Notre Dame Journal of Formal Logic

The Finitistic Consistency of Heck’s Predicative Fregean System

Luís Cruz-Filipe and Fernando Ferreira

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Abstract

Frege’s theory is inconsistent (Russell’s paradox). However, the predicative version of Frege’s system is consistent. This was proved by Richard Heck in 1996 using a model-theoretic argument. In this paper, we give a finitistic proof of this consistency result. As a consequence, Heck’s predicative theory is rather weak (as was suspected). We also prove the finitistic consistency of the extension of Heck’s theory to Δ11-comprehension and of Heck’s ramified predicative second-order system.

Article information

Source
Notre Dame J. Formal Logic, Volume 56, Number 1 (2015), 61-79.

Dates
First available in Project Euclid: 24 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1427202974

Digital Object Identifier
doi:10.1215/00294527-2835110

Mathematical Reviews number (MathSciNet)
MR3326589

Zentralblatt MATH identifier
1356.03101

Subjects
Primary: 03F25: Relative consistency and interpretations 03F35: Second- and higher-order arithmetic and fragments [See also 03B30]
Secondary: 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35] 03F05: Cut-elimination and normal-form theorems 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}

Keywords
Fregean arithmetic strict predicativity consistency

Citation

Cruz-Filipe, Luís; Ferreira, Fernando. The Finitistic Consistency of Heck’s Predicative Fregean System. Notre Dame J. Formal Logic 56 (2015), no. 1, 61--79. doi:10.1215/00294527-2835110. https://projecteuclid.org/euclid.ndjfl/1427202974


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References

  • [1] Burgess, J. P., “On a consistent subsystem of Frege’s Grundgesetze,” Notre Dame Journal of Formal Logic, vol. 39 (1998), pp. 274–78.
    Mathematical Reviews (MathSciNet): MR1715015
    Digital Object Identifier: doi:10.1305/ndjfl/1039293068
    Project Euclid: euclid.ndjfl/1039293068
  • [2] Burgess, J. P., Fixing Frege, Princeton Monographs in Philosophy, Princeton University Press, Princeton, 2005.
    Mathematical Reviews (MathSciNet): MR2157847
  • [3] Burgess, J. P., and A. P. Hazen, “Predicative logic and formal arithmetic,” Notre Dame Journal of Formal Logic, vol. 39 (1998), pp. 1–17.
    Mathematical Reviews (MathSciNet): MR1671801
    Digital Object Identifier: doi:10.1305/ndjfl/1039293068
    Project Euclid: euclid.ndjfl/1039293018
  • [4] Buss, S. R., “An introduction to proof theory,” pp. 1–78 in Handbook of Proof Theory, edited by S. R. Buss, vol. 137 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1998.
    Mathematical Reviews (MathSciNet): MR1640325
    Digital Object Identifier: doi:10.1016/S0049-237X(98)80016-5
  • [5] Ferreira, F., and G. Ferreira, “Interpretability in Robinson’s $\mathsf{Q}$,” Bulletin of Symbolic Logic, vol. 19 (2013), pp. 289–317.
    Mathematical Reviews (MathSciNet): MR3134895
    Digital Object Identifier: doi:10.2178/bsl.1903010
    Project Euclid: euclid.bsl/1389017495
  • [6] Ferreira, F., and K. F. Wehmeier, “On the consistency of the $\Delta^{1}_{1}$-CA fragment of Frege’s Grundgesetze,” Journal of Philosophical Logic, vol. 31 (2002), pp. 301–11.
    Mathematical Reviews (MathSciNet): MR1923976
    Digital Object Identifier: doi:10.1023/A:1019919403797
  • [7] Frege, G., Basic Laws of Arithmetic, Vols. I, II, a translation of Grundgesetze der Arithmetik, translated and edited by P. A. Ebert and M. Rossberg with a foreword by C. Wright, Oxford University Press, Oxford, 2013.
    Mathematical Reviews (MathSciNet): MR675436
  • [8] Heck, R. G., Jr., “The consistency of predicative fragments of Frege’s Grundgesetze der Arithmetik,” History and Philosophy of Logic, vol. 17 (1996), pp. 209–20.
    Mathematical Reviews (MathSciNet): MR1468610
    Digital Object Identifier: doi:10.1080/01445349608837265
  • [9] Parsons, T., “On the consistency of the first-order portion of Frege’s logical system,” Notre Dame Journal of Formal Logic, vol. 28 (1987), pp. 161–68.
    Mathematical Reviews (MathSciNet): MR871007
    Digital Object Identifier: doi:10.1305/ndjfl/1093636853
    Project Euclid: euclid.ndjfl/1093636853
  • [10] Schwichtenberg, H., “Proof theory: Some applications of cut-elimination,” pp. 867–95 in Handbook of Mathematical Logic, edited by J. Barwise, vol. 90 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1977.
    Mathematical Reviews (MathSciNet): MR491063
    Digital Object Identifier: doi:10.1016/S0049-237X(08)71122-4
  • [11] Tait, W. W., “Finitism,” Journal of Philosophy, vol. 78 (1981), pp. 524–46.
  • [12] Takeuti, G., Proof Theory, 2nd edition, vol. 81 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1987.
    Mathematical Reviews (MathSciNet): MR882549
  • [13] Tarski, A., A. Mostowski, and R. M. Robinson, Undecidable Theories, Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1953.
    Mathematical Reviews (MathSciNet): MR244048
  • [14] Visser, A., “The predicative Fregean hierarchy,” Annals of Pure and Applied Logic, vol. 160 (2009), pp. 129–53.
    Mathematical Reviews (MathSciNet): MR2541469
    Digital Object Identifier: doi:10.1016/j.apal.2009.02.001

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Notre Dame Journal of Formal Logic

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