Normann, Dag; Sanders, Sam On the uncountability of \(\mathbb{R}\). (English) Zbl 1523.03004 J. Symb. Log. 87, No. 4, 1474-1521 (2022). This article continues the authors’ exploration of higher-order reverse mathematics and computability theory. The focus here is on principles asserting the non-existence of injections (or bijections) from \([0,1]\) to the natural numbers. These principles are independent of surprisingly strong comprehension axioms. For example, \(\mathsf{NIN}\) (non-existence of an injection) is not provable in \(\mathbb{Z}^\omega_2\), a higher-order extension of full second-order arithmetic. On the other hand, \(\mathsf{NIN}\) is a consequence of higher-order formulations of well-known theorems, for example, the Baire category theorem as stated in Section 6 of [the authors, J. Log. Comput. 30, No. 8, 1639–1679 (2020; Zbl 1472.03012)]. The results here lay the foundation for analysis of other results in measure and category, as shown in the preprint of S. Sanders [“Big in reverse mathematics: measure and category”, Preprint, arXiv:2303.00493]. Reviewer: Jeffry L. Hirst (Boone) Cited in 1 ReviewCited in 6 Documents MSC: 03B30 Foundations of classical theories (including reverse mathematics) 03F35 Second- and higher-order arithmetic and fragments 03D65 Higher-type and set recursion theory 03D55 Hierarchies of computability and definability 03D30 Other degrees and reducibilities in computability and recursion theory Keywords:uncountability of \(\mathbb{R}\); reverse mathematics; Kleene S1–S9; higher-order computability theory; NIN; NBI Citations:Zbl 1472.03012 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Aharoni, R., Magidor, M., and Shore, R. A., On the strength of König’s duality theorem for infinite bipartite graphs. Journal of Combinatorial Theory, Series B, vol. 54 (1992), no. 2, pp. 257-290. · Zbl 0754.05053 [2] Arzelà, C., Sulla integrazione per serie. Atti della Accademia Nazionale dei Lincei. Rendiconti Lincei. Matematica e Applicazioni, vol. 1 (1885), pp. 532-537. · JFM 17.0256.02 [3] Avigad, J. and Feferman, S., Gödel’s functional “Dialectica” interpretation, Handbook of Proof Theory (Sam Buss, editor), Studies in Logic and the Foundations of Mathematics, vol. 137, Elsevier, Amsterdam, 1998, pp. 337-405. · Zbl 0913.03053 [4] Baire, R., Sur les fonctions de variables réelles. Annali di Matematica, vol. 3 (1899), no. 3, pp. 1-123. · JFM 30.0359.01 [5] Baire, R., Leçons Sur les fonctions discontinues, Les Grands Classiques Gauthier-Villars, Éditions Jacques Gabay, Sceaux, 1995 (in French). Reprint of the 1905 original. · Zbl 0897.01040 [6] Barbu, V. and Precupanu, T., Convexity and Optimization in Banach Spaces, fourth ed., Springer Monographs in Mathematics, Springer, Dordrecht, 2012. · Zbl 1244.49001 [7] Bauer, A., An injection from the Baire space to natural numbers. Mathematical Structures in Computer Science, vol. 25 (2015), no. 7, pp. 1484-1489. · Zbl 1362.03053 [8] Beth, E. W., Semantic Entailment and Formal Derivability, Mededelingen der koninklijke Nederlandse Akademie van Wetenschappen, Afd. Letterkunde, Nieuwe Reeks, Deel 18, Vol. 13, N. V. Noord-Hollandsche Uitgevers Maatschappij, Elsevier, Amsterdam, 1955. [9] Bishop, E., Foundations of Constructive Analysis, McGraw-Hill, New York, 1967. · Zbl 0183.01503 [10] Blumberg, H., New properties of all real functions. Transactions of the American Mathematical Society, vol. 24 (1922), no. 2, pp. 113-128. · JFM 49.0176.04 [11] Du Bois-Reymond, P., Die allgemeine Functionentheorie I, Wissenschaftliche Buchgesellschaft, Darmstadt, 1968 (in German). Part I, reproduction of the 1882 original with afterword and selected bibliography by Detlef Laugwitz. [12] Borel, E., Leçons Sur la théorie Des Fonctions, Gauthier-Villars, Paris, 1898. · JFM 