Project Euclid

References

• [1] Brattka, V., and T. Rakotoniaina, “On the uniform computational content of Ramsey’s theorem,” Journal of Symbolic Logic, vol. 82 (2017), pp. 1278–316.
  Mathematical Reviews (MathSciNet): MR3743611
  Zentralblatt MATH: 06824519
  Digital Object Identifier: doi:10.1017/jsl.2017.43
  
• [2] Cholak, P. A., C. G. Jockusch, Jr., and T. A. Slaman, “On the strength of Ramsey’s theorem for pairs,” Journal of Symbolic Logic, vol. 66 (2001), pp. 1–55.
  Zentralblatt MATH: 0977.03033
  Digital Object Identifier: doi:10.2307/2694910
  
• [3] Chong, C. T., and L. Yu, Recursion Theory: Computational Aspects of Definability, vol. 8 of De Gruyter Series in Logic and its Applications, De Gruyter, Berlin, 2015.
  Mathematical Reviews (MathSciNet): MR3381097
  
• [4] Dorais, F. G., D. D. Dzhafarov, J. L. Hirst, J. R. Mileti, and P. Shafer, “On uniform relationships between combinatorial problems,” Transactions of the American Mathematical Society, vol. 368 (2016), pp. 1321–59.
  Zentralblatt MATH: 06560459
  Digital Object Identifier: doi:10.1090/tran/6465
  
• [5] Downey, R. G., and D. R. Hirschfeldt, Algorithmic Randomness and Complexity, Springer, New York, 2010.
  Zentralblatt MATH: 1221.68005
  
• [6] Dzhafarov, D. D., and C. G. Jockusch, Jr., “Ramsey’s theorem and cone avoidance,” Journal of Symbolic Logic, vol. 74 (2009), pp. 557–78.
• [7] Dzhafarov, D. D., L. Patey, D. R. Solomon, and L. B. Westrick, “Ramsey’s theorem for singletons and strong computable reducibility,” Proceedings of the American Mathematical Society, vol. 145 (2017), pp. 1343–55.
  Zentralblatt MATH: 06668907
  Digital Object Identifier: doi:10.1090/proc/13315
  
• [8] Galvin, F., and K. Prikry, “Borel sets and Ramsey’s theorem,” Journal of Symbolic Logic, vol. 38 (1973), pp. 193–98.
  Zentralblatt MATH: 0276.04003
  Digital Object Identifier: doi:10.2307/2272055
  
• [9] Gandy, R. O., G. Kreisel, and W. W. Tait, “Set existence,” Bulletin L’Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 8 (1960), pp. 577–82.
• [10] Groszek, M. J., and T. A. Slaman, “Moduli of computation,” conference lecture on Logic, Computability and Randomness, Buenos Aires, Argentina, 2007.
• [11] Hinman, P. G., “A survey of Mučnik and Medvedev degrees,” Bulletin of Symbolic Logic, vol. 18 (2012), pp. 161–229.
  Zentralblatt MATH: 1248.03063
  Digital Object Identifier: doi:10.2178/bsl/1333560805
  Project Euclid: euclid.bsl/1333560805
  
• [12] Hirschfeldt, D. R., and C. G. Jockusch, Jr., “On notions of computability-theoretic reduction between $\Pi^{1}_{2}$ principles,” Journal of Mathematical Logic, vol. 16 (2016), no. 1, art. ID 1650002.
• [13] Hirschfeldt, D. R., C. G. Jockusch, Jr., B. Kjos-Hanssen, S. Lempp, and T. A. Slaman, “The strength of some combinatorial principles related to Ramsey’s theorem for pairs,” pp. 143–61 in Computational Prospects of Infinity, Part II: Presented Talks, vol. 15 of Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, World Scientific, Hackensack, NJ, 2008.
  Zentralblatt MATH: 1167.03009
  
