Abstract
We prove under V = L that the inclusion modulo the non-stationary ideal is a Σ11-complete quasi-order in the generalized Borel-reducibility hierarchy (κ > ω). This improvement to known results in L has many new consequences concerning the Σ11completeness 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 ∆11, then it is Σ11-complete. We also study the case V 6= L and prove Σ11-completeness results for weakly ineffable and weakly compact κ.
| Original language | English |
|---|---|
| Pages (from-to) | 245-268 |
| Number of pages | 24 |
| Journal | Fundamenta Mathematicae |
| Volume | 251 |
| Issue number | 3 |
| DOIs | |
| State | Published - 2020 |
Bibliographical note
Publisher Copyright:© Instytut Matematyczny PAN, 2020
Funding
of Finland for the support through its grant number 285203 as well as both Aalto University and University of Helsinki for providing suitable research environment during the academic year 2018–2019. The third author wishes to thank the University of Helsinki and its Doctoral Programme in Mathematics and Statistics (DOMAST) for support. He also thank Bar-Ilan University and Assaf Rinot for the support provided by the European Research Council through its grant agreement ERC-2018-StG 802756. The second author wishes to thank the Academy of Finland for the support through its grant number 285203 as well as both Aalto University and University of Helsinki for providing suitable research environment during the academic year 2018-2019. The third author wishes to thank the University of Helsinki and its Doctoral Programme in Mathematics and Statistics (DOMAST) for support. He also thank Bar-Ilan University and Assaf Rinot for the support provided by the European Research Council through its grant agreement ERC-2018StG 802756.
| Funders | Funder number |
|---|---|
| Helsingin Yliopisto | |
| Horizon 2020 Framework Programme | 802756 |
| European Commission | |
| Suomen Akatemia | 285203 |
| Aalto-Yliopisto |
Keywords
- Completeness-completeness
- Embeddability
- Generalized Baire space
- Generalized descriptive set theory
- Quasi-orders
- Reducibility