45.0664.01 [13] Bressoud, D. M., A Radical Approach to Lebesgue’s Theory of Integration, MAA Textbooks, Cambridge University Press, Cambridge, 2008. · Zbl 1165.00001 [14] Brown, D. K., Notions of compactness in weak subsystems of second order arithmetic, Reverse Mathematics 2001 (Stephen Simson, editor), Lecture Notes in Logic, vol. 21, Association for Symbolic Logic, La Jolla, 2005, pp. 47-66. · Zbl 1087.03039 [15] Brown, D. K., Giusto, M., and Simpson, S. G., Vitali’s theorem and WWKL. Archive for Mathematical Logic, vol. 41 (2002), no. 2, pp. 191-206. · Zbl 1030.03044 [16] Buchholz, W., Feferman, S., Pohlers, W., and Sieg, W., Iterated Inductive Definitions and Subsystems of Analysis, Lecture Notes in Mathematics, vol. 897, Springer, New York, 1981. · Zbl 0489.03022 [17] Cantor, G., Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen. Journal für die Reine und Angewandte Mathematik, vol. 77 (1874), pp. 258-262. · JFM 06.0057.01 [18] Cantor, G., Ein Beitrag zur Mannigfaltigkeitslehre. Journal für die reine und angewandte Mathematik, vol. 84 (1877), pp. 242-258. · JFM 09.0379.01 [19] Cantor, G., Ueber unendliche, lineare Punktmannichfaltigkeiten. Mathematische Annalen, vols. 17-23. Published in parts: 1879-1884. [20] Cantor, G., Mitteilungen zur Lehre vom Transfiniten, Pfeffer, 1887. · JFM 19.0044.02 [21] Cantor, G., Gesammelte Abhandlungen Mathematischen und Philosophischen Inhalts, Springer, Berlin, 1980. Reprint of the 1932 original. · Zbl 0441.04001 [22] Carslaw, H. S., Term-by-term integration of infinite series. The Mathematical Gazette, vol. 13 (1927), pp. 437-441. · JFM 53.0216.04 [23] Cohen, P., The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences of the United States of America, vol. 50 (1963), pp. 1143-1148. · Zbl 0192.04401 [24] Cohen, P., The independence of the continuum hypothesis. II. Proceedings of the National Academy of Sciences of the United States of America, vol. 51 (1964), pp. 105-110. [25] Cousin, P., Sur les fonctions de \(n\) variables complexes. Acta Mathematica, vol. 19 (1895), pp. 1-61. · JFM 26.0456.02 [26] Dauben, J. W. and Cantor, G., editors, His Mathematics and Philosophy of the Infinite, Princeton University Press, Princeton, 1990. · Zbl 0858.01028 [27] Devlin, K. J., Constructibility, Perspectives in Mathematical Logic, Springer, New York, 1984. · Zbl 0542.03029 [28] Diener, H., Variations on a theme by Ishihara. Mathematical Structures in Computer Science, vol. 25 (2015), no. 7, pp. 1569-1577. · Zbl 1362.03054 [29] Diestel, J. and Swart, J., The Riesz theorem, Handbook of Measure Theory,vol. I (E. Pap, editor), North-Holland, Amsterdam, 2002, pp. 401-447. · Zbl 1028.46024 [30] Dini, U., Fondamenti per la Teorica Delle Funzioni di Variabili Reali, Nistri, Pisa, 1878. · JFM 10.0274.01 [31] Dzhafarov, D. D., Reverse Mathematics Zoo.http://rmzoo.uconn.edu/. [32] Ewald, W., editor, From Kant to Hilbert: A Source Book in the Foundations of Mathematics,vols.I and II, Oxford Science Publications, Oxford University Press, Oxford, 1996. · Zbl 0859.01002 [33] Feferman, S., How a little bit goes a long way: Predicative foundations of analysis, 2013, unpublished notes from 1977 to 1981 with updated introduction. Available at https://math.stanford.edu/feferman/papers/pfa(1).pdf. [34] Ferreirós, J., Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics, second ed., Birkhäuser, Basel, 2007. · Zbl 1119.03044 [35] Fréchet, M., Sur quelques points du Calcul Fonctionel. Rendiconti del Circolo Matematico di Palermo, vol. XXII (1906), pp. 1-72. · JFM 37.0348.02 [36] Friedman, H., Some systems of second order arithmetic and their use, Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), vol. 1, 1975, pp. 235-242. · Zbl 0344.02022 [37] Friedman, H., Systems of second order arithmetic