### Abstract

Let a be a Kleene's ordinal notation of a nonzero computable ordinal. We give a sufficient condition on a, so that for every -computable family of two embedded sets, i.e., two sets A, B, with A properly contained in B, the Rogers semilattice of the family is infinite. This condition is satisfied by every notation of ω; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satisfies this condition. On the other hand, we also give a sufficient condition on a, that yields that there is a -computable family of two embedded sets, whose Rogers semilattice consists of exactly one element; this condition is satisfied by all notations of every successor ordinal bigger than 1, and by all notations of the ordinal ω + ω; moreover every computable ordinal that is sum of two smaller ordinals has a notation that satisfies this condition. We also show that for every nonzero n ∈ ω, or n = ω, and every notation a of a nonzero ordinal there exists a -computable family of cardinality n, whose Rogers semilattice consists of exactly one element.

Original language | English |
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Pages (from-to) | 366-376 |

Number of pages | 11 |

Journal | Mathematical Logic Quarterly |

Volume | 58 |

Issue number | 4-5 |

DOIs | |

Publication status | Published - Aug 1 2012 |

### Keywords

- Computable numbering
- Computable ordinal
- Ershov's hierarchy
- Ordinal notation
- Rogers semilattice

### ASJC Scopus subject areas

- Logic

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## Cite this

*Mathematical Logic Quarterly*,

*58*(4-5), 366-376. https://doi.org/10.1002/malq.201100114