No CrossRef data available.
Article contents
FROM REAL ANALYSIS TO THE SORITES PARADOX VIA REVERSE MATHEMATICS
Part of:
General logic
Proof theory and constructive mathematics
Philosophical aspects of logic and foundations
Published online by Cambridge University Press: 12 March 2025
Abstract
This paper presents a reverse mathematical analysis of several forms of the sorites paradox. We first illustrate how traditional discrete formulations are reliant on Hölder’s representation theorem for ordered Archimedean groups. While this is provable in 
$\mathsf {RCA}_0$, we also consider two forms of the sorites which rest on non-constructive principles: the continuous sorites of Weber & Colyvan [35] and a variant we refer to as the covering sorites. We show in the setting of second-order arithmetic that the former depends on the existence of suprema and thus on arithmetical comprehension (
$\mathsf {ACA}_0$) while the latter depends on the Heine–Borel theorem and thus on Weak König’s lemma (
$\mathsf {WKL}_0$). We finally illustrate how recursive counterexamples to these principles provide resolutions to the corresponding paradoxes which can be contrasted with supervaluationist, epistemicist, and constructivist approaches.
Information
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
References
BIBLIOGRAPHY
Apéry, R. (1979). Irrationalité de
$\zeta (2)$
et
$\zeta (3)$
. Astérisque, 61(11–13), 1.Google Scholar
$\zeta (2)$
et
$\zeta (3)$
. Astérisque, 61(11–13), 1.Google ScholarBernays, P. (1935). Sur le platonisme dans les mathématiques. L’Enseignement Mathématique, 34, 52–69.Google Scholar
Bobzien, S., & Rumfitt, I. (2020). Intuitionism and the modal logic of vagueness. Journal of Philosophical Logic, 49(2), 221–248.Google Scholar
Borel, É. (1895). Sur quelques points de la théorie des fonctions. Annales Scientifiques de l’École Normale Supérieure, 12, 9–55.Google Scholar
Cajori, F. (1923). Grafting of the theory of limits on the calculus of Leibniz. American Mathematical Monthly, 30(5), 223–234.Google Scholar
Clark, P. L. (2019). The instructor’s guide to real induction. Mathematics Magazine, 92(2), 136–150.Google Scholar
Coquand, T. (1992). Constructive topology and combinatorics. In Meyers, J. and O’Donnell, M., editors. Constructivity in Computer Science: Summer Symposium, 1991. Springer, pp. 159–164.Google Scholar
Dean, W. (2018). Strict finitism, feasibility, and the sorites. The Review of Symbolic Logic, 11(2), 295–346.Google Scholar
Dean, W., & Walsh, S. (2017). The prehistory of the subsystems of second-order arithmetic. The Review of Symbolic Logic, 10(2), 357–396.Google Scholar
Dean, W. (2019). Computational complexity theory and the philosophy of mathematics. Philosophia Mathematica, 27(3), 381–439.Google Scholar
Dugac, P. (1989). Sur la correspondance de Borel et le théorème de Dirichlet-Heine-Weierstrass-Borel-Schoenflies-Lebesgue. Archives internationales d’histoire des sciences, 39(122), 69–110.Google Scholar
Dzhafarov, D. D. (2019). A note on the reverse mathematics of the sorites. The Review of Symbolic Logic, 12(1), 30–36.Google Scholar
Dzhafarov, D. D., & Mummert, C. (2022). Reverse Mathematics: Problems, Reductions, and Proofs. Berlin: Springer.Google Scholar
Eastaugh, B. (2024). Reverse mathematics. In Zalta, E. N., & Nodelman, U., editors. The Stanford Encyclopedia of Philosophy (Spring 2024 ed.). Metaphysics Research Lab, Stanford University.Google Scholar
Hölder, O. (1901). Die axiome der quantität und die lehre vom mass. Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft, Mathematisch-Physikaliche, 53, 1–64.Google Scholar
Krantz, D. (1968). A survey of measurement theory. In Dantzig, G., & Cwinorr, D., editors, Mathematics of the Decision Sciences, Vol. 2. Providence, Rhode Island: American Mathematical Society, pp. 314–350.Google Scholar
Krantz, D., Luce, R., Suppes, P., & Tversky, A. (1971). Foundations of Measurement: Additive and Polynomial Representations, Vol. 1. New York: Academic Press.Google Scholar
Lebesgue, H. (1904). Leçons sur l’intégration et la Recherche des Fonctions Primitives. Paris: Gauthier-Villars.Google Scholar
Leibniz, G. W. (1920). The Early Mathematical Manuscripts of Leibniz. Chicago: Open Court.Google Scholar
Lorenzen, P. (1971). Differential and Integral: A Constructive Introduction to Classical Analysis. Austin, Texas: University of Texas Press.Google Scholar
Moss, R., & Roberts, G. (1968). A creeping lemma. American Mathematical Monthly, 75(6), 649–652.Google Scholar
Normann, D., & Sanders, S. (Oct. 2020). Open sets in computability theory and reverse mathematics. Journal of Logic and Computation, 30(8), 1639–1679.Google Scholar
Oms, S., & Zardini, E., editors (2019). The Sorites Paradox. Cambridge: Cambridge University Press.Google Scholar
Priest, G. (2006). In Contradiction: A Study of the Transconsistent. Oxford: Clarendon Press.Google Scholar
Rudin, W. (1976). Principles of Mathematical Analysis (third edition). New York, NY: McGraw.Google Scholar
Simpson, S. (2009). Subsystems of Second Order Arithmetic. Cambridge: Cambridge University Press.Google Scholar
Solomon, R. (1998). Reverse mathematics and fully ordered groups. Notre Dame Journal of Formal Logic, 39(2), 157–189.Google Scholar
Troelstra, A., & van Dalen, D. (1988). Constructivism in Mathematics, An Introduction, Vol. 1. Amsterdam: North-Holland.Google Scholar
Truesdell, C. (1960). The Rational Mechanics of Flexible or Elastic Bodies: 1638-1788, Introduction to L. Euler’s Opera Omnia, Vols. 10–11 of 2. Leipzig: Turici.Google Scholar
Weber, Z. (2021). Paradoxes and Inconsistent Mathematics. Cambridge: Cambridge University Press.Google Scholar
Weber, Z., & Colyvan, M. (2010). A topological sorites. The Journal of Philosophy, 107(6), 311–325.Google Scholar
Wright, C. (2019). Intuitionism and the sorites. In Oms, S. and Zardini, E., editors. Oms & Zardini (2019). Cambridge: Cambridge University Press, pp. 95–117.Google Scholar