Skip to main content Accessibility help
×
Home

COMPUTABILITY THEORY, NONSTANDARD ANALYSIS, AND THEIR CONNECTIONS

  • DAG NORMANN (a1) and SAM SANDERS (a2)

Abstract

We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related.

(T.1) A basic property of Cantor space $2^ $ is Heine–Borel compactness: for any open covering of $2^ $ , there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any $G:2^ \to $ , output a finite sequence $\langle f_0 , \ldots ,f_n \rangle $ in $2^ $ such that the neighbourhoods defined from $\overline {f_i } G\left( {f_i } \right)$ for $i \le n$ form a covering of $2^ $ .

(T.2) A basic property of Cantor space in Nonstandard Analysis is Abraham Robinson’s nonstandard compactness, i.e., that every binary sequence is “infinitely close” to a standard binary sequence. We analyse the strength of this nonstandard compactness property of Cantor space, compared to the other axioms of Nonstandard Analysis and usual mathematics.

Our study of (T.1) yields exotic objects in computability theory, while (T.2) leads to surprising results in Reverse Mathematics. We stress that (T.1) and (T.2) are highly intertwined, i.e., our study is holistic in nature in that results in computability theory yield results in Nonstandard Analysis and vice versa.

Copyright

References

Hide All
[1]Avigad, J. and Feferman, S., Gödel’s functional (“Dialectica”) interpretation, Handbook of proof theory. Studies in Logic and the Foundations of Mathematics, vol. 137 (1998), pp. 337405.
[2]Avigad, J., Dean, E., and Rute, J., Algorithmic randomness, reverse mathematics, and the dominated convergence theorem. Annals of Pure and Applied Logic, vol. 163 (2012), pp. 18541864.
[3]Aczel, P., An introduction to inductive definitions, Handbook of Mathematical Logic (Barwise, J., editor), Studies in Logic and the Foundations of Mathematics, vol. 90, North-Holland, Amsterdam, 1977, pp. 739782.
[4]Beeson, M. J., Foundations of Constructive Mathematics, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 6, Springer, Berlin, 1985.
[5]Berger, U., Uniform heyting arithmetic. Annals of Pure and Applied Logic, vol. 133 (2005), pp. 125148.
[6]Bienvenu, L., Patey, L., and Shafer, P., On the logical strengths of partial solutions to mathematical problems. Transactions of the London Mathematical Society, vol. 4 (2017), no. 1, pp. 3071.
[7]Blass, A., End extensions, conservative extensions, and the rudin-frolík ordering. Transactions of the American Mathematical Society, vol. 225 (1977), pp. 325340.
[8]Brouwer, L. E. J., Collected Works, Vol. 1: Philosophy and Foundations of Mathematics (Heyting, A., editor), North-Holland, Amsterdam, 1975.
[9]Borel, E., Sur quelques points de la théorie des fonctions. Annales Scientifiques de l’École Normale Supérieure, vol. 12 (1895), pp. 955.
[10]Cousin, P., Sur les fonctions de n variables complexes. Acta Mathematica, vol. 19 (1895), pp. 161.
[11]Dzhafarov, D. D., Reverse mathematics zoo, http://rmzoo.uconn.edu/.
[12]Escardó, M. and Xu, C., The inconsistency of a Brouwerian continuity principle with the Curry-Howard interpretation, 13th International Conference on Typed Lambda Calculi and Applications (Altenkirch, T., editor), Leibniz International Proceedings in Informatics, vol. 38, Dagstuhl, Saarbrücken, 2015, pp. 153164.
[13]Ferreira, F. and Gaspar, J., Nonstandardness and the bounded functional interpretation. Annals of Pure and Applied Logic, vol. 166 (2015), no. 6, pp. 701712.
[14]Flood, S., Reverse mathematics and a Ramsey-type König’s lemma, this Journal, vol. 77 (2012), no. 4, pp. 12721280.
[15]Friedman, H., Some systems of second order arithmetic and their use, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), vol. 1, Canadian Mathematical Congress, Montreal, 1975, pp. 235242.
[16]Friedman, H., Systems of second order arithmetic with restricted induction, I & II (abstracts), this Journal, vol. 41 (1976), pp. 557559.
[17]Friedman, H., Computational nonstandard analysis, FOM mailing list, (Sept. 1st, 2015). https://www.cs.nyu.edu/pipermail/fom/2015-September/018984.html.
[18]Gandy, R., Proof of Mostowski’s conjecture. Bulletin L’Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 8 (1960). (English, with Russian summary).
