Effectiveness for infinite variable words and the Dual Ramsey Theorem
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We examine the Dual Ramsey Theorem and two related combinatorial principles VW(k,l) and OVW(k,l) from the perspectives of reverse mathematics and effective mathematics. We give a statement of the Dual Ramsey Theorem for open colorings in second order arithmetic and formalize work of Carlson and Simpson  to show that this statement implies ACA 0 over RCA 0 . We show that neither VW(2,2) nor OVW(2,2) is provable in WKL 0 . These results give partial answers to questions posed by Friedman and Simpson .
KeywordsPartial Answer Variable Word Effective Mathematic Open Coloring Combinatorial Principle
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- 2.Ershov, Y.L., Goncharov, S.S., Nerode, A., Remmel, J.B.: Ed., ‘‘Handbook of Recursive Model Theory (Two Volumes)’’ Elsevier, Amsterdam 1998Google Scholar
- 3.Friedman, H., Simpson, S.G.: Issues and problems in reverse mathematics, in ‘‘Computability Theory and Its Applications: Current Trends and Open Problems’’ (P.A. Cholak, S. Lempp, M. Lerman & R.A. Shore, Ed.) American Mathematical Society, Providence 2000, 127–144Google Scholar
- 4.Jockusch, C.G.: Jr., Ramsey’s theorem and recursion theory. Journal of Symbolic Logic 37, 268–280 (1972)Google Scholar
- 6.Simpson, S.G.: Recursion theoretic aspects of the Dual Ramsey Theorem, in ‘‘Recursion Theory Week’’ (H.-D. Ebbinghaus, G.H. Müller & G.E. Sacks, Ed.) Springer- Verlag, New York 1985, pp. 356–371Google Scholar
- 7.Simpson, S.G.: ‘‘Subsystems of second order arithmetic’’, Springer-Verlag, New York 1999Google Scholar
- 8.Slaman, T.A.: A note on Dual Ramsey Theorem, 4 pages, unpublished, January 1997Google Scholar
- 9.Soare, R.I.: ‘‘Recursively enumerable sets and degrees’’, Springer-Verlag, New York 1987Google Scholar
- 10.Specker, E.: Ramsey’s theorem does not hold in recursive set theory, in ‘‘Logic Colloquium ‘69 (Proc. Summer School and Colloq., Manchester, 1969)’’, North-Holland, Amsterdam 1971, pp. 439–442Google Scholar