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On the strength of Ramsey's theorem for pairs

Published online by Cambridge University Press:  12 March 2014

Peter A. Cholak ,
Carl G. Jockusch  and
Theodore A. Slaman
Peter A. Cholak
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556-5683, USA, E-mail: [email protected]
Carl G. Jockusch
Affiliation:
Department of Mathematics, University of Illinois, at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801-2975, USA, E-mail: [email protected]
Theodore A. Slaman
Affiliation:
Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA, E-mail: [email protected]
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Abstract

We study the proof–theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RTkn denote Ramsey's theorem for k–colorings of n–element sets, and let RT<∞n denote (∀k)RTkn. Our main result on computability is: For any n ≥ 2 and any computable (recursive) k–coloring of the n–element sets of natural numbers, there is an infinite homogeneous set X with X″ ≤T 0(n). Let IΣn and BΣn denote the Σn induction and bounding schemes, respectively. Adapting the case n = 2 of the above result (where X is low2) to models of arithmetic enables us to show that RCA0 + IΣ2 + RT22 is conservative over RCA0 + IΣ2 for Π11 statements and that RCA0 + IΣ3 + RT<∞2 is Π11-conservative over RCA0 + IΣ3. It follows that RCA0 + RT22 does not imply BΣ3. In contrast, J. Hirst showed that RCA0 + RT<∞2 does imply BΣ3, and we include a proof of a slightly strengthened version of this result. It follows that RT<∞2 is strictly stronger than RT22 over RCA0.

Keywords

Ramsey's Theoremconservationreverse mathematicsrecursion theorycomputability theory
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 1 , March 2001 , pp. 1 - 55
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

