Hostname: page-component-6766d58669-6mz5d Total loading time: 0 Render date: 2026-05-17T17:09:58.879Z Has data issue: false hasContentIssue false
  • English
  • Français

ON ROBUST THEOREMS DUE TO BOLZANO, WEIERSTRASS, JORDAN, AND CANTOR

Part of: General logic Proof theory and constructive mathematics Computability and recursion theory

Published online by Cambridge University Press:  03 October 2022

DAG NORMANN  and
SAM SANDERS
DAG NORMANN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF OSLO OSLO, NORWAY E-mail: dnormann@math.uio.no
SAM SANDERS*
Affiliation:
INSTITUTE FOR PHILOSOPHY II RUB BOCHUM BOCHUM, GERMANY
*
E-mail: sasander@me.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics where the aim is to identify the minimal axioms needed to prove a given theorem from ordinary, i.e., non-set theoretic, mathematics. This program has unveiled surprising regularities: the minimal axioms are very often equivalent to the theorem over the base theory, a weak system of ‘computable mathematics’, while most theorems are either provable in this base theory, or equivalent to one of only four logical systems. The latter plus the base theory are called the ‘Big Five’ and the associated equivalences are robust following Montalbán, i.e., stable under small variations of the theorems at hand. Working in Kohlenbach’s higher-order RM, we obtain two new and long series of equivalences based on theorems due to Bolzano, Weierstrass, Jordan, and Cantor; these equivalences are extremely robust and have no counterpart among the Big Five systems. Thus, higher-order RM is much richer than its second-order cousin, boasting at least two extra ‘Big’ systems.

Keywords

countable setsBolzano–Weierstrass theoremReverse Mathematicshigher-order computabilityKleene’s S1–S9bounded variationregulated functions

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics)
Secondary: 03F35: Second- and higher-order arithmetic and fragments 03D55: Hierarchies 03D30: Other degrees and reducibilities

Information

Type
Article
Information
The Journal of Symbolic Logic , Volume 89 , Issue 3 , September 2024 , pp. 1077 - 1127
Check for updates
Copyright
© The Author(s), 2022. 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

