Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Reverse formalism 16

  • 175 Accesses

  • 2 Citations

Abstract

In his remarkable paper Formalism 64, Robinson defends his eponymous position concerning the foundations of mathematics, as follows:

  1. (i)

    Any mention of infinite totalities is literally meaningless.

  2. (ii)

    We should act as if infinite totalities really existed.

Being the originator of Nonstandard Analysis, it stands to reason that Robinson would have often been faced with the opposing position that ‘some infinite totalities are more meaningful than others’, the textbook example being that of infinitesimals (versus less controversial infinite totalities). For instance, Bishop and Connes have made such claims regarding infinitesimals, and Nonstandard Analysis in general, going as far as calling the latter respectively a debasement of meaning and virtual, while accepting as meaningful other infinite totalities and the associated mathematical framework. We shall study the critique of Nonstandard Analysis by Bishop and Connes, and observe that these authors equate ‘meaning’ and ‘computational content’, though their interpretations of said content vary. As we will see, Bishop and Connes claim that the presence of ideal objects (in particular infinitesimals) in Nonstandard Analysis yields the absence of meaning (i.e. computational content). We will debunk the Bishop–Connes critique by establishing the contrary, namely that the presence of ideal objects (in particular infinitesimals) in Nonstandard Analysis yields the ubiquitous presence of computational content. In particular, infinitesimals provide an elegant shorthand for expressing computational content. To this end, we introduce a direct translation between a large class of theorems of Nonstandard Analysis and theorems rich in computational content (not involving Nonstandard Analysis), similar to the ‘reversals’ from the foundational program Reverse Mathematics. The latter also plays an important role in gauging the scope of this translation.

This is a preview of subscription content, log in to check access.

Notes

  1. 1.

    Note that Robinson still held the formalist view nearly ten years after Formalism 64.

    Subsequent exchanges, both oral and in published writings, have not induced me to change my views [from Formalism 64]. Moreover, I believed then and I still believe that the well-known recent developments in set theory represent evidence favoring these views. (Robinson 1975, p. 42)

    Moreover, Dauben discusses the origin of Robinson’s ‘change of heart’ in (Dauben 1988, §5).

  2. 2.

    See Sect. 4.1.1 (resp. Sect. 4.2) for a discussion of Bishop’s (resp. Connes’) philosophy of mathematics.

  3. 3.

    In Constructive Reverse Mathematics (Ishihara 2006), intuitionistic logic is used instead.

  4. 4.

    \({\textsf {RCA}}_{0}\) consists of induction \(I{\varSigma }_{1}\), and the recursive comprehension axiom \({\varDelta }_{1}^{0}\)-CA.

  5. 5.

    Exceptions are classified in the so-called Reverse Mathematics Zoo ([37]). Most of these are ‘combinatorial in nature’, another vague notion.

  6. 6.

    The acronym \({\textsf {ZFC}}\) stands for Zermelo-Fraenkel set theory with the axiom of choice; see Hrbacek and Jech (1999) for an introduction to set theory.

  7. 7.

    The superscript ‘fin’ in (I) means that x is finite, i.e. its number of elements are bounded by a natural number.

  8. 8.

    By Example 3.1, we may assume f is nonstandard continuous. Let N be a nonstandard natural number and let \(j\le N\) be the least number such that \(f\left( \frac{j}{N}\right) f\left( \frac{j+1}{N}\right) \le 0\). Then \(f(j/N)\approx 0\) by nonstandard continuity, and we are done.

  9. 9.

    Let \(y\in [0,1]\) be such that \(f(y)\approx 0\) and consider the set of rationals \(z=\{ q_{1}, q_{1}, q_{2}, \dots , q_{N}\}\) where \(q_{i}\) is a rational such that \(|y-q_{i}|<\frac{1}{i}\) and \(q_{i}=\frac{j}{2^{i}}\) for some \(j\le 2^{i}\), and N is a nonstandard number. Applying S, there is a standard set w such that \((\forall ^{\text {st}}i)(q_{i}\in w)\). The standard sequence \(q_{i}\) converges to a standard real \(x\approx y\).

  10. 10.

    Recall that ‘\(f(x)\approx 0\)’ is short for \((\forall ^{\text {st}}k)(|f(x)|<\frac{1}{k})\).

  11. 11.

    Like for \({\textsf {ZFC}}\) and \({\textsf {IST}}\), if the system \({\textsf {P}}\) (resp. \({\textsf {H}}\)) proves an internal sentence, then this sentence is provable in \(\textsf {E-PA}^{\omega }\) (resp. E-HA \(^{\omega }\)).

  12. 12.

    The full version of Theorem 3.4 is Corollary 7.6 in Sect. 7.

  13. 13.

    See Sect. 4.1.1 (resp. Sect. 4.2) for a discussion of Bishop’s (resp. Connes’) philosophy of mathematics.

  14. 14.