[19]Gandy, R., General recursive functionals of finite type and hierarchies of functions. Annales de la Faculté des Sciences de Toulouse Clermont-Ferrand No, vol. 35 (1967), pp. 524.
[20]Gandy, R. and Hyland, M., Computable and Recursively Countable Functions of Higher Type, Studies in Logic and the Foundations of Mathematics, vol. 87, North-Holland, Amsterdam, 1977, pp. 407438.
[21]Gordon, E. I., Relatively standard elements in Nelson’s internal set theory. Siberian Mathematical Journal, vol. 30, no. 1, pp. 6873.
[22]Hadzihasanovic, A. and van den Berg, B., Nonstandard functional interpretations and categorical models. Notre Dame Journal of Formal Logic, vol. 58 (2017), no. 3, pp. 343380.
[23]Harrison, J., Recursive pseudo-well-orderings. Transactions of the American Mathematical Society, vol. 131 (1968), pp. 526543.
[24]Hirschfeldt, D. R., Slicing the Truth, Lecture Notes Series. Institute for Mathematical Sciences, vol. 28, World Scientific Publishing, National University of Singapore, 2015.
[25]Hurd, A. E. and Loeb, P. A., An Introduction to Nonstandard Real Analysis, Pure and Applied Mathematics, vol. 118, Academic Press, Orlando, FL, 1985.
[26]Ishihara, H., Reverse mathematics in Bishop’s constructive mathematics. Philosophia Scientiae (Cahier Spécial), vol. 6 (2006), pp. 4359.
[27]Keisler, H. J., Nonstandard arithmetic and reverse mathematics. Bulletin of Symbolic Logic, vol. 12 (2006), pp. 100125.
[28]Keisler, H. J., Nonstandard arithmetic and recursive comprehension. Annals of Pure and Applied Logic, vol. 161 (2010), no. 8, pp. 10471062.
[29]Kjos-Hanssen, B., Miller, J. S., and Solomon, R., Lowness notions, measure and domination. Journal of the London Mathematical Society (2), vol. 85, (2012), no. 3, pp. 869888.
[30]Kleene, S. C., Recursive functionals and quantifiers of finite types. I. Transactions of the American Mathematical Society, vol. 91 (1959), pp. 152.
[31]Kleene, S. C. and Vesley, R. E., The Foundations of Intuitionistic Mathematics, Especially in Relation to Recursive Functions, North–Holland, Amsterdam, 1965.
[32]Kleiner, I., Excursions in the History of Mathematics, Birkhäuser/Springer, New York, 2012.
[33]Kohlenbach, U., Applied Proof Theory: Proof Interpretations and their Use in Mathematics, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2008.
[34]Kohlenbach, U., Higher order reverse mathematics, Reverse Mathematics 2001 (Simpson, S., editor), Lecture Notes in Mathematics, vol. 21, Peters, A. K., Wellesley, MA, 2005, pp. 281295.
[35]Kohlenbach, U., Foundational and mathematical uses of higher types, Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman (Sieg, W., Sommer, R., and Talcott, C., editors), Lecture Notes in Mathematics, vol. 15, Peters, A. K., Wellesley, MA, 2002, pp. 92116.
[36]Kohlenbach, U., On uniform weak König’s lemma. Annals of Pure and Applied Logic, vol. 114 (2002), no. 1–3, pp. 103116. Commemorative Symposium Dedicated to Anne S. Troelstra (Noordwijkerhout, 1999).
[37]Lindelöf, E., Sur quelques points de la théorie des ensembles. Comptes Rendus, vol. 137 (1903), pp. 697700.
[38]Longley, J. and Normann, D., Higher-order Computability, Theory and Applications of Computability, Springer, Berlin, 2015.
[39]Martin-Löf, P., An intuitionistic theory of types: Predicative part, Logic Colloquium ’73 (Rose, H. E. and Shepherdson, J. C., editors), Studies in Logic and the Foundations of Mathematics, vol. 80, North-Holland, Amsterdam, 1975, pp. 73118.
[40]Montalbán, A., Open questions in reverse mathematics. The Bulletin of Symbolic Logic, vol. 17 (2011), no. 3, pp. 431454.
[41]Mummert, C. and Simpson, S. G., Reverse mathematics and ${\rm{\Pi }}_2^1 $ comprehension . The Bulletin of Symbolic Logic, vol. 11 (2005), no. 4, pp. 526533.
[42]Nelson, E., Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society, vol. 83 (1977), no. 6, pp. 11651198.
[43]Normann, D., Recursion on the Countable Functionals, Lecture Notes in Mathematics, vol. 811, Springer, Berlin, 1980.
[44]Normann, D., Functionals of type 3 as realisers of classical theorems in analysis, Proceedings of CiE18, Lecture Notes in Computer Science, Springer, Berlin, 2018, pp. 318327.