Appell, J., Banaś, J., and Merentes, N., Bounded Variation and Around , De Gruyter Series in Nonlinear Analysis and Applications, vol. 17, De Gruyter, Berlin, 2014.Google Scholar
Avigad, J. and Feferman, S., Gödel’s functional ‘Dialectica’ interpretation , Handbook of Proof Theory (Samuel R. Buss, editor), Studies in Logic and the Foundations of Mathematics, vol. 137, Elsevier, Amsterdam, 1998, pp. 337–405.CrossRefGoogle Scholar
Borel, E., Leçons Sur la théorie Des Fonctions , Gauthier-Villars, Paris, 1898.Google Scholar
Bourbaki, N., Élements de mathématique, Livre IV: Fonctions d’une Variable réelle (Théorie élémentaire) , Actualités Scientifiques et Industrielles, vol. 1132, Hermann et Cie, Paris, 1951 (in French).Google Scholar
Bridges, D., A constructive look at functions of bounded variation . The Bulletin of the London Mathematical Society , vol. 32 (2000), no. 3, pp. 316–324.CrossRefGoogle Scholar
Bridges, D. and Mahalanobis, A., Bounded variation implies regulated: A constructive proof, this Journal, vol. 66 (2001), no. 4, pp. 1695–1700.Google Scholar
Bridges, D., Richman, F., and Schuster, P., A weak countable choice principle . Proceedings of the American Mathematical Society , vol. 128 (2000), no. 9, pp. 2749–2752.CrossRefGoogle Scholar
Buchholz, W., Feferman, S., Pohlers, W., and Sieg, W., Iterated Inductive Definitions and Subsystems of Analysis , Lecture Notes in Mathematics, vol. 897, Springer, Berlin, Heidelberg, 1981.Google Scholar
Cantor, G., Ueber eine Eigenschaft des Inbegriffs Aller reellen algebraischen Zahlen . Journal fur die Reine und Angewandte Mathematik , vol. 77 (1874), pp. 258–262.Google Scholar
Cantor, G., Ein Beitrag zur Mannigfaltigkeitslehre . Journal für die reine und angewandte Mathematik , vol. 84 (1878), pp. 242–258.Google Scholar
Cantor, G., Ueber unendliche, lineare Punktmannichfaltigkeiten . Mathematische Annalen , vol. 21 (1883), no. 4, pp. 545–591.CrossRefGoogle Scholar
Cantor, G., Zur Lehre Vom Transfiniten: Gesammelte Abhandlungen Aus der Zeitschrift für Philosophie Und Philosophische Kritik, Vom Jahre 1887 , Pfeffer, Halle, 1890.Google Scholar
Cantor, G., Beiträge zur Begründung der transfiniten Mengenlehre . Mathematische Annalen , vol. 46 (1895), pp. 481–512.CrossRefGoogle Scholar
Cantor, G., Gesammelte Abhandlungen Mathematischen Und Philosophischen Inhalts , Springer, Berlin, Heidelberg, 1980, p. vii + 489. Reprint of the 1932 original.Google Scholar
Cenzer, D. and Remmel, J. B., Proof-theoretic strength of the stable marriage theorem and other problems , Reverse Mathematics 2001 (Stephen G. Simpson, editor), Lecture Notes in Logic, vol. 21, Association for Symbolic Logic, Wellesley, 2005, pp. 67–103.Google Scholar
Clote, P., The metamathematics of scattered linear orderings . Archive for Mathematical Logic , vol. 29 (1989), no. 1, pp. 9–20.CrossRefGoogle Scholar
Coolidge, J. L., The lengths of curves . The American Mathematical Monthly , vol. 60 (1953), pp. 89–93.CrossRefGoogle Scholar
Cousin, P., Sur les fonctions de $n$ variables complexes . Acta Mathematica (A. Kino, John Myhill, R.E. Vesley, editors), vol. 19 (1895), pp. 1–61.CrossRefGoogle Scholar
Dirichlet, L. P. G., Über Die Darstellung Ganz willkürlicher Funktionen Durch Sinus- Und Cosinusreihen , Repertorium der Physik, bd. 1 (Dove, H. W. und Moser, L., herausg), 1837.Google Scholar
Dirksen, E., Ueber die Anwendung der analysis auf die rectification der Curven . Akademie der Wissenschaften zu Berlin (1833), pp. 123–168.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.CrossRefGoogle 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.CrossRefGoogle Scholar
Dunham, J. M. C., Des méthodes Dans les Sciences de Raisonnement. Application des méthodes générales à la Science Des Nombres et à la Science de l’étendue , vol. II, Gauthier-Villars, Paris, 1866.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. Available at https://math.stanford.edu/~feferman/papers/pfa(1).pdf.Google Scholar
Fraïssé, R., Theory of Relations , Studies in Logic and the Foundations of Mathematics, vol. 145, North-Holland, Amsterdam, 2000. With an appendix by Norbert Sauer.Google Scholar
Friedman, H., Some systems of second order arithmetic and their use , Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974) (Ralph D. James, editor), vol. 1, Canadian Mathematical Congress, Vancouver, 1975, pp. 235–242.Google Scholar