    Note that outside of the context of \({\textsf {IST}}\), ‘mainstream’ mathematics just has its usual meaning in this paper.

  15. 15.

    To prove that \((\exists x)A(x)\) by contradiction in classical mathematics, one assumes \((\forall x)\lnot A(x)\) and derives a contradiction, i.e. one shows that \(\lnot [(\forall x)\lnot A(x)]\). Using the law of excluded middle \(B\vee \lnot B\), one then concludes that \((\exists x)A(x)\).

  16. 16.

    The axiom of choice implies LEM under certain conditions (Diaconescu (1975)), but fragments of the axiom of choice are considered acceptable in constructive mathematics (See Bridges and Vîţă 2006, §1.4).

  17. 17.

    Brouwer’s intuitionism is discussed at length by Dummett in Dummett (2000), including Brouwer’s theorem that all total functions on the unit interval must be (uniformly) continuous (Dummett 2000, Theorem 3.19), which contradicts classical mathematics.

  18. 18.

    The name ‘recursive’ mathematics betrays that all mathematical objects must be recursive (called ‘computable’ nowadays), as captured in the axiom Church’s Thesis (See Bridges and Richman 1987, Chap. 3 or Beeson 1985, I.8 for details), which contradicts classical mathematics.

  19. 19.

    There are a number of constructive approaches to Nonstandard Analysis (See e.g. Moerdijk and Palmgren 1997; Palmgren 1996a, b, 1997, 2000; Bell 2010) which we briefly discuss in Sect. 3.3.

  20. 20.

    The usual development of Robinson’s Nonstandard Analysis proceeds via the construction of a nonstandard model using a free ultrafilter. The existence of the latter is only slightly weaker than the axiom of choice of \({\textsf {ZFC}}\) (Wolff and Loeb 2015).

  21. 21.

    Note that Connes uses the word ‘constructive’ as synonymous with ‘effective’ and ‘explicit’ from mainstream mathematics, i.e. no connection with constructivism seems present.

  22. 22.

    As noted in Sect. 3.2.3, terms obtained from Theorem 3.4 are indeed algorithms, and we use these two words interchangeably in the context of system \({\textsf {P}}\).

  23. 23.

    Here, a ‘reasonable’ system is one which can prove the usual properties of finite lists, for which the presence of the exponential function suffices. In particular, a subsystem of primitive recursive arithmetic, where the latter is claimed to correspond to Hilbert’s finitist mathematics (Tait (1981)), suffices.

  24. 24.

    It is easy to prove (5.6) in \({\textsf {P}}\) using Bayes’ theorem \(P(A|B)=\frac{ P(B|A)P(A)}{P(B)}\) for \(P(B)\ne 0\), and basic properties of infinitesimals.

  25. 25.

    As noted in Sect. 3.2.3, terms obtained from Theorem 3.4 are indeed algorithms, and we use these two words interchangeably.

  26. 26.

    Bishop uses the exact words ‘indispensable part’ with regard to moduli in (Bishop 1967, p. 34).

  27. 27.

    The predicate ‘\(x\approx y\)’ is usually read as ‘the distance between x and y is infinitesimal’.

  28. 28.

    We stress that item v should be interpreted in a specific narrow technical sense (beyond the scope of this paper), namely as discussed in Sanders (2016b).

  29. 29.

    A space is F-compact in Nonstandard Analysis if there is a discrete grid which approximates every point of the space up to infinitesimal error, i.e. the intuitive notion of compactness from physics and engineering.

  30. 30.

    The language of \(\textsf {E-PA}_{\text {st}}^{\omega *}\) contains a symbol \(\text {st}_{\sigma }\) for each finite type \(\sigma \), but the subscript is essentially always omitted. Hence \(\mathcal {T}^{*}_{\text {st}}\) is an axiom schema and not an axiom.

  31. 31.

    A term is called closed in van den Berg et al. (2012) (and in this paper) if all variables are bound via lambda abstraction. Thus, if \(\underline{x}, \underline{y}\) are the only variables occurring in the term t, the term \((\lambda \underline{x})(\lambda \underline{y})t(\underline{x}, \underline{y})\) is closed while \((\lambda \underline{x})t(\underline{x}, \underline{y})\) is not. The second axiom in Definition 7.3 thus expresses that \(\text {st}_{\tau }\big ((\lambda \underline{x})(\lambda \underline{y})t(\underline{x}, \underline{y})\big )\) if \((\lambda \underline{x})(\lambda \underline{y})t(\underline{x}, \underline{y})\) is of type \(\tau \). We essentially always omit lambda abstraction for brevity.

  32. 32.

    Recall the definition of closed terms from van den Berg et al. (2012) as sketched in Footnote 31.

  33. 33.

    Here, a ‘reasonable’ system is one which can prove the usual properties of finite lists, for which the presence of the exponential function suffices.

