Computable Reductions and Reverse Mathematics

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9709)

Abstract

Recent work in reverse mathematics on combinatorial principles below Ramsey’s theorem for pairs has made use of a variety of computable reductions to give a finer analysis of the relationships between these principles. We use three concrete examples to illustrate this work, survey the known results and give new negative results concerning \(\mathsf {RT}^1_k\), \(\mathsf {SRT}^2_\ell \) and \(\mathsf {COH}\). Motivated by these examples, we introduce several variations of \(\mathsf {ADS}\) and describe the relationships between these principles under Weihrauch and strong Weihrauch reductions.

References

  1. 1.
    Astor, E.P., Dzhafarov, D.D., Solomon, R., Suggs, J.: The uniform content of partial and linear orders (in preparation)Google Scholar
  2. 2.
    Cholak, P.A., Jockusch Jr., C.G., Slaman, T.A.: On the strength of Ramsey’s theorem for pairs. J. Symb. Log. 66, 1–55 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chong, C.T., Lempp, S., Yang, Y.: On the role of collection principles for \(\Sigma ^0_2\) formulas in second-order reverse mathematics. Proc. Am. Math. Soc. 138, 1093–1100 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chong, C.T., Slaman, T.A., Yang, Y.: The metamathematics of stable Ramsey’s theorem for pairs. J. Am. Math Soc. 27, 863–892 (2014)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dorais, F.G., Dzhafarov, D.D., Hirst, J.L., Mileti, J.R., Shafer, P.: On uniform relationships between combinatorial problems. Trans. Am. Math. Soc. 368(2), 1321–1359 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Downey, R., Hirschfeldt, D.R., Lempp, S., Solomon, R.: A \(\Delta ^0_2\) set with no infinite low set in either it or its complement. J. Symb. Log. 66, 1371–1381 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dzhafarov, D.D.: Strong reducibilities between combinatorial principles. J. Symb. Log. (to appear)Google Scholar
  8. 8.
    Dzhafarov, D.D., Patey, L., Solomon, R., Westrick, L.B.: Ramsey’s theorem for singletons and strong computable reducibility (submitted)Google Scholar
  9. 9.
    Hirschfeldt, D.R., Jockusch Jr., C.G.: On notions of computability theoretic reduction between \(\Pi ^1_2\) principles. J. Math. Logic (to appear)Google Scholar
  10. 10.
    Hirschfeldt, D.R., Shore, R.A.: Combinatorial principles weaker than Ramsey’s theorem for pairs. J. Symb. Log. 72, 171–206 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Patey, L.: The weakness of being cohesive, thin or free in reverse mathematics. Isr. J. Math. (to appear)Google Scholar
  12. 12.
    Rakotoniaina, T.: The computational strength of Ramsey’s theorem. Ph.D. thesis, University of Cape Town (2015)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.University of ConnecticutStorrsUSA

Personalised recommendations