## Uniform Martin’s conjecture, locally

HTML articles powered by AMS MathViewer

- by Vittorio Bard PDF
- Proc. Amer. Math. Soc.
**148**(2020), 5369-5380 Request permission

## Abstract:

We show that part I of the uniform Martin’s conjecture follows from a local phenomenon, namely that every non-constant uniformly Turing invariant $f:[x]_{\equiv _T}\to [y]_{\equiv _T}$ satisfies $x\le _T y$. Besides improving our knowledge about part I of the 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 by raising the question:*Is part II of the uniform Martin’s conjecture implied by local phenomena, too?*and briefly indicating possible directions.

## References

- Scot Adams and Alexander S. Kechris,
*Linear algebraic groups and countable Borel equivalence relations*, J. Amer. Math. Soc.**13**(2000), no. 4, 909–943. MR**1775739**, DOI 10.1090/S0894-0347-00-00341-6 - Howard Becker,
*A characterization of jump operators*, J. Symbolic Logic**53**(1988), no. 3, 708–728. MR**960994**, DOI 10.2307/2274567 - Samuel Coskey, Joel David Hamkins, and Russell Miller,
*The hierarchy of equivalence relations on the natural numbers under computable reducibility*, Computability**1**(2012), no. 1, 15–38. MR**3068302**, DOI 10.3233/com-2012-004 - C. T. Chong, Wei Wang, and Liang Yu,
*The strength of the projective Martin conjecture*, Fund. Math.**207**(2010), no. 1, 21–27. MR**2576277**, DOI 10.4064/fm207-1-2 - Su Gao,
*Invariant descriptive set theory*, Pure and Applied Mathematics (Boca Raton), vol. 293, CRC Press, Boca Raton, FL, 2009. MR**2455198** - S. Jackson, A. S. Kechris, and A. Louveau,
*Countable Borel equivalence relations*, J. Math. Log.**2**(2002), no. 1, 1–80. MR**1900547**, DOI 10.1142/S0219061302000138 - Takayuki Kihara and Antonio Montalbán,
*The uniform Martin’s conjecture for many-one degrees*, Trans. Amer. Math. Soc.**370**(2018), no. 12, 9025–9044. MR**3864404**, DOI 10.1090/tran/7519 - Donald A. Martin,
*The axiom of determinateness and reduction principles in the analytical hierarchy*, Bull. Amer. Math. Soc.**74**(1968), 687–689. MR**227022**, DOI 10.1090/S0002-9904-1968-11995-0 - Andrew Marks, Theodore A. Slaman, and John R. Steel,
*Martin’s conjecture, arithmetic equivalence, and countable Borel equivalence relations*, Ordinal definability and recursion theory: The Cabal Seminar. Vol. III, Lect. Notes Log., vol. 43, Assoc. Symbol. Logic, Ithaca, NY, 2016, pp. 493–519. MR**3469180** - Gerald E. Sacks,
*On suborderings of degrees of recursive unsolvability*, Z. Math. Logik Grundlagen Math.**7**(1961), 46–56. MR**131973**, DOI 10.1002/malq.19610070109 - Theodore A. Slaman and John R. Steel. Definable functions on degrees. In Alexander S. Kechris, Donald A. Martin, and John R. Steel, editors,
*Cabal Seminar 81–85*, pages 37–55, Berlin, Heidelberg, 1988. Springer Berlin Heidelberg. - John R. Steel,
*A classification of jump operators*, J. Symbolic Logic**47**(1982), no. 2, 347–358. MR**654792**, DOI 10.2307/2273146 - Simon Thomas,
*Martin’s conjecture and strong ergodicity*, Arch. Math. Logic**48**(2009), no. 8, 749–759. MR**2563815**, DOI 10.1007/s00153-009-0148-0 - Jay Williams,
*Universal countable Borel quasi-orders*, J. Symb. Log.**79**(2014), no. 3, 928–954. MR**3305823**, DOI 10.1017/jsl.2013.35

## Additional Information

**Vittorio Bard**- Affiliation: Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, via Carlo Alberto 10, 10121 Torino, Italy
- ORCID: 0000-0003-3840-2059
- Email: vittorio.bard@unito.it
- Received by editor(s): July 24, 2019
- Received by editor(s) in revised form: April 10, 2020
- Published electronically: September 17, 2020
- Communicated by: Heike Mildenberger
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**148**(2020), 5369-5380 - MSC (2010): Primary 03D28; Secondary 03E15, 03E60
- DOI: https://doi.org/10.1090/proc/15159
- MathSciNet review: 4163848