References

  1. Agda. (2016). a dependently typed functional programming language, http://wiki.portal.chalmers.se/agda/pmwiki.php.

  2. Artigue, M. (1994). Analysis. In O. T. David (Ed.), Advanced mathematical thinking. Berlin: Springer.

  3. Avigad, Jeremy, & Feferman, Solomon. (1998). Gödel’s functional (“Dialectica”) interpretation. Handbook of proof theory. Studies in Logic and the Foundations of Mathematics, 137, 337–405.

  4. Barwise, J. (Ed.). (1977). Handbook of mathematical logic, Studies in Logic and the Foundations of Mathematics vol. 90, North-Holland. With the cooperation of H. J. Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra.

  5. Beeson, M. J. (1985). Foundations of constructive mathematics, Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin: Springer Metamathematical Studies.

  6. Bell, John L. (2010). Continuity and infinitesimals, continuity and infinitesimals. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy http://plato.stanford.edu/archives/fall2010/entries/continuity/.

  7. Bernstein, A. R., & Wattenberg, F. (1969). Nonstandard measure theory, applications of model theory to algebra, analysis, and probability (Internat. Sympos., Pasadena, Calif., 1967). Holt, Rinehart and Winston, New York, 1969, 171–185.

  8. Bishop, E. (1970). Mathematics as a numerical language, Intuitionism and Proof Theory, (Proc. Conf., Buffalo, N.Y., 1968), North-Holland, pp. 53–71.

  9. Bishop, E. (1972). Aspects of constructivism, Notes on the lectures delivered at the Tenth Holiday Mathematics Symposium, New Mexico State University, Las Cruces, December 27–31.

  10. Bishop, E. (1975). The crisis in contemporary mathematics, Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics, pp. 507–517.

  11. Bishop, E. A. (1985). Schizophrenia in contemporary mathematics, E. Bishop: reflections on him and his research, Contemp. Math., vol. 39, American Mathematical Society, pp. 1–32.

  12. Bishop, E. (1967). Foundations of constructive analysis. New York: McGraw-Hill Book Co.

  13. Bishop, E. (1977). Book review: Elementary calculus. Bulletin of the American Mathematical Society, 83(2), 205–208.

  14. Bishop, Errett, & Bridges, Douglas S. (1985). Constructive analysis, Grundlehren der Mathematischen Wissenschaften (Vol. 279). Berlin: Springer-Verlag.

  15. Bourgine, P., & Raiman, O. (1986). Economics as reasoning on a qualitative model, 1986. Aix-en-Provence, France: IFAC Economies and Artificial Intelligence.

  16. Bridges, D. (1998). Constructive truth in practice, Truth in mathematics, (Mussomeli, 1995), Oxford University Press, Oxford , pp. 53–69.

  17. Brouwer, L.E.J. (1975). Collected works. Vol. 1, North-Holland Publishing Co., Amsterdam. Philosophy and foundations of mathematics; Edited by A. Heyting.

  18. Bridges, D., & Palmgren, E. Constructive Mathematics, Winter 2013 (Edward N. Zalta ed.), The Stanford Encyclopedia of Philosophy.

  19. Bridges, Douglas S. (1999). Constructive mathematics: A foundation for computable analysis. Theoretical Computer Science, 219(1–2), 95–109.

  20. Bridges, D., & Richman, F. (1987). Varieties of constructive mathematics, London Mathematical Society Lecture Note Series (Vol. 97). Cambridge: Cambridge University Press.

  21. Bridges, Douglas S., & Vîţă, Luminiţa Simona. (2006). Techniques of constructive analysis, Universitext. New York: Springer.

  22. Buss, S. R. (1998). An introduction to proof theory, Handbook of proof theory, Stud. Logic Found. Math., vol. 137, North-Holland, Amsterdam, pp. 1–78.

  23. Changeux, J.-P., & Connes, A. (1995). Conversations on mind, matter, and mathematics, Princeton University Press, Princeton, NJ. Edited and translated from the 1989 French original by M. B. DeBevoise.

  24. Connes, A. (1997). Brisure de symétrie spontanée et géométrie du point de vue spectral. J. Geom. Phys., 23(3-4), 206–234, (French, with English summary).

  25. Connes, A. (2007a). Non-standard stuff, Alain Connes’ blog. http://noncommutativegeometry.blogspot.com/2007/07/non-standard-stuff.html.

  26. Connes, A. (2007b). An interview with Alain Connes, Part I, EMS Newsletter, 63, 25–30. http://www.mathematics-in-europe.eu/maths-as-a-profession/interviews.

  27. Connes, A. (1995). Noncommutative geometry and reality. Journal of Mathematical Physics, 36(11), 6194–6231.

  28. Dague, P., Raiman, O., & Deves, P. (1987). Troubleshooting: When Modeling Is the Trouble, 1987. In Proceedings of the 6th National Conference on Artificial Intelligence. Seattle, WA.

