Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2507.12631 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909692442181632 |
|---|---|
| author | Shelah, Saharon Usvyatsov, Alexander |
| author_facet | Shelah, Saharon Usvyatsov, Alexander |
| contents | We consider the following property of a first order theory T with a distinguished unary predicate P: every model of the theory of P occurs as the P-part of some model of T. We call this property the Gaifman property. Gaifman conjectured that if T is relatively categorical over P, then it has the Gaifman property. We propose a generalized version of this conjecture: if T fails the Gaifman property, then it exhibits non-structure over P, i.e., has many non-isomorphic models over P in many cardinalities. We address this conjecture for countable theories. Motivated by ideas from Classification Theory, we separate this conjecture into two parts: 1) stability over P (a structure property of theories) implies the Gaifman property, and 2) instability over P implies non-structure. In this paper prove the first part of this conjecture. In fact, we prove a stronger statement: an appropriate version of stability implies higher stable amalgamation properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12631 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stable amalgamation over a predicate and the Gaifman property Shelah, Saharon Usvyatsov, Alexander Logic We consider the following property of a first order theory T with a distinguished unary predicate P: every model of the theory of P occurs as the P-part of some model of T. We call this property the Gaifman property. Gaifman conjectured that if T is relatively categorical over P, then it has the Gaifman property. We propose a generalized version of this conjecture: if T fails the Gaifman property, then it exhibits non-structure over P, i.e., has many non-isomorphic models over P in many cardinalities. We address this conjecture for countable theories. Motivated by ideas from Classification Theory, we separate this conjecture into two parts: 1) stability over P (a structure property of theories) implies the Gaifman property, and 2) instability over P implies non-structure. In this paper prove the first part of this conjecture. In fact, we prove a stronger statement: an appropriate version of stability implies higher stable amalgamation properties. |
| title | Stable amalgamation over a predicate and the Gaifman property |
| topic | Logic |
| url | https://arxiv.org/abs/2507.12631 |