Friedman, H., Systems of second order arithmetic with restricted induction, I & II (abstracts), this Journal, vol. 41 (1976), pp. 557–559.Google Scholar
Friedman, H., Remarks on reverse mathematics/1, FOM mailing list (September 21, 2021). Available at https://cs.nyu.edu/pipermail/fom/2021-September/022875.html.Google Scholar
Fujiwara, M., Higuchi, K., and Kihara, T., On the strength of marriage theorems and uniformity . Mathematics Logic Quarterly , vol. 60 (2014), no. 3, pp. 136–153.CrossRefGoogle Scholar
Fujiwara, M. and Yokoyama, K., A note on the sequential version of  ${\Pi}_2^1$ statements, The Nature of Computation (Paola Bonizzoni, Vasco Brattka, Benedikt Löwe, editors), Lecture Notes in Computer Science, vol. 7921, Springer, Heidelberg, 2013, pp. 171–180.Google Scholar
Greenberg, N., Miller, J. S., and Nies, A., Highness properties close to PA-completeness . Israel Journal of Mathematics , vol. 244 (2021), pp. 419–465.CrossRefGoogle Scholar
Herrlich, H., Axiom of Choice , Lecture Notes in Mathematics, vol. 1876, Springer, Berlin, 2006.Google Scholar
Heyting, A., Recent progress in intuitionistic analysis , Intuitionism and Proof Theory (Proceedings of the Summer Conference at Buffalo, NY, 1968) , North-Holland, Amsterdam, 1970, pp. 95–100.CrossRefGoogle Scholar
Hilbert, D., Über das Unendliche . Mathematische Annalen , vol. 95 (1926), no. 1, pp. 161–190 (in German).CrossRefGoogle Scholar
Hirschfeldt, D. R., Slicing the Truth , Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, vol. 28, World Scientific, Singapore, 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Hrbacek, K. and Jech, T., Introduction to Set Theory , third ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 220, Marcel Dekker, New York, 1999.Google Scholar
Hunter, J., Higher-order reverse topology , Ph.D. thesis, The University of Wisconsin–Madison, ProQuest LLC, Ann Arbor, 2008.Google Scholar
Huxley, M. N., Area, Lattice Points, and Exponential Sums , London Mathematical Society Monographs. New Series, vol. 13, The Clarendon Press and Oxford University Press, New York, 1996 (Oxford Science Publications).CrossRefGoogle Scholar
Jech, T. J., The Axiom of Choice , Studies in Logic and the Foundations of Mathematics, vol. 75, North-Holland, Amsterdam, 1973.Google Scholar
Jordan, C., Sur la série de Fourier . Comptes rendus de l’Académie des Sciences, Paris, Gauthier-Villars , vol. 92 (1881), pp. 228–230.Google Scholar
Jordan, C., Cours d’analyse de l’École Polytechnique. Tome I , Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Paris, 1991. Reprint of the third (1909) edition; first edition: 1883.Google Scholar
Jullien, P., Contribution à l’étude Des Types d’ordres dispersés , Ph.D. thesis, University of Marseilles, 1969.Google Scholar
Kleene, S. C., Recursive functionals and quantifiers of finite types. I . Transactions of the American Mathematical Society , vol. 91 (1959), pp. 1–52.Google Scholar
Kohlenbach, U., Foundational and mathematical uses of higher types , Reflections on the Foundations of Mathematics (Wilfried Sieg, Richard Sommer, editors), Lecture Notes in Logic, vol. 15, Association for Symbolic Logic, Wellesley, 2002, pp. 92–116.Google Scholar
Kohlenbach, U., Higher order reverse mathematics , Reverse Mathematics 2001 (Stephen G. Simpson, editor), Lecture Notes in Logic, vol. 21, Association for Symbolic Logic, Wellesley, 2005, pp. 281–295.Google Scholar
Kolmogorov, A. N. and Fomin, S. V., Elements of the Theory of Functions and Functional Analysis. Vol. 1. Metric and Normed Spaces , Graylock Press, Rochester, 1957. Translated from the first Russian edition by Leo F. Boron.Google Scholar
Kreuzer, A. P., Bounded variation and the strength of Helly’s selection theorem . Logical Methods in Computer Science , vol. 10 (2014), no. 4:16, pp. 1–23.Google Scholar
Kunen, K., Set Theory , Studies in Logic, vol. 34, College Publications, London, 2011.Google Scholar
Lakatos, I., Proofs and Refutations: The Logic of Mathematical Discovery , Cambridge Philosophy Classics, Cambridge University Press, Cambridge (UK), 2015; Edited by J. Worrall and E. Zahar; With a new preface by Paolo Mancosu; Originally published in 1976.CrossRefGoogle Scholar