  29. Dauben, J. W., (1988). Abraham Robinson and nonstandardanalysis: History, philosophy, and foundations of mathematics,History and philosophy of modern mathematics (Minneapolis, MN,1985), Minnesota Stud. Philos. Sci., XI, pp. 177–200.

  30. Dauben, J. W. (1996). Arguments, logic and proof: mathematics, logic and the infinite, History of mathematics and education: ideas and experiences (Essen, 1992). Stud. Wiss. Soz. Bildungsgesch. Math., 11, 113–148.

  31. Davis, E. (1990). Order of magnitude reasoning in qualitative differential equations, 11–39. Readings in Qualitative Reasoning about Physical Systems.

  32. Davis, M., & Hausner, M. (1978). Book review. The Joy of Infinitesimals. J. Keisler’s Elementary Calculus. Mathematical Intelligencer, 1, 168–170.

  33. Diaconescu, R. (1975). Axiom of choice and complementation. Proceedings of the American Mathematical Society, 51, 176–178.

  34. Diener, H., & Loeb, Iris. (2011). Constructive reverse investigations into differential equations, Journal of Logic and Analysis, 3, Paper 8, pp. 26.

  35. Dummett, M. (2000). Elements of intuitionism, 2nd ed., Oxford Logic Guides, vol. 39, Oxford University Press.

  36. Dzhafarov, D. D. (2017) Reverse mathematics zoo, http://rmzoo.uconn.edu/.

  37. Ferreira, F., & Gaspar, J. (2015). Nonstandardness and the bounded functional interpretation. Annals of Pure and Applied Logic, 166(6), 701–712.

  38. Friedman, H. (1975). Some systems of second order arithmetic and their use. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pp. 235–242.

  39. Friedman, H. (1976). Systems of second order arithmetic with restricted induction, I & II (Abstracts). Journal of Symbolic Logic, 41, 557–559.

  40. Gödel, K. (1958). Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica 12, 280–287 (German, with English summary).

  41. Goldblatt, R. (1998). Lectures on the hyperreals, Graduate Texts in Mathematics (Vol. 188). Berlin: Springer-Verlag. An introduction to nonstandard analysis.

  42. Heijerman, T. A. F. (1981). Intuition and the Intellect: On the Relation Between Mathematics, Philosophy and Mysticism in the Work of L. E. J. Brouwer and Including a Comparison with Nicholas of Cusa, Department of Mathematics, Univ. Utrecht.

  43. Hesseling, D. E. (2003). Gnomes in the fog, Science Networks. Historical Studies, vol. 28, Birkhäuser Verlag, Basel. The reception of Brouwer’s intuitionism in the 1920s.

  44. Heyting, A. (1973). Address to Professor A. Robinson. At theoccasion of the Brouwer memorial lecture given by Prof. A.Robinsonon the 26th April 1973. Nieuw Archief voor Wiskunde, 21(3), 134–137.

  45. Hrbacek, K., & Jech, T. (1999). Introduction to set theory, 3rd ed., Monographs and Textbooks in pure and applied mathematics, vol. 220, Marcel Dekker, Inc., New York.

  46. Hurd, A. E., & Loeb, P. A. (1985). An introduction to nonstandard real analysis, pure and applied mathematics (Vol. 118). Orlando, FL: Academic Press Inc.

  47. Ishihara, H. (2006). Reverse mathematics in Bishop’s constructive mathematics. Philosophia Scientiae (Cahier Spécial), 6, 43–59.

  48. Kanovei, V., Katz, M. G., & Mormann, T. (2013). Tools, objects, and chimeras: Connes on the role of hyperreals in mathematics. Foundations of Science, 18(2), 259–296.

  49. Katz, K. U., & Katz, M. G. (2011). Meaning in classical mathematics: Is it at odds with intuitionism. Intellectica, 56(2), 223–302.

  50. Katz, M. G., & Leichtnam, E. (2013). Commuting and noncommuting infinitesimals. American Mathematical Monthly, 120(7), 631–641.

  51. Kaye, R. (1991). Models of Peano arithmetic, Oxford Logic Guides (Vol. 15). Oxford: The Clarendon Press.

  52. Keisler, H. J. (1976). Elementary calculus. Boston: Prindle, Weber and Schmidt.

  53. Kohlenbach, U. (2002). Foundational and mathematical uses of higher types, Reflections on the foundations of mathematics (Stanford, CA, 1998), Lect. Notes Log., vol. 15, ASL, pp. 92–116.

  54. Kohlenbach, U. (2005). Higher order reverse mathematics, Reverse mathematics 2001, Lect. Notes Log., vol. 21, ASL, pp. 281–295.

  55. Kohlenbach, U. (2008). Applied proof theory: Proof interpretations and their use in mathematics, Springer Monographs in Mathematics. Berlin: Springer-Verlag.

  56. Moerdijk, I., & Palmgren, E. (1997). Minimal models of Heyting arithmetic. The Journal of Symbolic Logic, 62(4), 1448–1460.

