Weakly Represented Families in Reverse Mathematics

  • Rupert Hölzl
  • Dilip Raghavan
  • Frank Stephan
  • Jing Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10010)


We study the proof strength of various second order logic principles that make statements about families of sets and functions. Usually, families of sets or functions are represented in a uniform way by a single object. In order to be able to go beyond the limitations imposed by this approach, we introduce the concept of weakly represented families of sets and functions. This allows us to study various types of families in the context of reverse mathematics that have been studied in set theory before. The results obtained witness that the concept of weakly represented families is a useful and robust tool in reverse mathematics.



The authors would like to thank C.T. Chong, Wei Li and Yue Yang for fruitful discussions and suggestions. They are also grateful to the anonymous referee for detailed and helpful comments, in particular for pointing out a simplification of the proof of Theorem 29.


  1. 1.
    Bartoszyński, T., Judah, H.: Set Theory: On the Structure of the Real Line. A K Peters Ltd., Wellesley (1995)MATHGoogle Scholar
  2. 2.
    Blass, A.: Combinatorial cardinal characteristics of the continuum. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, pp. 395–489. Springer, Dordrecht (2010). doi: 10.1007/978-1-4020-5764-9_7. vols. 1, 2, 3CrossRefGoogle Scholar
  3. 3.
    Brendle, J., Brooke-Taylor, A., Ng, K.M., Nies, A.: An analogy between cardinal characteristics and highness properties of oracles. In: Proceedings of the 13th Asian Logic Conference, Guangzhou, China, 16–20 September 2013, pp. 1–28. World Scientific (2013)Google Scholar
  4. 4.
    Cholak, P.A., Jockusch Jr., C.G., Slaman, T.A.: On the strength of Ramsey’s theorem for pairs. J. Symbolic Logic 66(1), 1–55 (2001)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Chong, C.T., Lempp, S., Yang, Y.: On the role of the collection principle for \({\rm \Sigma ^{0}_{2}}\) formulas in second order reverse mathematics. Proc. Am. Math. Soc. 138, 1093–1100 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Chong, C.T., Slaman, T.A., Yang, Y.: \(\Pi ^1_1\)-conservation of combinatorial principles weaker than Ramsey’s theorem for pairs. Adv. Math. 230, 1060–1077 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Demuth, O., Kučera, A.: Remarks on \(1\)-genericity, semigenericity and related concepts. Commentationes Math. Univ. Carol. 028(1), 85–94 (1987)MathSciNetMATHGoogle Scholar
  8. 8.
    Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Springer, Heidelberg (2010)CrossRefMATHGoogle Scholar
  9. 9.
    Friedberg, R.: A criterion for completeness of degrees of unsolvability. J. Symbolic Logic 22, 159–160 (1957)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Friedberg, R.: Three theorems on recursive enumeration. J. Symbolic Logic 23, 309–316 (1958)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hechler, S.H.: On the existence of certain cofinal subsets of \(^{\omega }\omega \). In: Axiomatic Set Theory Proceedings of Symposia in Pure Mathematics, Part II, University of California, Los Angeles, California, 1967, vol. 13, pp. 155–173. American Mathematical Society, Providence (1974)Google Scholar
  12. 12.
    Hirschfeldt, D.R.: Slicing the Truth. World Scientific, River Edge (2015)Google Scholar
  13. 13.
    Hirschfeldt, D.R., Shore, R.A., Slaman, T.A.: The atomic model theorem and type omitting. Trans. Am. Math. Soc. 361, 5805–5837 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hölzl, R., Jain, S., Stephan, F.: Inductive inference and reverse mathematics. Ann. Pure Appl. Logic (2016).
  15. 15.
    Jockusch Jr., C.G.: Upward closure and cohesive degrees. Isr. J. Math. 15, 332–335 (1973)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jockusch, C.G., Soare, R.I.: \(\Pi _1^0\) classes and degrees of theories. Trans. Am. Math. Soc. 173, 33–56 (1972)MATHGoogle Scholar
  17. 17.
    Jockusch, C.G., Stephan, F.: A cohesive set which is not high. Math. Logic Q. 39, 515–530 (1993)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kjos-Hanssen, B., Merkle, W., Stephan, F.: Kolmogorov complexity and recursion theorem. Trans. Am. Math. Soc. 263, 5465–5480 (2011)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kleene, S.C., Post, E.L.: The uppersemilattice of degrees of recursive unsolvability. Ann. Math. 59, 379–407 (1954)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lacombe, D.: Sur le semi-réseau constitué par les degrès d’indecidabilité récursive. Comptes rendus hebdomadaires des séances de l’Académie des Sciences 239, 1108–1109 (1954)MATHGoogle Scholar
  21. 21.
    van Lambalgen, M.: The axiomatization of randomness. J. Symbolic Logic 55, 1143–1167 (1990)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Martin, D.A.: Classes of recursively enumerable sets and degrees of unsolvability. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 12, 295–310 (1966)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Mileti, J.: Partition theorems and computability theory, Ph.D. dissertation, University of Illinois at Urbana-Champaign (2004)Google Scholar
  24. 24.
    Miller, A.W.: Some properties of measure and category. Trans. Am. Math. Soc. 266, 93–114 (1981)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Nies, A.: Computability and Randomness. Oxford Science Publications, Oxford (2009)CrossRefMATHGoogle Scholar
  26. 26.
    Odifreddi, P.: Classical Recursion Theory. Elsevier, North Holland (1989)MATHGoogle Scholar
  27. 27.
    Odifreddi, P.: Classical Recursion Theory, vol. 2. Elsevier, North Holland (1999)MATHGoogle Scholar
  28. 28.
    Rupprecht, N.: Relativized Schnorr tests with universal behavior. Arch. Math. Logic 49(5), 555–570 (2010)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Shelah, S.: Proper and Improper Forcing. Perspectives in Mathematical Logic, 2nd edn. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  30. 30.
    Simpson, S.G.: Subsystems of Second Order Arithmetic, 2nd edn. Cambridge University Press, Cambridge (2006)MATHGoogle Scholar
  31. 31.
    Spector, C.: On degrees of unsolvability. Ann. Math. 64, 581–592 (1956)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)CrossRefMATHGoogle Scholar
  33. 33.
    Stephan, F.: Recursion theory, lecture notes, school of computing, National University of Singapore, Technical report TR10/12 (2012)Google Scholar
  34. 34.
    Liang, Y.: Lowness for genericity. Arch. Math. Logic 45, 233–238 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Rupert Hölzl
    • 1
  • Dilip Raghavan
    • 2
  • Frank Stephan
    • 2
    • 3
  • Jing Zhang
    • 4
  1. 1.Faculty of Computer ScienceUniversität der Bundeswehr MünchenNeubibergGermany
  2. 2.Department of MathematicsThe National University of SingaporeSingaporeRepublic of Singapore
  3. 3.Department of Computer ScienceNational University of SingaporeSingaporeRepublic of Singapore
  4. 4.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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