with restricted induction, I & II (abstracts), this Journal, vol. 41 (1976), pp. 557-559. [38] Gandy, R., General recursive functionals of finite type and hierarchies of functions. Annales scientifiques de l’Université de Clermont. Mathématiques, vol. 35 (1967), pp. 5-24. [39] Gödel, K., The consistency of the axiom of choice and of the generalized continuum-hypothesis. Proceedings of the National Academy of Science, vol. 24 (1938), no. 12, pp. 556-557. · JFM 64.0035.01 [40] Gordon, R. A., A convergence theorem for the Riemann integral. Mathematics Magazine, vol. 73 (2000), no. 2, pp. 141-147. [41] Gray, R., Georg Cantor and transcendental numbers. American Mathematical Monthly, vol. 101 (1994), no. 9, pp. 819-832. · Zbl 0827.01004 [42] Hankel, H., Untersuchungen über die unendlich oft oscillirenden und unstetigen Functionen. Mathematische Annalen, vol. 20 (1882), no. 1, pp. 63-112 (in German). · JFM 14.0320.02 [43] Harnack, A., Vereinfachung der Beweise in der Theorie der Fourier’schen Reihe. Mathematische Annalen, vol. 19 (1881), no. 2, pp. 235-279 (in German). · JFM 13.0182.02 [44] Harnack, A., Ueber den Inhalt von Punktmengen. Mathematische Annalen, vol. 25 (1885), pp. 241-250. · JFM 17.0506.01 [45] Hartley, J. P., The countably based functionals, this Journal, vol. 48 (1983), no. 2, pp. 458-474. · Zbl 0548.03024 [46] Hawkins, T., Lebesgue’s Theory of Integration: Its Origins and Development, second ed., AMS Chelsea, Providence, 2001. · Zbl 0986.26002 [47] Helly, E., Über lineare Funktionaloperationen. Sitzungsberichte der Mathematisch-Naturwissenschaftlichen Klasse der Kaiserlichen Akademie der Wissenschaften, Wien, vol. 121 (1912), pp. 265-297. · JFM 43.0418.02 [48] Hilbert, D., Über das Unendliche. Mathematische Annalen, vol. 95 (1926), no. 1, pp. 161-190 (in German). · JFM 51.0044.02 [49] Hilbert, D., Mathematical problems. Bulletin of the American Mathematical Society, vol. 37 (2000), no. 4, pp. 407-436. Reprinted from Bulletin of the American Mathematical Society, vol. 8 (1902), pp. 437-479. · JFM 33.0976.07 [50] Howard, P. and Rubin, J. E., Consequences of the Axiom of Choice, Mathematical Surveys and Monographs, vol. 59, American Mathematical Society, Providence, 1998. · Zbl 0947.03001 [51] 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. · Zbl 1045.03521 [52] Hunter, J., Higher-Order Reverse Topology, ProQuest LLC, Ann Arbor, 2008, Ph.D. thesis, The University of Wisconsin, Madison. [53] Hunter, J. K. and Nachtergaele, B., Applied Analysis, World Scientific, River Edge, 2001. · Zbl 0981.46002 [54] Katz, M. and Reimann, J., An Introduction to Ramsey Theory: Fast Functions, Infinity, and Metamathematics, Student Mathematical Library, vol. 87, American Mathematical Society, Providence; Mathematics Advanced Study Semesters, University Park, 2018. · Zbl 1404.05210 [55] Kelley, J. L., General Topology, Graduate Texts in Mathematics, vol. 27, Springer, Berlin, 1975. Reprint of the 1955 edition. · Zbl 0306.54002 [56] Kestelman, H., Riemann integration of limit functions. American Mathematical Monthly, vol. 77 (1970), no. 2, pp. 182-187. · Zbl 0195.34302 [57] Kihara, T., Marcone, A., and Pauly, A., Searching for an analogue of \({\mbox{ATR}}_0\) in the Weihrauch lattice, this Journal, vol. 85 (2020), pp. 1006-1043. · Zbl 1473.03026 [58] Kleene, S. C., Recursive functionals and quantifiers of finite types. I. Transactions of the American Mathematical Society, vol. 91 (1959), pp. 1-52. · Zbl 0088.01301 [59] Kleiner, I., Excursions in the History of Mathematics, Birkhäuser/Springer, Berlin, 2012. · Zbl 1230.01003 [60] Kohlenbach, U., Foundational and mathematical uses of higher types, Reflections on the Foundations of Mathematics (Wilfied Sieg, Richard Sommer, and Carolyn Talcott, editors), Lecture Notes