## Notre Dame Journal of Formal Logic

### Classifying Dini's Theorem

#### Abstract

Dini's theorem says that compactness of the domain, a metric space, ensures the uniform convergence of every simply convergent monotone sequence of real-valued continuous functions whose limit is continuous. By showing that Dini's theorem is equivalent to Brouwer's fan theorem for detachable bars, we provide Dini's theorem with a classification in the recently established constructive reverse mathematics propagated by Ishihara. As a complement, Dini's theorem is proved to be equivalent to the analogue of the fan theorem, weak König's lemma, in the original classical setting of reverse mathematics started by Friedman and Simpson.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 47, Number 2 (2006), 253-262.

Dates
First available in Project Euclid: 25 July 2006

https://projecteuclid.org/euclid.ndjfl/1153858650

Digital Object Identifier
doi:10.1305/ndjfl/1153858650

Mathematical Reviews number (MathSciNet)
MR2240623

Zentralblatt MATH identifier
1156.03055

#### Citation

Berger, Josef; Schuster, Peter. Classifying Dini's Theorem. Notre Dame J. Formal Logic 47 (2006), no. 2, 253--262. doi:10.1305/ndjfl/1153858650. https://projecteuclid.org/euclid.ndjfl/1153858650

#### References

• [1] Bishop, E., Foundations of Constructive Analysis, McGraw-Hill Book Co., New York, 1967.
• [2] Bishop, E., and D. Bridges, Constructive Analysis, vol. 279 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1985.
• [3] Bridges, D., and F. Richman, Varieties of Constructive Mathematics, vol. 97 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1987.
• [4] Bridges, D. S., "Dini's theorem: A constructive case study", pp. 69--80 in Combinatorics, Computability and Logic (Constanţa, 2001), edited by C. S. Calude, M. J. Dinneen, and S. Sburlan, Springer Series in Discrete Mathematics and Theoretical Computer Science, Springer-Verlag London Ltd., London, 2001.
• [5] Dummett, M., Elements of Intuitionism, 2d edition, vol. 39 of Oxford Logic Guides, The Clarendon Press, New York, 2000.
• [6] Ishihara, H., "Informal constructive reverse mathematics", Sūrikaisekikenkyūsho Kōkyūroku, vol. 1381 (2004), pp. 108--117.
• [7] Ishihara, H., Reverse mathematics in Bishop's constructive mathematics,'' forthcoming in Philosophia Scientiae, cahier spécial 6 (2006).
• [8] Ishihara, H., "Weak König's lemma implies Brouwer's fan theorem: A direct proof", Notre Dame Journal of Formal Logic, vol. 47 (2006), pp. 249--52
• [9] Ishihara, H., "Constructive reverse mathematics: Compactness properties", pp. 245--67 in From Sets and Types to Topology and Analysis, edited by L. Crosilla and P. Schuster, vol. 48 of Oxford Logic Guides, Oxford University Press, Oxford, 2005.
• [10] Kamo, H., "Effective Dini's theorem on effectively compact metric spaces", in Proceedings of the 6th Workshop on Computability and Complexity in Analysis (CCA 2004), vol. 120 of Electronic Notes Theoretical Computer Science, pp. 73--82, Amsterdam, 2005. Elsevier,
• [11] Kohlenbach, U., "Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals", Archive for Mathematical Logic, vol. 36 (1996), pp. 31--71.
• [12] Kohlenbach, U., "The use of a logical principle of uniform boundedness in analysis", pp. 93--106 in Logic and Foundations of Mathematics (Florence, 1995), edited by A. Cantini, vol. 280 of Synthese Library, Kluwer Academic Publishers, Dordrecht, 1999.
• [13] Loeb, I., "Equivalents of the (weak) fan theorem", Annals of Pure and Applied Logic, vol. 132 (2005), pp. 51--66.
• [14] Richman, F., "Constructive mathematics without choice", pp. 199--205 in Reuniting the Antipodes---Constructive and Nonstandard Views of the Continuum (Venice, 1999), edited by P. Schuster, U. Berger, and H. Osswald, vol. 306 of Synthese Library, Kluwer Academic Publishers, Dordrecht, 2001.
• [15] Richman, F., "Intuitionism as generalization", Philosophia Mathematica, vol. 5 (1990), pp. 124--28.
• [16] Richman, F., "The fundamental theorem of algebra: A constructive development without choice", Pacific Journal of Mathematics, vol. 196 (2000), pp. 213--30.
• [17] Simpson, S. G., Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1999.
• [18] Troelstra, A. S., and D. van Dalen, Constructivism in Mathematics. Vols. I and II, vols. 121, 123 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1988.
• [19] Veldman, W., "Brouwer's fan theorem as an axiom and as a contrast to Kleene's alternative", preprint, 2005.