  57. Martin-Löf, Per. (1975). An intuitionistic theory of types: predicative part, Logic Colloquium ’73, (Bristol, 1973), North-Holland, Amsterdam, pp. 73–118. Studies in Logic and the Foundations of Mathematics, Vol. 80.

  58. Montalbán, A. (2011). Open questions in reverse mathematics. Bulletin of Symbolic Logic, 17(3), 431–454.

  59. Nelson, E. (1977). Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society, 83(6), 1165–1198.

  60. Nerode, A., Metakides, G., & Constable, R., (1985). Remembrances of Errett Bishop, Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), Contemp. Math., vol. 39, Amer. Math. Soc., Providence, RI, pp. 79–84.

  61. Palmgren, E. (1996a). Overview of Constructive Nonstandard Mathematics, Website: http://www2.math.uu.se/~palmgren/biblio/nonstd.html.

  62. Palmgren, E. (1996b). Constructive nonstandard analysis, Méthodes et analyse non standard, Cahiers Centre Logique (Vol. 9). Louvain-la-Neuve: Acad.-Bruylant.

  63. Palmgren, E. (1998). Developments in constructive nonstandard analysis. Bulletin of Symbolic Logic, 4, 233–272.

  64. Palmgren, E. (1997). A sheaf-theoretic foundation for nonstandard analysis. Annals of Pure and Applied Logic, 85(1), 69–86.

  65. Palmgren, E. (2000). Constructive nonstandard representations of generalized functions. Indagationes Mathematicae, 11(1), 129–138.

  66. Raiman, O. (1986). Order of magnitude reasoning. In Proceedings of the 5th National Conference on Artificial Intelligence. Philadelphia, PA, August 11–15, Volume 1: Science.

  67. Raiman, O. (1991). Order of magnitude reasoning. Artificial Intelligence, 51(1–3), 11–38.

  68. Richman, Fred. (1990). Intuitionism as a generalization. Philosophia Math, 5, 124–128.

  69. Richman, F. (1996). Interview with a constructive mathematician. Modern Logic, 6(3), 247–271.

  70. Robert, A. M. (2003). Nonstandard analysis. Mineola: Dover Publications.

  71. Robinson, A. (1965). Formalism 64, Logic, Methodology and Philos. Sci. (Proc. 1964 Internat. Congr.), North-Holland, pp. 228–246.

  72. Robinson, A., (1975). Concerning progress in the philosophy of mathematics, Logic Colloquium ’73, (Bristol, 1973), North-Holland, pp. 41–52. Studies in Logic and the Foundations of Mathematics, Vol. 80.

  73. Robinson, A. (1966). Non-standard analysis. Amsterdam: North-Holland.

  74. Robinson, A. (1968). Reviews: Foundations of constructive analysis. American Mathematical Monthly, 75(8), 920–921.

  75. Rosenblatt, M. (Ed.). (1985). Errett Bishop: reflections on him and his research, Contemporary Mathematics (Vol. 39). Providence, RI: American Mathematical Society.

  76. Sakamoto, N., & Yamazaki, T. (2004). Uniform versions of some axioms of second order arithmetic. Mathematical Logic Quarterly, 50(6), 587–593.

  77. Sanders, S. (2015). The taming of the Reverse Mathematics zoo, Submitted, arxiv:1412.2022.

  78. Sanders, S., (2016a) The unreasonable effectiveness of Nonstandard Analysis, Submitted to APAL special issue of LFCS, arxiv:1508.07434.

  79. Sanders, S. (2016b). The computational content of the Loeb measure, arxiv:1609.01945.

  80. Sanders, S., (2016c). The refining of the taming of the Reverse Mathematics zoo, To appear in Notre Dame Journal for Formal Logic, arxiv:1602.02270.

  81. Sanders, S. (2017) From nonstandard analysis to various flavours of computability theory, To appear in Proceedings of TAMC17, Lecture Notes in Computer Science, Springer.

  82. Simpson, S. G. (2005). Reverse mathematics 2001, Lecture Notes in Logic (Vol. 21). La Jolla, CA: ASL.

  83. Simpson, S. G., (2009). Subsystems of second orderarithmetic, 2nd ed. Perspectives in Logic, CUP.

  84. Stern, J. (1985). Le problème de la mesure, Astérisque, 121–122, 325–346 (French). Seminar Bourbaki, Vol. 1983/84.

  85. Suenaga, K., Sekine, H., & Hasuo, I. (2013). Hyperstream processing systems: nonstandard modeling of continuous-time signals, The 40th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL ’13, Rome, Italy–January 23–25, 2013.

  86. Tait, W. W. (1981). Finitism. The Journal of Philosophy, 78, 524–564.

  87. Tall, D. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48, 2–3.

  88. Troelstra, AS, & van Dalen, D. (1988). Constructivism in mathematics. Vol. I, Studies in Logic and the Foundations of Mathematics, vol. 121, North-Holland.