in Logic, vol. 15, Association for Symbolic Logic, Urbana, 2002, pp. 92-116. · Zbl 1022.03044 [61] Kohlenbach, U., Higher order reverse mathematics, Reverse Mathematics 2001 (Stephen Simson, editor), Lecture Notes in Logic, vol. 21, Association for Symbolic Logic, La Jolla, 2005, pp. 281-295. · Zbl 1097.03053 [62] Kohlenbach, U., Applied Proof Theory: Proof Interpretations and Their Use in Mathematics, Springer Monographs in Mathematics, Springer, Berlin, 2008. · Zbl 1158.03002 [63] Kolmogorov, A. N., Foundations of the Theory of Probability, Chelsea, New York, 1950. · Zbl 0037.08101 [64] König, D., Über eine Schlussweise aus dem Endlichen ins Unendliche. Acta Litterarum ac Scientarum (Sectio Scientiarum Mathematicarum) Szeged, vol. 3 (1927), pp. 121-130. · JFM 53.0170.04 [65] Kreuzer, A. P., Bounded variation and the strength of Helly’s selection theorem, Logical Methods in Computer Science, vol. 10 (2014), no. 4, p. 4:16, 15. · Zbl 1322.03012 [66] Kreuzer, A. P., Measure theory and higher order arithmetic. Proceedings of the American Mathematical Society, vol. 143 (2015), no. 12, pp. 5411-5425. · Zbl 1386.03016 [67] Kunen, K., Set Theory, Studies in Logic, vol. 34, College Publications, London, 2011. · Zbl 1262.03001 [68] Longley, J. and Normann, D., Higher-Order Computability, Theory and Applications of Computability, Springer, Berlin, 2015. · Zbl 1471.03002 [69] Luckhardt, H., The real elements in a consistency proof for simple type theory I, \({\vDash}\) ISILC Proof Theory Symposion (Proceedings of the International Summer Institute and Logic Colloquium, Kiel, 1974) (Justus Diller and Gert H. Müller, editors), Lecture Notes in Mathematics, vol. 500, Springer, Berlin, 1975, pp. 233-256. · Zbl 0326.02019 [70] Luxemburg, W. A. J., Arzelà’s dominated convergence theorem for the Riemann integral. American Mathematical Monthly, vol. 78 (1971), pp. 970-979. · Zbl 0225.26013 [71] Moore, E. H., On a form of general analysis with application to linear differential and integral equations. Atti IV Cong. Inter. Mat. (Roma, 1908), vol. 2 (1909), pp. 98-114. · JFM 40.0396.01 [72] Moore, E. H., Introduction to a Form of General Analysis, Yale University Press, New Haven, 1910. · JFM 41.0376.01 [73] Moore, E. H., Definition of limit in general integral analysis. Proceedings of the National Academy of Sciences of the United States of America, vol. 1 (1915), no. 12, pp. 628-632. · JFM 45.0426.03 [74] Moore, E. H., On power series in general analysis, Festschrift David Hilbert zu seinem Sechzigsten Geburtstag am 23. Januar 1922, Springer, 1922, pp. 355-364. · JFM 48.0026.03 [75] Moore, E. H., On power series in general analysis. Mathematische Annalen, vol. 86 (1922), nos. 1-2, pp. 30-39. · JFM 48.0488.02 [76] Moore, E. H. and Smith, H., A general theory of limits. American Journal of Mathematics, vol. 44 (1922), pp. 102-121. · JFM 48.1254.01 [77] 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, New York, 1987. · Zbl 0623.28008 [78] Mummert, C., On the Reverse Mathematics of General Topology, ProQuest LLC, Ann Arbor, 2005, Ph.D. thesis, The Pennsylvania State University. [79] Mummert, C., Reverse mathematics of MF spaces. Journal of Mathematical Logic, vol. 6 (2006), no. 2, pp. 203-232. · Zbl 1122.03005 [80] Mummert, C. and Simpson, S. G., Reverse mathematics and \({\Pi}_2^1\) comprehension. Bulletin of Symbolic Logic, vol. 11 (2005), no. 4, pp. 526-533. · Zbl 1106.03050 [81] Nies, A., Triplett, M. A., and Yokoyama, K., The reverse mathematics of theorems of Jordan and Lebesgue, this Journal, vol. 86 (2021), pp. 1657-1675. · Zbl 07457794 [82] Normann, D., Functionals of type 3 as realisers of classical theorems in analysis, Proceedings of CiE18 (Florin Manea, Russell G. Miller, and Dirk Nowotka, editors), Lecture Notes in