Lebesgue, H., Comptes rendus et analyses: Review of young and young, the theory of sets of points . Bulletin des Sciences Mathématiques , vol. 31 (1907), no. 2, pp. 132–134.Google Scholar
Lindelöf, E., Sur quelques points de la théorie des ensembles . Comptes Rendus (1903), PP. 697–700.Google Scholar
Longley, J. and Normann, D., Higher-Order Computability , Theory and Applications of Computability, Springer, Heidelberg, 2015.CrossRefGoogle Scholar
Mal’cev, A. I., On ordered groups . Izvestiya Akad. Nauk SSSR. Ser. Mat. , vol. 13 (1949), pp. 473–482 (in Russian).Google Scholar
Martin-Löf, P., The Hilbert–Brouwer controversy resolved? , One Hundred Years of Intuitionism (1907–2007) (Mark Atten, Pascal Boldini, Michel Bourdeau, Gerhard Heinzmann, editors), Birkhäuser (Springer), 1967, pp. 243–256.CrossRefGoogle Scholar
Moll, V. H., Numbers and Functions , Student Mathematical Library, vol. 65, American Mathematical Society, Providence, 2012.Google Scholar
Montalbán, A., Indecomposable linear orderings and hyperarithmetic analysis . Journal of Mathematical Logic , vol. 6 (2006), no. 1, pp. 89–120.CrossRefGoogle Scholar
Montalbán, A., Open questions in reverse mathematics . Bulletin of Symbolic Logic , vol. 17 (2011), no. 3, pp. 431–454.CrossRefGoogle Scholar
Mostowski, A., Foundational Studies. Selected Works, vol. I, Studies in Logic and the Foundations of Mathematics, 93, North-Holland and PWN-Polish Scientific, Amsterdam, Warsawa, 1979.Google Scholar
Muldowney, P., A General Theory of Integration in Function Spaces, Including Wiener and Feynman Integration , Pitman Research Notes in Mathematics Series, vol. 153, Longman Scientific & Technical, Harlow; Wiley, Hoboken, 1987.Google Scholar
Mummert, C., On the reverse mathematics of general topology , Ph.D. thesis, The Pennsylvania State University, ProQuest LLC, Ann Arbor, 2005.Google Scholar
Mummert, C., Reverse mathematics of MF spaces . Journal of Mathematical Logic , vol. 6 (2006), no. 2, pp. 203–232.CrossRefGoogle Scholar
Mummert, C. and Simpson, S. G., Reverse mathematics and ${\Pi}_2^1$ comprehension . The Bulletin of Symbolic Logic , vol. 11 (2005), no. 4, pp. 526–533.CrossRefGoogle Scholar
Nies, A., Triplett, M. A., and Yokoyama, K., The reverse mathematics of theorems of Jordan and Lebesgue, this Journal, vol. 86 (2021), pp. 1–18.Google Scholar
Normann, D. and Sanders, S., Nonstandard analysis, computability theory, and their connections, this Journal, vol. 84 (2019), no. 4, pp. 1422–1465.Google Scholar
Normann, D. and Sanders, S., The strength of compactness in computability theory and nonstandard analysis . Annals of Pure and Applied Logic , vol. 170 (2019), no. 11, Article no. 102710.CrossRefGoogle Scholar
Normann, D. and Sanders, S., On the mathematical and foundational significance of the uncountable . Journal of Mathematical Logic , vol. 19 (2019), p. 1950001.CrossRefGoogle Scholar
Normann, D. and Sanders, S., Representations in measure theory, preprint, 2019. arXiv:1902.02756.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.CrossRefGoogle 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, Article no. 102788, 41 pp.CrossRefGoogle Scholar
Normann, D. and Sanders, S., The axiom of choice in computability theory and reverse mathematics . Journal of Logic and Computation , vol. 31 (2021), no. 1, pp. 297–325.CrossRefGoogle Scholar
Normann, D. and Sanders, S., On the uncountability of  $\mathbb{R}$ , this Journal, to appear, 2022, pp. 1–45. https://doi.org/10.1017/jsl.2022.27.CrossRefGoogle Scholar
Normann, D. and Sanders, S., Betwixt Turing and Kleene , Logical Foundations of Computer Science (Sergei Artemov, Anil Nerode, editors), Lecture Notes in Computer Science, vol. 13137, Springer Nature, Cham, 2022, pp. 281–300.Google Scholar
Normann, D. and Sanders, S., On the computational properties of basic mathematical notions. Journal of Logic and Computation , to appear, 2022, p. 44, arXiv:2203.05250.CrossRefGoogle Scholar
Rathjen, M., The Art of Ordinal Analysis , International Congress of Mathematicians, vol. II, European Mathematical Society, Zürich, 2006.Google Scholar
Richman, F., Omniscience principles and functions of bounded variation . Mathematical Logic Quarterly , vol. 48 (2002), pp. 111–116.3.0.CO;2-6>CrossRefGoogle Scholar
Roitman, J., Introduction to Modern Set Theory , Pure and Applied Mathematics, Wiley, New York, 1990.Google Scholar