  89. Tzouvaras, A. (1998). Modeling vagueness by nonstandardness. Fuzzy Sets and Systems, 94(3), 385–396.

  90. van den Berg, Benno, Briseid, Eyvind, & Safarik, Pavol. (2012). A functional interpretation for nonstandard arithmetic. Annals of Pure and Applied Logic, 163(12), 1962–1994.

  91. Vopenka, P. (1991). The philosophical foundations of alternative set theory. International Journal of General Systems, 20(1), 115–126.

  92. Weld, D. S. (1990). Exaggeration. Artificial Intelligence, 43(3), 311–368.

  93. Wolff, M., & Loeb, P. A., (Ed.) (2015). Nonstandard analysis for the working mathematician, Mathematics and its Applications, vol. 510, Kluwer, Second edition.

  94. Xu, C., & Sanders, S. (2015). Extracting the computational content of Nonstandard Analysis, In preparation; Agda code: http://cj-xu.github.io/agda/nonstandard_dislectica/Dialectica.html.

Download references

Acknowledgements

This research was supported by the following entities: FWO Flanders, the John Templeton Foundation, the Alexander von Humboldt Foundation, LMU Munich (via the Excellence Initiative), and the Japan Society for the Promotion of Science. The author expresses his gratitude towards these institutions. The author would also like to thank the two referees for their helpful remarks which greatly improved this paper.

Author information

Correspondence to Sam Sanders.

Appendices

Appendix 1: The formal systems \({\textsf {P}}\) and \({\textsf {H}}\) in full detail

Gödel’s system T

In this section, we briefly introduce Gödel’s system T and the associated systems \(\textsf {E-PA}^{\omega }\) and \(\textsf {E-PA}^{\omega *}\). In his famous Dialectica paper (Gödel (1958)), Gödel defines an interpretation of intuitionistic arithmetic into a quantifier-free calculus of functionals. This calculus is now known as ‘Gödel’s system T’, and is essentially just primitive recursive arithmetic (Buss 1998, §1.2.10) with the schema of recursion expanded to all finite types. The set of all finite types \(\pmb {T}\) is:

figurep

where 0 is the type of natural numbers, and \(\sigma \rightarrow \tau \) is the type of mappings from objects of type \(\sigma \) to objects of type \(\tau \). Hence, Gödel’s system T includes ‘recursor’ constants \(\pmb {R}^{\rho }\) for every finite type \(\rho \in \pmb {T}\), defining primitive recursion as follows:

figureq

for \(f^{\rho }\) and \(g^{0\rightarrow ( \rho \rightarrow \rho )}\). The system \(\textsf {E-PA}^{\omega }\) is a combination of Peano Arithmetic and system T, and the full axiom of extensionality E. The detailed definition of \(\textsf {E-PA}^{\omega }\) may be found in (Kohlenbach 2008, §3.3); We do introduce the notion of equality and extensionality in \(\textsf {E-PA}^{\omega }\), as these notions are needed below.

Definition 7.3

[Equality] The system \(\textsf {E-PA}^{\omega }\) includes equality between natural numbers ‘\(=_{0}\)’ as a primitive. Equality ‘\(=_{\tau }\)’ for type \(\tau \)-objects xy is then:

$$\begin{aligned}{}[x=_{\tau }y] \equiv (\forall z_{1}^{\tau _{1}}\dots z_{k}^{\tau _{k}})[xz_{1}\dots z_{k}=_{0}yz_{1}\dots z_{k}] \end{aligned}$$
(7.1)

if the type \(\tau \) is composed as \(\tau \equiv (\tau _{1}\rightarrow \dots \rightarrow \tau _{k}\rightarrow 0)\). The usual inequality predicate ‘\(\le _{0}\)’ between numbers has an obvious definition, and the predicate ‘\(\le _{\tau }\)’ is just ‘\(=_{\tau }\)’ with ‘\(=_{0}\)’ replaced by ‘\(\le _{0}\)’ in (7.1). The axiom of extensionality is the statement that for all \(\rho , \tau \in \pmb {T}\), we have:

figurer

Next, we introduce \(\textsf {E-PA}^{\omega *}\), a definitional extension of \(\textsf {E-PA}^{\omega }\) from van den Berg et al. (2012) with a type for finite sequences. In particular, the set \(\pmb {T}^{*}\) is defined as:

figures

where \(\sigma ^{*}\) is the type of finite sequences of objects of type \(\sigma \). The system \(\textsf {E-PA}^{\omega *}\) includes PR for all \(\rho \in \pmb {T}^{*}\), as well as dedicated ‘list recursors’ to handle finite sequences for any \(\rho ^{*}\in \pmb {T}^{*}\). A detailed definition of \(\textsf {E-PA}^{\omega *}\) may be found in (van den Berg et al. 2012, §2.1). We now introduce some notations specific to \(\textsf {E-PA}^{\omega *}\), as also used in van den Berg et al. (2012).

