Uniform Martin’s conjecture, locally
| Autor: | Vittorio Bard |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Pure mathematics
Determinacy Conjecture Turing degree Computable reducibility Martin’s conjecture Uniformly invariant functions Applied Mathematics General Mathematics Structure (category theory) Physics::Classical Physics Raising (linguistics) Mathematics::Logic symbols.namesake Phenomenon symbols Equivalence relation Turing computer Mathematics computer.programming_language |
| Zdroj: | Proceedings of the American Mathematical Society. 148:5369-5380 |
| ISSN: | 1088-6826 0002-9939 |
| DOI: | 10.1090/proc/15159 |
| Popis: | We show that part I of uniform Martin's conjecture follows from a local phenomenon, namely that if a non-constant Turing invariant function goes from the Turing degree $\boldsymbol x$ to the Turing degree $\boldsymbol y$, then $\boldsymbol x \le_T \boldsymbol y$. Besides improving our knowledge about part I of uniform Martin's conjecture (which turns out to be equivalent to Turing determinacy), the discovery of such local phenomenon also leads to new results that did not look strictly related to Martin's conjecture before. In particular, we get that computable reducibility $\le_c$ on equivalence relations on $\mathbb N$ has a very complicated structure, as $\le_T$ is Borel reducible to it. We conclude raising the question "Is part II of uniform Martin's conjecture implied by local phenomena, too?" and briefly indicating a possible direction. |
| Databáze: | OpenAIRE |
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