Computer Science, vol. 10936, Springer, New York, 2018, pp. 318-327. · Zbl 1509.03135 [83] Normann, D., Computability and non-monotone induction, preprint, 2020, arXiv:2006.03389, p. 41. [84] Normann, D. and Sanders, S., Nonstandard analysis, computability theory, and their connections, this Journal, vol. 84 (2019), no. 4, pp. 1422-1465. · Zbl 1454.03018 [85] 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, 102710. · Zbl 1430.03035 [86] Normann, D. and Sanders, S., On the mathematical and foundational significance of the uncountable. Journal of Mathematical Logic, vol. 19 (2019), 1950001. · Zbl 1484.03018 [87] Normann, D. and Sanders, S., Open sets in reverse mathematics and computability theory. Journal of Logic and Computability, vol. 30 (2020), no. 8, p. 40. · Zbl 1472.03012 [88] 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, 102788, p. 41. · Zbl 1443.03008 [89] Normann, D. and Sanders, S., The axiom of choice in computability theory and reverse mathematics with a cameo for the continuum hypothesis. Journal of Logic and Computation, vol. 31 (2021), pp. 297-325. · Zbl 1509.03037 [90] Normann, D. and Sanders, S., On robust theorems due to Bolzano, Weierstrass, and Cantor in reverse mathematics, preprint, 2021, arXiv:2102.04787, p. 30. [91] Normann, D. and Sanders, S., On the logical and computational properties of the Vitali covering theorem, preprint, 2022, arXiv:1902.02756. · Zbl 07638209 [92] Normann, D. and Sanders, S., Betwixt Turing and Kleene, Logical Foundations of Computer Science (Artemov, S. and Nerode, A., editors), Lecture Notes in Computer Science, vol. 13137, Springer, Cham, 2022, pp. 236-252. · Zbl 1490.03004 [93] Normann, D. and Sanders, S., On the computational properties of basic mathematical notions, preprint, 2022, arxiv:2203.05250, p. 43. · Zbl 07638209 [94] Oliva, P. and Powell, T., Bar recursion over finite partial functions. Annals of Pure and Applied Logic, vol. 168 (2017), no. 5, pp. 887-921. · Zbl 1422.03091 [95] Rathjen, M., The Art of Ordinal Analysis, International Congress of Mathematicians, vol. II, European Mathematical Society, Zürich, 2006. · Zbl 1101.03035 [96] Royden, H. L. and Fitzpatrick, P. M., Real Analysis, fourth ed., Pearson, New York, 2010. · Zbl 1191.26002 [97] Sakamoto, N. and Yamazaki, T., Uniform versions of some axioms of second order arithmetic. MLQ, vol. 50 (2004), no. 6, pp. 587-593. · Zbl 1063.03045 [98] Sanders, S., Plato and the foundations of mathematics, preprint, 2020, arxiv:1908.05676. [99] Sanders, S., Lifting recursive counterexamples to higher-order arithmetic, Logical Foundations of Computer Science. LFCS 2020 (Artemov, S. and Nerode, A., editors), Lecture Notes in Computer Science, vol. 11972, Springer, Cham, 2020, pp. 249-267. · Zbl 1485.03243 [100] Sanders, S., Reverse mathematics of topology: Dimension, paracompactness, and splittings. Notre Dame J. for Formal Logic, vol. 61 (2020), no. 4, pp. 537-559. · Zbl 1486.03023 [101] Sanders, S., Lifting countable to uncountable mathematics, Information and Computation, 104762 (2021). https://doi.org/10.1016/j.ic.2021.104762. [102] Sanders, S., Representations and the foundations of mathematics, Notre Dame Journal of Formal Logic, vol. 63 (2022), no. 1, pp. 1-28. · Zbl 07522852 [103] Sanders, S., Reverse mathematics of the uncountability of \(\mathbb{R} \) , preprint, 2022, arXiv:2203.05292, p. 12. · Zbl 07627935 [104] Sanders, S. and Yokoyama, K., The Dirac delta function in two settings of reverse mathematics. Archive for Mathematical Logic, vol. 51 (2012), no. 1, pp. 99-121. · Zbl 1248.03023 [105] Shafer, P., The strength of compactness for countable complete linear orders. Computability, vol. 9 (2020), no. 1, pp. 25-36. · Zbl 1485.03028 [106] Sierpiński, W., Sur un problème de la théorie