The weakness of being cohesive, thin or free in reverse mathematics
- 30 Downloads
Informally, a mathematical statement is robust if its strength is left unchanged under variations of the statement. In this paper, we investigate the lack of robustness of Ramsey’s theorem and its consequence under the frameworks of reverse mathematics and computable reducibility. To this end, we study the degrees of unsolvability of cohesive sets for different uniformly computable sequence of sets and identify different layers of unsolvability. This analysis enables us to answer some questions of Wang about how typical sets help computing cohesive sets.
We also study the impact of the number of colors in the computable reducibility between coloring statements. In particular, we strengthen the proof by Dzhafarov that cohesiveness does not strongly reduce to stable Ramsey’s theorem for pairs, revealing the combinatorial nature of this nonreducibility and prove that whenever k is greater than l, stable Ramsey’s theorem for n-tuples and k colors is not computably reducible to Ramsey’s theorem for n-tuples and l colors. In this sense, Ramsey’s theorem is not robust with respect to his number of colors over computable reducibility. Finally, we separate the thin set and free set theorem from Ramsey’s theorem for pairs and identify an infinite decreasing hierarchy of thin set theorems in reverse mathematics. This shows that in reverse mathematics, the strength of Ramsey’s theorem is very sensitive to the number of colors in the output set. In particular, it enables us to answer several related questions asked by Cholak, Giusto, Hirst and Jockusch.
Unable to display preview. Download preview PDF.
- L. Bienvenu, L. Patey and P. Shafer, On the logical strengths of partial solutions to mathematical problems, ArXiv e-prints (2014).Google Scholar
- A. Bovykin and A. Weiermann, The strength of infinitary Ramseyan principles can be accessed by their densities, Ann. Pure Appl. Logic, to appear (2005).Google Scholar
- V. Brattka and T. Rakotoniaina, On the Uniform Computational Content of Ramsey’s Theorem, ArXiv e-prints (2015).Google Scholar
- P. A. Cholak, M. Giusto, J. L. Hirst and C. G. Jockusch, Free sets and reverse mathematics, in Reverse mathematics 2001, Lect. Notes Log., Vol. 21, Assoc. Symbol. Logic, La Jolla, CA, 2005, pp. 104–119.Google Scholar
- D. D. Dzhafarov, Cohesive avoidance and arithmetical sets, ArXiv e-prints (2012).Google Scholar
- D. D. Dzhafarov, Strong reductions between combinatorial principles, Journal of Symbolic Logic, to appear.Google Scholar
- H. M. Friedman, Fom:53:free sets and reverse math and fom:54:recursion theory and dynamics, Available at https://www.cs.nyu.edu/pipermail/fom/.Google Scholar
- H. M. Friedman, Boolean relation theory and incompleteness, Lecture Notes in Logic, to appear (2013).Google Scholar
- D. R. Hirschfeldt, Slicing the truth, Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore, Vol. 28, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015, On the computable and reverse mathematics of combinatorial principles, Edited and with a foreword by Chitat Chong, Qi Feng, Theodore A. Slaman, W. Hugh Woodin and Yue Yang.Google Scholar
- D. R. Hirschfeldt and C. G. Jockusch, On notions of computability theoretic reduction between 12 principles, To appear.Google Scholar
- S. M. Kautz, Degrees of random sets, ProQuest LLC, Ann Arbor, MI, 1991, Thesis (Ph.D.)–Cornell University.Google Scholar
- M. Khan and J. S. Miller, Forcing with bushy trees, Preprint (2014).Google Scholar
- L. Patey, Controlling iterated jumps of solutions to combinatorial problems, Computability, to appear.Google Scholar
- L. Patey, Combinatorial weaknesses of Ramseyan principles, In preparation (2015), Available athttp://ludovicpatey.com/media/research/combinatorial-weaknesses-draft.pdf.Google Scholar
- L. Patey, Iterative forcing and hyperimmunity in reverse mathematics, in Evolving computability, Lecture Notes in Comput. Sci., Vol. 9136, Springer, Cham, 2015, pp. 291–301.Google Scholar
- T. Rakotoniaina, The computational strength of Ramseys theorem, 2015, Thesis (Ph.D.)–University of Cape Town.Google Scholar
- J. G. Rosenstein, Linear orderings, Pure and Applied Mathematics, Vol. 98, Academic Press, Inc.[Harcourt Brace Jovanovich, Publishers], New York-London, 1982.Google Scholar
- F. Stephan, Martin-Löf random and PA-complete sets, in Logic Colloquium’ 02, Lect. Notes Log., Vol. 27, Assoc. Symbol. Logic, La Jolla, CA, 2006, pp. 342–348.Google Scholar
- W. Wang, The Definability Strength of Combinatorial Principles, ArXiv e-prints (2014).Google Scholar