Avigad, Jeremy, [1996], Formalizing forcing arguments in subsystems of second-order arithmetic, Annals of Pure and Applied Logic, vol. 82, no. 2, pp. 165–191.CrossRefGoogle Scholar
Downey, R., Hirschfeldt, Denis R., Lempp, S., and Solomon, R., [to appear], A Δ20 set withno infinite low set in either it or its complement, this Journal.Google Scholar
Fairtlough, Matt and Wainer, Stanley S. [1998], Hierarchies of provably recursive functions, Handbook of proof theory, Studies in Logic and the Foundations of Mathematics, vol. 137, North-Holland, Amsterdam.CrossRefGoogle Scholar
Friedman, Harvey, [1975], Some systems of second order arithmetic and their use, Proceedings of the international congress of mathematicians (Vancouver, B.C., 1974), vol. 1, Canadian Mathematical Congress, Montreal, Quebec, pp. 235–242.Google Scholar
Friedman, Harvey, [1976], Systems of second order arithmetic with restricted induction, I, II (abstracts), this Journal, vol. 41, pp. 557–559.Google Scholar
Graham, Ronald L., Rothschild, Bruce L., and Spencer, Joel H. [1990], Ramsey theory, second ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, A Wiley-Interscience Publication.Google Scholar
Hájek, Petr, [1993], Interpretability and fragments of arithmetic, Arithmetic, proof theory, and computational complexity (Prague, 1991), Oxford Univ. Press, New York, pp. 185–196.Google Scholar
Hájek, Petr and Pudlák, Pavel, [1993], Metamathematics of first-order arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Herrmann, E., [to appear], Infinite chains and antichains in recursive partial orderings, this Journal.Google Scholar
Hirst, Jeffry L., [1987], Combinatorics in Subsystems of Second Order Arithmetic, Ph.D. thesis, The Pennsylvania State University.Google Scholar
Hummel, Tamara Lakins, [1994], Effective versions of Ramsey's theorem: Avoiding the cone above 0′, this Journal, vol. 59, pp. 1301–1325.Google Scholar
Hummel, Tamara Lakins and Jockusch, Carl G. Jr. [1999], Generalized cohesiveness, this Journal, vol. 64, pp. 489–516.Google Scholar
Jockusch, Carl G. Jr. [1968], Semirecursive sets and positive reducibility, Transactions of the American Mathematical Society, vol. 131, pp. 420–436.CrossRefGoogle Scholar
Jockusch, Carl G. Jr. [1972], Ramsey's theorem and recursion theory, this Journal, vol. 37, pp. 268–280.Google Scholar
Jockusch, Carl G. Jr. [1973], Upward closure and cohesive degrees, Israel Journal of Mathematics, vol. 15, pp. 332–335.CrossRefGoogle Scholar
Jockusch, Carl G. Jr., and Soare, Robert I., [1972], Π10 classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173, pp. 33–56.Google Scholar
Jockusch, Carl G. Jr. and Stephan, Frank [1993], A cohesive set which is not high, Mathematical Logic Quarterly, vol. 39, pp. 515–530.CrossRefGoogle Scholar
Jockusch, Carl G. Jr. and Stephan, Frank [1997], Correction to “A cohesive set which isnothigh”, Mathematical Logic Quarterly, vol. 43, p. 569.CrossRefGoogle Scholar
Kaye, Richard, [1991], Models of Peano arithmetic, Oxford Logic Guides, vol. 15, The Clarendon Press Oxford University Press, New York, Oxford Science Publications.CrossRefGoogle Scholar
Mints, G. E. [1973], Quantifer-free and one quantifier systems, Journal of Soviet Mathematics, vol. 1, pp. 71–84.CrossRefGoogle Scholar
Mytilinaios, Michael E. and Slaman, Theodore A. [1996], On a question of Brown and Simpson, Computability, enumerability, unsolvability, directions in recursion theory (Cooper, S. B., Slaman, T. A., and Wainer, S. S., editors), Cambridge University Press, Cambridge, UK, pp. 205–218.CrossRefGoogle Scholar
Odifreddi, Piergiorgio, [1989], Classical recursion theory (volume I), North-Holland Publishing Co., Amsterdam.Google Scholar
Paris, J. B., [1980], A hierarchy of cuts in models of arithmetic, Model theory of algebra and arithmetic (Proceedigs, Karpacz, Poland 1979), Lecture Notes in Mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, New York, pp. 312–337.CrossRefGoogle Scholar
Parsons, Charles, [1970], On a number theoretic choice schema and its relation to induction, Intuitionism and proof theory (Proceedings of a Conference, Buffalo, N.Y., 1968), North-Holland, Amsterdam, pp. 459–473.CrossRefGoogle Scholar
Posner, David B. and Robinson, Robert W., [1981], Degrees joining to 0′, this Journal, vol. 46, no. 4, pp. 714–722.Google Scholar
Ramsey, F. P., [1930], On a problem informal logic, Proceedings of London Mathematical Society (3), vol. 30, pp. 264–286.CrossRefGoogle Scholar
Scott, Dana, [1962], Algebras of sets binumerable in complete extensions of arithmetic, Recursive function theory (Providence, R.I.), Proceedings of Symposia in Pure Mathematics, no. 5, American Mathematical Society, pp. 117–121.CrossRefGoogle Scholar
Seetapun, David and Slaman, Theodore A., [1995], On the strength of Ramsey's theorem, Notre Dame Journal of Formal Logic, vol. 36, no. 4, pp. 570–582, Special Issue: Models of arithmetic.CrossRefGoogle Scholar
Simpson, Stephen G., [1977], Degrees of unsolvability: a survey of results, Handbook of mathematical logic (Barwise, Jon, editor), North-Holland, Amsterdam, pp. 1133–1142.Google Scholar
Simpson, Stephen G., [1999], Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, XIV + 445 pages.CrossRefGoogle Scholar
Soare, Robert I., [1987], Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Omega Series, Springer-Verlag, Heidelberg.CrossRefGoogle Scholar
Specker, E., [1971], Ramsey's Theorem does not hold in recursive set theory, Logic Colloquium; 1969 Manchester, pp. 439–442.Google Scholar
Spector, C., [1956], On the degrees of recursive unsolvability, Annals of Mathematics (2), vol. 64, pp. 581–592.CrossRefGoogle Scholar
Takeuti, Gaisi, [1987], Proof theory, second ed., Studies in Logic and the Foundations of Mathematics, vol. 81, North-Holland Publishing Co., Amsterdam, With an appendix containing contributions by Georg Kreisel, Wolfram Pohlers, Stephen G. Simpson and Solomon Feferman.Google Scholar