Skip to main content



We prove that the statement “there is a k such that for every f there is a k-bounded diagonally nonrecursive function relative to f” does not imply weak König’s lemma over ${\rm{RC}}{{\rm{A}}_0} + {\rm{B\Sigma }}_2^0$ . This answers a question posed by Simpson. A recursion-theoretic consequence is that the classic fact that every k-bounded diagonally nonrecursive function computes a 2-bounded diagonally nonrecursive function may fail in the absence of ${\rm{I\Sigma }}_2^0$ .

Hide All
[1]Ambos-Spies, Klaus, Kjos-Hanssen, Bjørn, Lempp, Steffen, and Slaman, Theodore A., Comparing DNR and WWKL, this Journal, vol. 69 (2004), no. 4, pp. 10891104.
[2]Austen, Jane, Pride and Prejudice, Egerton, T., Whitehall, 1813.
[3]Bean, Dwight R., Effective coloration, this Journal, vol. 41 (1976), no. 2, pp. 469480.
[4]Chong, C. T., Li, Wei, and Yang, Yue, Nonstandard models in recursion theory and reverse mathematics. Bulletin of Symbolic Logic, vol. 20 (2014), no. 2, pp. 170200.
[5]Chong, C. T. and Mourad, K. J., Σndefinable sets without Σninduction. Transactions of the American Mathematical Society, vol. 334 (1992), no. 1, pp. 349363.
[6]Chong, C. T., Qian, Lei, Slaman, Theodore A., and Yang, Yue, Σ2induction and infinite injury priority arguments, part III: Prompt sets, minimal pairs and Shoenfield’s conjecture. Israel Journal of Mathematics, vol. 121 (2001), pp. 128.
[7]Chong, C. T., Slaman, Theodore A., and Yang, Yue, ${\rm{\Pi }}_1^1$- conservation of combinatorial principles weaker than Ramsey’s theorem for pairs. Advances in Mathematics, vol. 230 (2012), no. 3, pp. 10601077.
[8]Chong, C. T., Slaman, Theodore A., and Yang, Yue, The metamathematics of stable Ramsey’s theorem for pairs. Journal of the American Mathematical Society, vol. 27 (2014), no. 3, pp. 863892.
[9]Chong, C. T., Slaman, Theodore A., and Yang, Yue, The inductive strength of Ramsey’s theorem for pairs, 2014, preprint.
[10]Chong, C. T., and Yang, Yue, Σ2induction and infinite injury priority arguments, Part II Tame Σ2coding and the jump operator. Annals of Pure and Applied Logic, vol. 87 (1997), no. 2, pp. 103116.
[11]Chong, C. T., and Yang, Yue, Σ2induction and infinite injury priority argument, Part I: Maximal sets and the jump operator, this Journal, vol. 63 (1998), no. 3, pp. 797814.
[12]Chubb, Jennifer, Hirst, Jeffry L., and McNicholl, Timothy H., Reverse mathematics, computability, and partitions of trees, this Journal, vol. 74 (2009), no. 1, pp. 201215.
[13]Corduan, Jared, Groszek, Marcia J., and Mileti, Joseph R., Reverse mathematics and Ramsey’s property for trees, this Journal, vol. 75 (2010), no. 3, pp. 945954.
[14]Friedman, Harvey, Some systems of second order arithmetic and their use, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, 1975, pp. 235242.
[15]Gasarch, William and Hirst, Jeffry L., Reverse mathematics and recursive graph theory. Mathematical Logic Quarterly. vol. 44 (1998), no. 4, pp. 465473.
[16]Groszek, Marcia J., Mytilinaios, Michael E., and Slaman, Theodore A., The Sacks density theorem and Σ2-bounding, this Journal, vol. 61 (1996), no. 2, pp. 450467.
[17]Groszek, Marcia J. and Slaman, Theodore A., On Turing reducibility, 1994, preprint.
[18]Hájek, Petr, Interpretability and fragments of arithmetic, Arithmetic, proof theory, and computational complexity, 1993, pp. 185196.
[19]Hájek, Petr, and Pudlák, Pavel, Metamathematics of First-Order Arithmetic. Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998.
[20]Hirst, Jeffry L., Marriage theorems and reverse mathematics, Logic and computation (Pittsburgh, PA, 1987), 1990, pp. 181196.
[21]Jockusch, Carl G. Jr., Degrees of functions with no fixed points. Logic, Methodology and Philosophy of Science VIII, (1989), pp. 191201.
[22]Jockusch, Carl G. Jr., and Soare, Robert I., ${\rm{\Pi }}_1^0$classes and degrees of theories. Transactions of the American Mathematical Society, vol. 173 (1972), pp. 3356.
[23]Mytilinaios, Michael E., Finite injury and Σ1-induction, this Journal, vol. 54 (1989), no. 1, pp. 3849.
[24]Schmerl, James H., Graph coloring and reverse mathematics. Mathematical Logic Quarterly, vol. 46 (2000), no. 4, pp. 543548.
[25]Schmerl, James H., Reverse mathematics and graph coloring: eliminating diagonalization, Reverse mathematics 2001, 2005, pp. 331348.
[26]Simpson, Stephen G., Why the recursion theorists ought to thank me, 2001.
[27]Simpson, Stephen G., Subsystems of Second Order Arithmetic, Cambridge University Press, Cambridge, 2009.
[28]Slaman, Theodore A., Σn-Bounding and Δn-Induction. Proceedings of the American Mathematical Society, vol. 132 (2004), no. 8, pp. 24492456.
[29]Slaman, Theodore A. and Hugh Woodin, W., Σ1-Collection and the finite injury priority method. Mathematical logic and applications, 1989, 178188.
[30]Soare, Robert I., Recursively Enumerable Sets and Degrees, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987.
[31]Yu, Xiaokang and Simpson, Stephen G., Measure theory and weak König’s lemma. Archive for Mathematical Logic, vol. 30 (1990), no. 3, pp. 171180.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 17 *
Loading metrics...

Abstract views

Total abstract views: 123 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd May 2018. This data will be updated every 24 hours.