Pi_5$-fragment of $T$ is hereditarily undecidable. Similar results are obtained for the structure of all computably enumerable equivalence relations on $\mathbb{N}$, equipped with composition.

UR - https://link.springer.com/article/10.1007/s00153-018-0647-y

U2 - DOI: 10.1007/s00153-018-0647-y

DO - DOI: 10.1007/s00153-018-0647-y

M3 - Article

JO - Archive for Mathematical Logic

JF - Archive for Mathematical Logic

SN - 0933-5846

M1 - 1

ER -