Notation 1.21 (Finite sequences) The system \(\textsf {E-PA}^{\omega *}\) has a dedicated type for ‘finite sequences of objects of type \(\rho \)’, namely \(\rho ^{*}\). Since the usual coding of finite sequences of natural numbers goes through in \(\textsf {E-PA}^{\omega *}\), we shall not always distinguish between 0 and \(0^{*}\). Similarly, we do not always distinguish between ‘\(s^{\rho }\)’ and ‘\(\langle s^{\rho }\rangle \)’, where the former is ‘the object s of type \(\rho \)’, and the latter is ‘the sequence of type \(\rho ^{*}\) with only element \(s^{\rho }\)’. The empty sequence for the type \(\rho ^{*}\) is denoted by ‘\(\langle \rangle _{\rho }\)’, usually with the typing omitted. Furthermore, we denote by ‘\(|s|=n\)’ the length of the finite sequence \(s^{\rho ^{*}}=\langle s_{0}^{\rho },s_{1}^{\rho },\dots ,s_{n-1}^{\rho }\rangle \), where \(|\langle \rangle |=0\), i.e. the empty sequence has length zero. For sequences \(s^{\rho ^{*}}, t^{\rho ^{*}}\), we denote by ‘\(s*t\)’ the concatenation of s and t, i.e. \((s*t)(i)=s(i)\) for \(i<|s|\) and \((s*t)(j)=t(|s|-j)\) for \(|s|\le j< |s|+|t|\). For a sequence \(s^{\rho ^{*}}\), we define \(\overline{s}N:{=}\langle s(0), s(1), \dots , s(N)\rangle \) for \(N^{0}<|s|\). For a sequence \(\alpha ^{0\rightarrow \rho }\), we also write \(\overline{\alpha }N=\langle \alpha (0), \alpha (1),\dots , \alpha (N)\rangle \) for any \(N^{0}\). By way of shorthand, \(q^{\rho }\in Q^{\rho ^{*}}\) abbreviates \((\exists i<|Q|)(Q(i)=_{\rho }q)\). Finally, we shall use \(\underline{x}, \underline{y},\underline{t}, \dots \) as short for tuples \(x_{0}^{\sigma _{0}}, \dots x_{k}^{\sigma _{k}}\) of possibly different type \(\sigma _{i}\).

We have used \(\textsf {E-PA}^{\omega }\) and \(\textsf {E-PA}^{\omega *}\) interchangeably in this paper. Our motivation is the ‘star morphism’ used in Robinson’s approach to Nonstandard Analysis, and the ensuing potential for confusion.

The classical system \({\textsf {P}}\)

In this section, we introduce the system \({\textsf {P}}\), a conservative extension of \(\textsf {E-PA}^{\omega }\) with fragments of Nelson’s \({\textsf {IST}}\).

To this end, we first introduce the base system \(\textsf {E-PA}_{\text {st}}^{\omega *}\). We use the same definition as (van den Berg et al. 2012, Def. 6.1), where E-PA \(^{\omega *}\) is the definitional extension of E-PA \(^{\omega }\) with types for finite sequences as in (van den Berg et al. 2012, §2). The set \(\mathcal {T}^{*}\) is defined as the collection of all the terms in the language of \(\textsf {E-PA}^{\omega *}\).

Definition 7.4

The system \( \textsf {E-PA}^{\omega *}_{\text {st}} \) is defined as \( \textsf {E-PA}^{\omega {*}} + \mathcal {T}^{*}_{\text {st}} + \textsf {IA}^{\text {st}}\), where \(\mathcal {T}^{*}_{\text {st}}\) consists of the following axiom schemas.

  1. 1.

    The schemaFootnote 30 \(\text {st}(x)\wedge x=y\rightarrow \text {st}(y)\),

  2. 2.

    The schema providing for each closedFootnote 31 term \(t\in \mathcal {T}^{*}\) the axiom \(\text {st}(t)\).

  3. 3.

    The schema \(\text {st}(f)\wedge \text {st}(x)\rightarrow \text {st}(f(x))\).

The external induction axiom IA \(^{\text {st}}\) is as follows.

figuret

Secondly, we introduce some essential fragments of \({\textsf {IST}}\) studied in van den Berg et al. (2012).

Definition 7.7

[External axioms of \({\textsf {P}}\)]

  1. 1.

    \({\textsf {HAC}}_{{\textsf {int}}}\): For any internal formula \(\varphi \), we have

    $$\begin{aligned} (\forall ^{\text {st}}x^{\rho })(\exists ^{\text {st}}y^{\tau })\varphi (x, y)\rightarrow \big (\exists ^{\text {st}}F^{\rho \rightarrow \tau ^{*}}\big )(\forall ^{\text {st}}x^{\rho })(\exists y^{\tau }\in F(x))\varphi (x,y), \end{aligned}$$
    (7.2)
  2. 2.

