Notre Dame Journal of Formal Logic

Combinatorial Unprovability Proofs and Their Model-Theoretic Counterparts

Mojtaba Aghaei and Amir Khamseh

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For a function f with domain [X]n, where XN, we say that HX is canonical for f if there is a υn such that for any x0,,xn1 and y0,,yn1 in H, f(x0,,xn1)=f(y0,,yn1) iff xi=yi for all iυ. The canonical Ramsey theorem is the statement that for any nN, if f:[N]nN, then there is an infinite HN canonical for f. This paper is concerned with a model-theoretic study of a finite version of the canonical Ramsey theorem with a largeness condition and also a version of the Kanamori–McAloon principle. As a consequence, we produce new indicators for cuts satisfying PA.

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Notre Dame J. Formal Logic, Volume 55, Number 2 (2014), 231-244.

First available in Project Euclid: 24 April 2014

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Zentralblatt MATH identifier

Primary: 03F30: First-order arithmetic and fragments
Secondary: 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35] 03C62: Models of arithmetic and set theory [See also 03Hxx]

unprovable statements Kanamori–McAloon principle Paris–Harrington principle canonical Ramsey theorem


Aghaei, Mojtaba; Khamseh, Amir. Combinatorial Unprovability Proofs and Their Model-Theoretic Counterparts. Notre Dame J. Formal Logic 55 (2014), no. 2, 231--244. doi:10.1215/00294527-2420654.

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  • [1] Bovykin, A., “Several proofs of PA-unprovability,” pp. 29–43 in Logic and Its Applications, vol. 380 of Contemporary Mathematics, American Mathematical Society, Providence, 2005.
  • [2] Bovykin, A., “Brief introduction to unprovability,” pp. 38–64 in Logic Colloquium 2006, Lecture Notes in Logic, Association for Symbolic Logic, Chicago, 2009.
  • [3] Bovykin, A., and A. Weiermann, “The strength of infinitary Ramseyan statements can be accessed by their densities,” to appear in Annals of Pure and Applied Logic.
  • [4] Carlucci, L., G. Lee, and A. Weiermann, “Sharp thresholds for hypergraph regressive Ramsey numbers,” Journal of Combinatorial Theory, Series A, vol. 118 (2011), pp. 558–85.
  • [5] Carlucci, L., G. Lee, and A. Weiermann, “Classifying the phase transition threshold for regressive Ramsey functions,” preprint, 2006.
  • [6] Carlucci, L., and A. Weiermann, “Classifying the phase transition for canonical Ramsey functions,” preprint, 2010.
  • [7] Erdös, P., and R. Rado, “A combinatorial theorem,” Journal of the London Mathematical Society, vol. 25 (1950), pp. 249–55.
  • [8] Kanamori, A., and K. McAloon, “On Gödel incompleteness and finite combinatorics,” Annals of Pure and Applied Logic, vol. 33 (1987), pp. 23–41.
  • [9] Kaye, R., Models of Peano Arithmetic, vol. 15 of Oxford Logic Guides, Oxford University Press, New York, 1991.
  • [10] Lee, G., “Phase transitions in axiomatic thought,” Ph.D. dissertation, University of Münster, Münster, Germany, 2005.
  • [11] Lefmann, H., and V. Rödl, “On canonical Ramsey numbers for complete graphs versus paths,” Journal of Combinatorial Theory, Series B, vol. 58 (1993), pp. 1–13.
  • [12] Mileti, J. R., “The canonical Ramsey theorem and computability theory,” Transactions of the American Mathematical Society, vol. 360 (2008), pp. 1309–40.
  • [13] Mills, G., “A tree analysis of unprovable combinatorial statements,” pp. 248–311 in Model Theory of Algebra and Arithmetic (Karpacz, Poland, 1978), vol. 834 of Lecture Notes in Mathematics, Springer, Berlin, 1980.
  • [14] Paris, J., and L. Harrington, “A mathematical incompleteness in Peano arithmetic,” pp. 1133–42 in Handbook of Mathematical Logic, edited by J. Barwise, vol. 90 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1977.
  • [15] Ramsey, F. P., “On a problem of formal logic,” Proceedings of the London Mathematical Society, vol. 30 (1930), pp. 264–86.
  • [16] Weiermann, A., “An application of graphical enumeration to PA,” Journal of Symbolic Logic, vol. 68 (2003), pp. 5–16.
  • [17] Weiermann, A., “A classification of rapidly growing Ramsey functions,” Proceedings of the American Mathematical Society, vol. 132 (2004), pp. 553–61.
  • [18] Weiermann, A., “Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results, Annals of Pure and Applied Logic, vol. 136 (2005), pp. 189–218.
  • [19] Weiermann, A., and W. Van Hoof, “Sharp phase transition thresholds for the Paris Harrington Ramsey numbers for a fixed dimension,” Proceedings of the American Mathematical Society, vol. 140 (2012), pp. 2913–27.