## Abstract

We prove under V = L that the inclusion modulo the non-stationary ideal is a Sigma(1)(1)-complete quasi-order in the generalized Borel-reducibility hierarchy (kappa > omega). This improvement to known results in L has many new consequences concerning the Sigma(1)(1)-completeness of quasi-orders and equivalence relations such as the embeddability of dense linear orders as well as the equivalence modulo various versions of the non-stationary ideal. This serves as a partial or complete answer to several open problems stated in the literature. Additionally the theorem is applied to prove a dichotomy in L: If the isomorphism of a countable first-order theory (not necessarily complete) is not Delta(1)(1), then it is Sigma(1)(1)-complete.

We also study the case V not equal L and prove Sigma(1)(1)-completeness results for weakly ineffable and weakly compact kappa.

Original language | English |
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Pages (from-to) | 245-268 |

Number of pages | 24 |

Journal | FUNDAMENTA MATHEMATICAE |

Volume | 251 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2020 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- generalized Baire space
- generalized descriptive set theory
- reducibility
- quasi-orders
- embeddability
- Sigma(1)(1)-completeness
- EQUIVALENT NONISOMORPHIC MODELS
- BOREL-REDUCIBILITY