des relations. Annali della Scuola Normale Superiore di Pisa—Classe di Scienze, Série 2, vol. 2 (1933), no. 3, pp. 285-287 (in French). · Zbl 0007.09702 [107] Simpson, S. G., editor, Reverse Mathematics 2001, Lecture Notes in Logic, vol. 21, Association for Symbolic Logic, La Jolla, 2005. · Zbl 0578.03005 [108] Simpson, S. G., Subsystems of Second Order Arithmetic, second ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009. · Zbl 1181.03001 [109] Simpson, S. G., The Gödel hierarchy and reverse mathematics. Kurt Gödel. Essays for His Centennial (Solomon Feferman, Charles Parsons, and Stephen G. Simpson, editors), Association for Symbolic Logic, Providence, 2010, pp. 109-127. · Zbl 1223.03006 [110] Smith, H. J. S., On the integration of discontinuous functions, Proceedings of the London Mathematical Society, vol. 6 (1874/75), pp. 140-153. · JFM 07.0247.01 [111] Sohrab, H. H., Basic Real Analysis, second ed., Birkhäuser/Springer, New York, 2014. · Zbl 1308.26006 [112] Stillwell, J., Roads to Infinity: The Mathematics of Truth and Proof, A K Peters, Natick, 2010. · Zbl 1196.00004 [113] Stillwell, J., The Real Numbers: An Introduction to Set Theory and Analysis, Undergraduate Texts in Mathematics, Springer, Cham, 2013. · Zbl 1292.26004 [114] Stillwell, J., Reverse Mathematics: Proofs from the Inside Out, Princeton University Press, Princeton, 2018. · Zbl 1386.00059 [115] Swartz, C., Introduction to Gauge Integrals, World Scientific, Singapore, 2001. · Zbl 0982.26006 [116] Tao, T., Structure and Randomness: Pages from Year One of a Mathematical Blog, American Mathematical Society, Providence, 2008. · Zbl 1245.00024 [117] Tao, T., An Epsilon of Room, I: Real Analysis, Graduate Studies in Mathematics, vol. 117, American Mathematical Society, Providence, 2010. · Zbl 1216.46002 [118] Tao, T., An Introduction to Measure Theory, Graduate Studies in Mathematics, vol. 126, American Mathematical Society, Providence, 2011. · Zbl 1231.28001 [119] Thomson, B., Monotone convergence theorem for the Riemann integral. American Mathematical Monthly, vol. 117 (2010), no. 6, pp. 547-550. · Zbl 1210.26010 [120] Troelstra, A. S., Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, Springer, Berlin, 1973. · Zbl 0275.02025 [121] Troelstra, A. S. and Van Dalen, D., Constructivism in Mathematics,vol. I, Studies in Logic and the Foundations of Mathematics, 121, Elsevier, Amsterdam, 1988. · Zbl 0661.03047 [122] Troelstra, A. S. and Van Dalen, D., Constructivism in Mathematics,vol. II, Studies in Logic and the Foundations of Mathematics, 123, Elsevier, Amsterdam, 1988. · Zbl 0661.03047 [123] Tukey, J. W., Convergence and Uniformity in Topology, Annals of Mathematics Studies, vol. 2, Princeton University Press, Princeton, 1940. · Zbl 0025.09102 [124] Turing, A., On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, vol. 42 (1936), pp. 230-265. · Zbl 0016.09701 [125] Vitali, G., Sui gruppi di punti e sulle funzioni di variabili reali. Atti della Accademia delle Scienze di Torino, vol. XLIII (1907), pp. 229-247. [126] Weaver, G., König’s infinity lemma and Beth’s tree theorem. History and Philosophy of Logic, vol. 38 (2017), no. 1, pp. 48-56. · Zbl 1417.03038 [127] Cantor’s First Set Theory Article, Wikipedia, the Free Encyclopedia, 2020. Available at https://en.wikipedia.org/wiki/Cantor [128] Yokoyama, K., Standard and Non-Standard Analysis in Second Order Arithmetic, Tohoku Mathematical Publications, vol. 34, Tohoku University, Sendai, 2009, Ph.D. thesis, Tohoku University, 2007. · Zbl 1178.03076 [129] Young, W. H., On non-uniform convergence and term-by-term integration of series. Proceedings of the London Mathematical Society. Second Series, vol. 1 (1904), pp. 89-102. · JFM 34.0284.01 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.