Hostname: page-component-cb9f654ff-fg9bn Total loading time: 0 Render date: 2025-08-14T17:47:15.786Z Has data issue: false hasContentIssue false
  • English
  • Français

CODING IS HARD

Part of: Proof theory and constructive mathematics General logic

Published online by Cambridge University Press:  04 February 2025

SAM SANDERS
SAM SANDERS*
Affiliation:
DEPARTMENT OF PHILOSOPHY II RUB, BOCHUM, GERMANY
*
E-mail: sasander@me.com
Get access Rights & Permissions [Opens in a new window]

Abstract

A central topic in mathematical logic is the classification of theorems from mathematics in hierarchies according to their logical strength. Ideally, the place of a theorem in a hierarchy does not depend on the representation (aka coding) used. In this paper, we show that the standard representation of compact metric spaces in second-order arithmetic has a profound effect. To this end, we study basic theorems for such spaces like a continuous function has a supremum and a countable set has measure zero. We show that these and similar third-order statements imply at least Feferman’s highly non-constructive projection principle, and even full second-order arithmetic or countable choice in some cases. When formulated with representations (aka codes), the associated second-order theorems are provable in rather weak fragments of second-order arithmetic. Thus, we arrive at the slogan that

$$ \begin{align*} {coding\ compact\ metric\ spaces\ in\ the\ language\ of\ second\text{-}order\ arithmetic\ can\ be\ as\ hard }\\ {as\ second\text{-}order\ arithmetic\ or\ countable\ choice}. \end{align*} $$

We believe every mathematician should be aware of this slogan, as central foundational topics in mathematics make use of the standard second-order representation of compact metric spaces. In the process of collecting evidence for the above slogan, we establish a number of equivalences involving Feferman’s projection principle and countable choice. We also study generalisations to fourth-order arithmetic and beyond with similar-but-stronger results.

Keywords

higher-order arithmeticsecond-order arithmeticcodingrepresentationsreal analysisReverse Mathematics

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics) 03F35: Second- and higher-order arithmetic and fragments

Information

Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 29
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

