Archive for Mathematical Logic

, Volume 56, Issue 5–6, pp 491–506 | Cite as

The reverse mathematics of non-decreasing subsequences

  • Ludovic PateyEmail author


Every function over the natural numbers has an infinite subdomain on which the function is non-decreasing. Motivated by a question of Dzhafarov and Schweber, we study the reverse mathematics of variants of this statement. It turns out that this statement restricted to computably bounded functions is computationally weak and does not imply the existence of the halting set. On the other hand, we prove that it is not a consequence of Ramsey’s theorem for pairs. This statement can therefore be seen as an arguably natural principle between the arithmetic comprehension axiom and stable Ramsey’s theorem for pairs.


Reverse mathematics Non-decreasing subsequence Ramsey’s theorem 

Mathematics Subject Classification

03B30 03F35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Brattka, V., Gherardi, G.: Weihrauch degrees, omniscience principles and weak computability. J. Symb. Log. 76(1), 143–176 (2011). doi: 10.2178/jsl/1294170993 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cholak, P.A., Jockusch, C.G., Slaman, T.A.: On the strength of Ramsey’s theorem for pairs. J. Symb. Log. 66(01), 1–55 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chong, C., Lempp, S., Yang, Y.: On the role of the collection principle for \(\varSigma ^0_2\)-formulas in second-order reverse mathematics. Proc. Am. Math. Soc. 138(3), 1093–1100 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, New York (2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dzhafarov, D.D., Jockusch, C.G.: Ramsey’s theorem and cone avoidance. J. Symb. Log. 74(2), 557–578 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dzhafarov, D.D., Schweber, N.: Finding limit-nondecreasing sets for certain functions (2016).
  7. 7.
    Hirschfeldt, D.R.: Some questions in computable mathematics. In: Computability and Complexity, pp. 22–55. Springer International Publishing (2017)Google Scholar
  8. 8.
    Hirschfeldt, D.R., Shore, R.A.: Combinatorial principles weaker than Ramsey’s theorem for pairs. J. Symb. Log. 72(1), 171–206 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hirschfeldt, D.R., Shore, R.A., Slaman, T.A.: The atomic model theorem and type omitting. Trans. Am. Math. Soc. 361(11), 5805–5837 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jockusch, C.G., Soare, R.I.: \(\varPi ^0_1\) classes and degrees of theories. Trans. Am. Math. Soc. 173, 33–56 (1972)zbMATHGoogle Scholar
  11. 11.
    Liu, L.: Cone avoiding closed sets. Trans. Am. Math. Soc. 367(3), 1609–1630 (2015). doi: 10.1090/S0002-9947-2014-06049-2 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Patey, L.: Iterative forcing and hyperimmunity in reverse mathematics. Computability (2015). doi: 10.3233/COM-160062
  13. 13.
    Simpson, S.G.: Subsystems of Second Order Arithmetic. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Dept. of MathematicsUniversity of California, BerkeleyBerkeleyUSA

Personalised recommendations