[45]Normann, D. and Sanders, S., Nonstandard analysis, computability theory, and their connections, preprint, arXiv: https://arxiv.org/abs/1702.06556.
[46]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.
[47]Normann, D. and Sanders, S., On the mathematical and foundational significance of the uncountable. Journal of Mathematical Logic, vol. 19 (2019), no. 1, p. 40.
[48]Normann, D. and Sanders, S., Representations in measure theory, submitted, 2019, arXiv: https://arxiv.org/abs/1902.02756.
[49]Nuprl, Main website, 2019, http://www.nuprl.org/.
[50]Rahli, V. and Bickford, M., A nominal exploration of intuitionism, Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs, Association for Computing Machinery, New York, 2016, pp. 130141.
[51]Robinson, A., Non-Standard Analysis, North-Holland, Amsterdam, 1966.
[52]Rogers, H., Theory of Recursive Functions and Effective Computability, MIT Press, Cambridge, MA, 1987.
[53]Sacks, G. E., Higher Recursion Theory, Perspectives in Mathematical Logic, Springer, Berlin, 1990.
[54]Sakamoto, N. and Yamazaki, T., Uniform versions of some axioms of second order arithmetic. MLQ Mathematical Logic Quarterly, vol. 50 (2004), no. 6, pp. 587593.
[55]Sanders, S., The Gandy-Hyland functional and a computational aspect of nonstandard analysis. Computability, vol. 7 (2018), no. 1, pp. 743.
[56]Sanders, S., Some nonstandard equivalences in reverse mathematics, Sailing Routes in the World of Computation (Manea, F., Miller, R. G., and Nowotka, D., editors), Lecture Notes in Computer Science, vol. 10936, Springer, Berlin, 2018, pp. 365375.
[57]Sanders, S., Metastability and higher-order computability, Proceedings of LFCS18, Lecture Notes in Computer Science, vol. 10703, Springer, Berlin, 2018.
[58]Sanders, S., The unreasonable effectiveness of nonstandard analysis, submitted, 2015, available from http://arxiv.org/abs/1508.07434.
[59]Sanders, S., From nonstandard analysis to various flavours of computability theory, Theory and Applications of Models of Computation (Gopal, T. V., Jäger, G., and Steila, S., editors), Lecture Notes in Computer Science, vol. 10185 Springer, Berlin, 2017, pp. 556570.
[60]Sanders, S., To be or not to be constructive. Indagationes Mathematicae and the Brouwer Volume L.E.J. Brouwer, Fifty Years Later, 2018, p. 69, https://doi.org/10.1016/j.indag.2017.05.005.
[61]Sanders, S., Refining the taming of the reverse mathematics zoo. Notre Dame Journal of Formal Logic, vol. 59 (2018), no. 4, pp. 579597.
[62]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. 99121.
[63]Simpson, S. G. (ed.), Reverse Mathematics 2001, Lecture Notes in Logic, vol. 21, ASL, La Jolla, CA, 2005.
[64]Simpson, S. G., Subsystems of Second Order Arithmetic, second ed., Perspectives in Logic, Cambridge University Press, Cambridge, 2009.
[65]Simpson, S. G. and Yokoyama, K., A nonstandard counterpart of WWKL. Notre Dame Journal of Formal Logic, vol. 52 (2011), no. 3, pp. 229243.
[66]Spector, C., Hyperarithmetical quantifiers. Fundamenta Mathematicae, vol. 48 (1959/60), pp. 313320.
[67]Stillwell, J., Reverse Mathematics, Proofs from the Inside Out, Princeton University Press, Princeton, NJ, 2018.
[68]Troelstra, A. S., Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, Springer, Berlin, 1973.
[69]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.
[70]van den Berg, B., Briseid, E., and Safarik, P., A functional interpretation for nonstandard arithmetic. Annals of Pure and Applied Logic, vol. 163 (2012), no. 12, pp. 19621994.
[71]van den Berg, B. and Sanders, S., Reverse mathematics and parameter-free transfer. Annals of Pure and Applied Logic, vol. 170 (2019), no. 3, pp. 273296.
[72]Vitali, G., Sui gruppi di punti e sulle funzioni di variabili reali.Atti della Accademia delle Scienze di Torino, vol. XLIII, no. 4, (1907), pp. 229247.
[73]Yu, X. and Simpson, S. G., Measure theory and weak König’s lemma. Archive for Mathematical Logic, vol. 30 (1990), no. 3, pp. 171180.

Keywords

COMPUTABILITY THEORY, NONSTANDARD ANALYSIS, AND THEIR CONNECTIONS

  • DAG NORMANN (a1) and SAM SANDERS (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.