Andreasson, N., Evgrafov, A., and Patriksson, M., An Introduction to Continuous Optimization: Foundations and Fundamental Algorithms, Dover Publications, 2020.Google Scholar
Assad, N. A. and Kirk, W. A., Fixed point theorems for set-valued mappings of contractive type . Pacific Journal of Mathematics, vol. 43 (1972), pp. 553–562.Google Scholar
Avigad, J. and Feferman, S., Gödel’s functional (“Dialectica”) interpretation , Handbook of Proof Theory, Studies in Logic and the Foundations of Mathematics, 137, 1998, pp. 337–405.Google Scholar
Avigad, J., Dean, E. T., and Rute, J., Algorithmic randomness, reverse mathematics, and the dominated convergence theorem . Annals of Pure and Applied Logic, vol. 163 (2012), no. 12, pp. 1854–1864.Google Scholar
Berger, J. and Schuster, P., Classifying Dini’s theorem . Notre Dame Journal of Formal Logic, vol. 47 (2006), no. 2, pp. 253–262.Google Scholar
Brattka, V., Gherardi, G., and Pauly, A., Weihrauch complexity in computable analysis , Handbook of Computability and Complexity in Analysis, Theory and Applications of Computability, Springer, Cham, pp. 367–417.Google Scholar
Brattka, V. and Gherardi, G., Effective choice and boundedness principles in computable analysis . Bulletin of Symbolic Logic, vol. 17 (2011), no. 1, pp. 73–117.Google Scholar
Brattka, V. and Presser, G., Computability on subsets of metric spaces . Theoretical Computer Science, vol. 305 (2003), nos. 1–3, pp. 43–76. Topology in computer science (Schloß Dagstuhl, 2000).Google Scholar
Brown, D. K., Notions of compactness in weak subsystems of second order arithmetic , Reverse Mathematics, Lecture Notes in Logic, 21, Association for Symbolic Logic, 2005, 2001, pp. 47–66.Google Scholar
Buchholz, W., Feferman, S., Pohlers, W., and Sieg, W., Iterated Inductive Definitions and Subsystems of Analysis, LNM 897, Springer, 1981.Google Scholar
Bukovsky, L., The Structure of the Real Line, vol. 71, Birkhäuser/Springer, 2011.Google Scholar
Carothers, N. L., Real Analysis, Cambridge University Press, 2000.Google Scholar
Cooper, S. B., Computability Theory, Chapman & Hall/CRC, Boca Raton, 2004.Google Scholar
Dorais, F. G., Classical consequences of continuous choice principles from intuitionistic analysis . Notre Dame Journal of Formal Logic, vol. 55 (2014), no. 1, pp. 25–39.Google Scholar
Dorais, F. G., Dzhafarov, D. D., Hirst, J. L., Mileti, J. R., and Shafer, P., On uniform relationships between combinatorial problems . Transactions of the American Mathematical Society, vol. 368 (2016), no. 2, pp. 1321–1359.Google Scholar
Dzhafarov, D. D. and Mummert, C., Reverse Mathematics: Problems, Reductions, and Proofs, Springer, Cham, 2022.Google Scholar
Engelking, R., General Topology, second ed., Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989.Google Scholar
Ershov, Y. L., Goncharov, S. S., Nerode, A., Remmel, J. B., and Marek, V. W. (eds.), Handbook of Recursive Mathematics: Recursive Model Theory, vol. 1, Studies in Logic and the Foundations of Mathematics, 138, North-Holland, Amsterdam, 1998.Google Scholar
Ershov, Y. L., Goncharov, S. S., Nerode, A., Remmel, J. B., and Marek, V. W. (eds.), Handbook of Recursive Mathematics, vol. 2, Studies in Logic and the Foundations of Mathematics, 139, North-Holland, Amsterdam, 1998. Recursive algebra, analysis and combinatorics.Google Scholar
Feferman, S., How a little bit goes a long way: Predicative foundations of analysis, 2013. Unpublished notes from 1977–1981 with updated introduction, https://math. stanford.edu/~feferman/papers/pfa(1).pdf.Google Scholar
Friedman, H., 897: Remarks on reverse mathematics/4, FOM mailing list, Oct. 2021. https://cs.nyu.edu/pipermail/fom/2021-October/022896.html.Google Scholar
Friedman, H., 901: Remarks on reverse mathematics 6, FOM mailing list, Oct. 2021. https://cs.nyu.edu/pipermail/fom/2021-October/022903.html.Google Scholar
Frink, O. Jr., Jordan measure and Riemann integration . Annals of Mathematics, vol. 34 (1933), no. 3.Google Scholar
Fujiwara, M. and Yokoyama, K., A Note on the Sequential Version of Π1 Statements, Lecture Notes in Computer Science, 7921, Springer, Heidelberg, 2013, pp. 171–180.Google Scholar
Fujiwara, M., Higuchi, K., and Kihara, T., On the strength of marriage theorems and uniformity . Mathematical Logic Quarterly, vol. 60 (2014), no. 3.Google Scholar
Herrlich, H., Axiom of Choice, Lecture Notes in Mathematics, 1876, Springer, 2006.Google Scholar
Hilbert, D. and Bernays, P., Grundlagen der Mathematik. II, Zweite Auflage, Die Grundlehren der Mathematischen Wissenschaften, 50, Springer, 1970.Google Scholar
Hirschfeldt, D. R., Slicing the Truth, Lecture Notes Series, Institute for Mathematical Sciences, 28, National University of Singapore, World Scientific Publishing, 2015.Google Scholar
Hirst, J. L., Representations of reals in reverse mathematics . Bulletin of the Polish Academy of Sciences. Mathematics, vol. 55 (2007), no. 4, pp. 303–316.Google Scholar
Hirst, J. L. and Mummert, C., Reverse mathematics and uniformity in proofs without excluded middle . Notre Dame Journal of Formal Logic, vol. 52 (2011), no. 2, pp. 149–162.Google Scholar