Rosenstein, J. G., Linear Orderings , Pure and Applied Mathematics, vol. 98, Academic Press, Cambridge (USA), 1982.Google Scholar
Royden, H. L., Real Analysis , Lecture Notes in Mathematics, Pearson Education, London, 1989.Google Scholar
Russ, S. B., A translation of Bolzano’s paper on the intermediate value theorem . Historia Mathematica , vol. 7 (1980), pp. 156–185.CrossRefGoogle 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.CrossRefGoogle Scholar
Sanders, S., Plato and the foundations of mathematics, preprint, 2019, arxiv:1908.05676.Google Scholar
Sanders, S., Reverse mathematics of topology: Dimension, paracompactness, and splittings . Notre Dame Journal for Formal Logic , vol. 61 (2020), no. 4, pp. 537–559.Google Scholar
Sanders, S., Lifting recursive counterexamples to higher-order arithmetic , Logical Foundations of Computer Science (Sergei Artemov, Anil Nerode, editors), Lecture Notes in Computer Science, vol. 11972, Springer, Cham, 2020, pp. 249–267.CrossRefGoogle Scholar
Sanders, S., Reverse Mathematics of the uncountability of $\mathbb{R}$ : Baire classes, metric spaces, and unordered sums, preprint, 2020, arXiv:2011.02915.Google Scholar
Sanders, S., Lifting countable to uncountable mathematics . Information and Computation , vol. 287 (2021), Article no. 104762, 25 pp.Google Scholar
Sanders, S., Countable sets versus sets that are countable in reverse mathematics . Computability , vol. 11 (2022), no. 1, pp. 9–39.CrossRefGoogle Scholar
Sanders, S., On the computational properties of the uncountability of the reals , Logic, Language, Information, and Computation: Proceedings of WoLLIC22 (Agata Ciabattoni, Elaine Pimentel, Ruy J.G.B.de Queiroz, editors), Lecture Notes in Computer Science, vol. 13468, Springer, Cham, 2022.CrossRefGoogle Scholar
Sanders, S., Big in reverse mathematics: The uncountability of the reals, preprint, 2022, arxiv:2208.03027.Google Scholar
Sanders, S., Finding points of continuity in real analysis, submitted, 2022.Google Scholar
Sanders, S., On the computational properties of the Baire category theorem, submitted, 2022, arxiv:2210.05251.Google Scholar
Scheeffer, L., Allgemeine Untersuchungen über rectification der Curven . Acta Mathematica , vol. 5 (1884), no. 1, pp. 49–82 (in German).CrossRefGoogle Scholar
Shafer, P., The strength of compactness for countable complete linear orders . Computability , vol. 9 (2020), no. 1, pp. 25–36.CrossRefGoogle Scholar
Sierpiński, W., Sur Une propriété des fonctions qui n’ont que des discontinuités de première espèce . Bulletin de la Section Scientifique de l’Académie Roumaine , vol. 16 (1933), pp. 1–4 (in French).Google Scholar
Simpson, S. G. (ed.), Reverse Mathematics 2001 , Lecture Notes in Logic, vol. 21, Association for Symbolic Logic, La Jolla, 2005.Google Scholar
Simpson, S. G. (ed.), Subsystems of Second Order Arithmetic , second ed., Perspectives in Logic, Cambridge University Press, Cambridge (UK), 2009.CrossRefGoogle Scholar
Solomon, R., ${\varPi}_1^1$ - ${CA}_0$ and order types of countable ordered groups, this Journal, vol. 66 (2001), no. 1, pp. 192–206.Google Scholar
Stillwell, J., Reverse Mathematics, Proofs from the Inside out , Princeton University Press, Princeton, 2018.Google Scholar
Suslin, M., Problème 3 . Fundamenta Mathematicae , vol. 1 (1920), p. 223.Google Scholar
Swartz, C., Introduction to Gauge Integrals , World Scientific, Singapore, 2001.CrossRefGoogle Scholar
Troelstra, A. S., Metamathematical Investigation of Intuitionistic Arithmetic and Analysis , Lecture Notes in Mathematics, vol. 344, Springer, Berlin, 1973.CrossRefGoogle Scholar
Troelstra, A. S. and van Dalen, D., Constructivism in Mathematics, vol. I, Studies in Logic and the Foundations of Mathematics, vol. 121, North-Holland, Amsterdam, 1988.Google Scholar
Vatssa, B. S., Discrete Mathematics , fourth ed., New Age International, New Delhi, 1993.Google Scholar
Weierstrass, K., Einleiting in Die Theorie der Analytischen Funktionen , Schriftenr. Math. Inst. Univ. Münster, 2. Ser. 38, 108 S., 1986.Google Scholar
Yokoyama, K., Standard and non-standard analysis in second order arithmetic , Tohoku Mathematical Publications, vol. 34, Sendai, 2009, Ph.D. thesis, Tohoku University, 2007.Google Scholar
Zheng, X. and Rettinger, R., Effective Jordan decomposition . Theory of Computing Systems , vol. 38 (2005), no. 2, pp. 189–209.CrossRefGoogle Scholar