• [14] Jockusch, C. G., Jr., “Ramsey’s theorem and recursion theory,” Journal of Symbolic Logic, vol. 37 (1972), pp. 268–80.
• [15] Jockusch, C. G., Jr., and R. I. Soare, “$\Pi^{0}_{1}$ classes and degrees of theories,” Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.
  Zentralblatt MATH: 0262.02041
  
• [16] Jockusch, C. G., Jr., and R. I. Soare, “Encodability of Kleene’s $O$, Journal of Symbolic Logic, vol. 38 (1973), pp. 437–40.
• [17] Montalbán, A., “Open questions in reverse mathematics,” Bulletin of Symbolic Logic, vol. 17 (2011), pp. 431–54.
  Zentralblatt MATH: 1233.03023
  Digital Object Identifier: doi:10.2178/bsl/1309952320
  Project Euclid: euclid.bsl/1309952320
  
• [18] Moschovakis, Y. N., Descriptive Set Theory, 2nd edition, vol. 155 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2009.
  Zentralblatt MATH: 1172.03026
  
• [19] Nies, A., Computability and Randomness, vol. 51 of Oxford Logic Guides, Oxford University Press, Oxford, 2009.
  Mathematical Reviews (MathSciNet): MR2548883
  Zentralblatt MATH: 1169.03034
  
• [20] Patey, L., “The weakness of being cohesive, thin or free in reverse mathematics,” Israel Journal of Mathematics, vol. 216 (2016), pp. 905–55.
  Zentralblatt MATH: 1368.03018
  Digital Object Identifier: doi:10.1007/s11856-016-1433-3
  
• [21] Sacks, G. E., Degrees of Unsolvability, vol. 55 of Annals of Mathematics Studies, Princeton University Press, Princeton, 1963.
• [22] Sacks, G. E., Higher Recursion Theory, Perspectives in Mathematical Logic, Springer, Berlin, 1990.
  Zentralblatt MATH: 0716.03043
  
• [23] Seetapun, D., and T. A. Slaman, “On the strength of Ramsey’s theorem,” Notre Dame Journal of Formal Logic, vol. 36 (1995), pp. 570–82.
  Zentralblatt MATH: 0843.03034
  Digital Object Identifier: doi:10.1305/ndjfl/1040136917
  Project Euclid: euclid.ndjfl/1040136917
  
• [24] Shore, R. A., “Reverse mathematics: The playground of logic,” Bulletin of Symbolic Logic, vol. 16 (2010), pp. 378–402.
  Zentralblatt MATH: 1218.03006
  Digital Object Identifier: doi:10.2178/bsl/1286284559
  Project Euclid: euclid.bsl/1286284559
  
• [25] Silver, J., “Every analytic set is Ramsey,” Journal of Symbolic Logic, vol. 35 (1970), pp. 60–64.
  Zentralblatt MATH: 0216.01304
  Digital Object Identifier: doi:10.1017/S0022481200092239
  
• [26] Simpson, S. G., Subsystems of Second Order Arithmetic, 2nd edition, Perspectives in Logic, Cambridge University Press, Cambridge, 2009.
  Zentralblatt MATH: 1181.03001
  
• [27] Soare, R. I., “Sets with no subset of higher degree,” Journal of Symbolic Logic, vol. 34 (1969), pp. 53–56.
  Zentralblatt MATH: 0182.01602
  Digital Object Identifier: doi:10.2307/2270981
  
• [28] Solovay, R. M., “Hyperarithmetically encodable sets,” Transactions of the American Mathematical Society, vol. 239 (1978), pp. 99–122.
  Zentralblatt MATH: 0411.03039
  Digital Object Identifier: doi:10.1090/S0002-9947-1978-0491103-7
  
• [29] Spector, C., “Recursive well-orderings,” Journal of Symbolic Logic, vol. 20 (1955), pp. 151–63.
  Zentralblatt MATH: 0067.00303
  Digital Object Identifier: doi:10.2307/2266902