Hunter, J., Higher-order reverse topology, Ph.D. thesis, The University of Wisconsin—Madison, 2008.Google Scholar
Jeyakumar, V. and Rubinov, A. (eds.), Continuous Optimization, Applied Optimization, 99, Springer, New York, 2005.Google Scholar
Jordan, C., Remarques sur les int´egrales d´efinies . Journal de Mathématiques Pures et Appliquées, vol. 8 (1892), pp. 69–99.Google Scholar
Kohlenbach, U., Foundational and mathematical uses of higher types , Reflections on the Foundations of Mathematics, Lecture Notes in Logic, 15, ASL, 2002, pp. 92–116.Google Scholar
Kohlenbach, U., On uniform weak K¨onig’s lemma . Annals of Pure and Applied Logic, vol. 114 (2002), nos. 1–3, pp. 103–116. Commemorative Symposium Dedicated to Anne S. Troelstra (Noordwijkerhout, 1999).Google Scholar
Kohlenbach, U., Higher order reverse mathematics , Reverse Mathematics 2001, Lecture Notes in Logic, 21, ASL, 2005, pp. 281–295.Google Scholar
Montalbán, A., Open questions in reverse mathematics . Bulletin of Symbolic Logic, vol. 17 (2011), no. 3, pp. 431–454.Google Scholar
Munkres, J. R., Topology, first ed., Prentice-Hall, 2000.Google Scholar
Nadler, S. B. Jr., Multi-valued contraction mappings . Pacific Journal of Mathematics, vol. 30 (1969), pp. 475–488.Google Scholar
Nadler, S. B. Jr., An embedding theorem for certain spaces with an equidistant property . Proceedings of the American Mathematical Society, vol. 59 (1976), no. 1, pp. 179–183.Google Scholar
Neeman, I., Necessary use of Σ1 induction in a reversal . Journal of Symbolic Logic, vol. 76 (2011), no. 2, pp. 561–574.Google Scholar
Nies, A., Triplett, M. A., and Yokoyama, K., The reverse mathematics of theorems of Jordan and Lebesgue . The Journal of Symbolic Logic, (2021), pp. 1–18.Google Scholar
Normann, D. and Sanders, S., Open sets in reverse mathematics and computability theory . Journal of Logic and Computation, vol. 30 (2020), no. 8, p. 40.Google Scholar
Normann, D. and Sanders, S., Pincherle’s theorem in reverse mathematics and computability theory . Annals of Pure and Applied Logic, vol. 171 (2020), no. 5, p. 102788, 41.Google Scholar
Normann, D. and Sanders, S., The biggest five of reverse mathematics . Journal for Mathematical Logic, (2023), p. 56, https://doi.org/10.1142/S0219061324500077.Google Scholar
Rudin, W., Principles of Mathematical Analysis, third ed., International Series in Pure and Applied Mathematics, McGraw-Hill, 1976.Google Scholar
Rudin, W., Real and Complex Analysis, third ed., McGraw-Hill, 1987.Google Scholar
Sakamoto, N. and Yamazaki, T., Uniform versions of some axioms of second order arithmetic . Mathematical Logic Quarterly, vol. 50 (2004), no. 6, pp. 587–593.Google Scholar
Sanders, S., Some contributions to higher-order reverse mathematics, Habilitationsschrift, TU Darmstadt, 2022.Google Scholar
Sanders, S., Big in reverse mathematics: Measure and category . Journal of Symbolic Logic, (2023), p. 44. https://doi.org/10.1017/jsl.2023.65.Google Scholar
Sanders, S., Sometimes tame, sometimes wild: Weak continuity . Bulletin of the London Mathematical Society, 2025, p. 15, to appear. arxiv: https://arxiv.org/abs/2405.06420 Google Scholar
Simpson, S. G., Subsystems of Second Order Arithmetic, second ed., Perspectives in Logic, Cambridge University Press, 2009.Google Scholar
Soare, R. I., Recursively Enumerable Sets and Degrees, Perspectives in Mathematical Logic, Springer, 1987.Google Scholar
Stein, E. M., Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Mathematics Studies, 63, Princeton University Press, University of Tokyo Press, Princeton, NJ, Tokyo, 1970.Google Scholar
Stein, E. M. (ed.), Beijing Lectures in Harmonic Analysis, Annals of Mathematics Studies, 112, Princeton University Press, Princeton, 1986.Google Scholar
Stein, E. M., Problems in harmonic analysis related to curvature and oscillatory integrals , Proceedings of the International Congress of Mathematicians, vols. 1, 2, American Mathematical Society, Providence, RI, 1987, 1986, pp. 196–221.Google Scholar
Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993.Google Scholar
Stillwell, J., Reverse Mathematics, Proofs from the Inside Out, Princeton University Press, 2018.Google Scholar
Tao, T., An Epsilon of Room, I: Real Analysis, Graduate Studies in Mathematics, 117, American Mathematical Society, Providence, 2010.Google Scholar
Tao, T., Compactness and Contradiction, American Mathematical Society, Providence, 2013.Google Scholar
Tao, T., Analysis, vol. 2, third ed., Texts and Readings in Mathematics, 38, Springer, 2014.Google Scholar
Väisälä, J., Gromov hyperbolic spaces . Expositiones Mathematicae, vol. 23 (2005), no. 3, pp. 187–231.Google Scholar
Weihrauch, K., Computable Analysis: An Introduction, Springer-Verlag, Berlin, 2000.Google Scholar
Wilder, R. L., Evolution of the topological concept of “connected” . The American Mathematical Monthly, vol. 85 (1978), no. 9, pp. 720–726.Google Scholar
Yokoyama, K., Standard and non-standard analysis in second order arithmetic, Ph.D. thesis, Tohoku University, Sendai, 2009, p. 2007.Google Scholar