    \(\textsf {I}\): For any internal formula \(\varphi \), we have

    $$\begin{aligned} (\forall ^{\text {st}} x^{\sigma ^{*}})(\exists y^{\tau } )(\forall z^{\sigma }\in x)\varphi (z,y)\rightarrow (\exists y^{\tau })(\forall ^{\text {st}} x^{\sigma })\varphi (x,y), \end{aligned}$$
  3. 3.

    The system \({\textsf {P}}\) is \(\textsf {E-PA}_{\text {st}}^{\omega *}+\textsf {I}+{\textsf {HAC}}_{{\textsf {int}}}\).

Note that I and \({\textsf {HAC}}_{{\textsf {int}}}\) are fragments of Nelson’s axioms Idealisation and Standard part. By definition, F in (7.2) only provides a finite sequence of witnesses to \((\exists ^{\text {st}}y)\), explaining its name Herbrandized Axiom of Choice.

The system \({\textsf {P}}\) is connected to \(\textsf {E-PA}^{\omega }\) by the following theorem. Here, the superscript ‘\(S_{\text {st}}\)’ is the syntactic translation defined in (van den Berg et al. 2012, Def. 7.1).

Theorem 7.5

Let \({\varPhi }(\underline{a})\) be a formula in the language of E-PA \(^{\omega *}_{\text {st}}\) and suppose \({\varPhi }(\underline{a})^{S_{\text {st}{}}}\equiv \forall ^{\text {st}{}}\underline{x} \, \exists ^{\text {st}{}}\underline{y} \, \varphi (\underline{x}, \underline{y}, \underline{a})\). If \({\varDelta }_{{{\textsf {int}}}}\) is a collection of internal formulas and

$$\begin{aligned} \mathrm{\mathsf{P}} + {\varDelta }_{{{\textsf {int}}}} \vdash {\varPhi }(\underline{a}), \end{aligned}$$
(7.3)

then one can extract from the proof a sequence of closedFootnote 32 terms t in \(\mathcal {T}^{*}\) such that

$$\begin{aligned} \mathrm{\mathsf{E}}\hbox {-}\mathrm{\mathsf{PA}}^{\omega *} + {\varDelta }_{{{\textsf {int}}}} \vdash \ \forall \underline{x} \, \exists \underline{y}\in \underline{t}(\underline{x})\ \varphi (\underline{x},\underline{y}, \underline{a}). \end{aligned}$$
(7.4)

Proof

Immediate by (van den Berg et al. 2012, Theorem 7.7). \(\square \)

The proofs of the soundness theorems in (van den Berg et al. 2012, §5-7) provide an algorithm \(\mathcal {A}\) to obtain the term t from the theorem. In particular, these terms can be ‘read off’ from the nonstandard proofs. The translation \(S_{\text {st}}\) can be formalised in any reasonableFootnote 33 system of constructive mathematics. In fact, the formalisation of the results in van den Berg et al. (2012) in the proof assistant Agda (based on Martin-Löf’s constructive type theory Martin-Löf (1975)) is underway in Xu and Sanders (2015).

In light of the results in Sanders (2016a), the following corollary (which is not present in van den Berg et al. (2012)) is essential to our results. Indeed, the following corollary expresses that we may obtain effective results as in (7.6) from any theorem of Nonstandard Analysis which has the same form as in (7.5). It was shown in Sanders (2016a), Sanders (2015), Sanders (2016c) that the scope of this corollary includes the Big Five systems of Reverse Mathematics and the associated ‘zoo’ ([37]).

Corollary 7.6

If \({\varDelta }_{{{\textsf {int}}}}\) is a collection of internal formulas and \(\psi \) is internal, and

$$\begin{aligned} \mathrm{\mathsf{P}} + {\varDelta }_{{{\textsf {int}}}} \vdash (\forall ^{\text {st}}\underline{x})(\exists ^{\text {st}}\underline{y})\psi (\underline{x},\underline{y}, \underline{a}), \end{aligned}$$
(7.5)

then one can extract from the proof a sequence of closed\(^{32}\) terms t in \(\mathcal {T}^{*}\) such that

$$\begin{aligned} \mathrm{\mathsf{E}}\hbox {-}\mathrm{\mathsf{PA}}^{\omega *} + {\varDelta }_{{{\textsf {int}}}} \vdash (\forall \underline{x})(\exists \underline{y}\in t(\underline{x}))\psi (\underline{x},\underline{y},\underline{a}). \end{aligned}$$
(7.6)

Proof

Clearly, if for internal \(\psi \) and \({\varPhi }(\underline{a})\equiv (\forall ^{\text {st}}\underline{x})(\exists ^{\text {st}}\underline{y})\psi (x, y, a)\), we have \([{\varPhi }(\underline{a})]^{S_{\text {st}}}\equiv {\varPhi }(\underline{a})\), then the corollary follows immediately from the theorem. A tedious but straightforward verification using the clauses (i)-(v) in (van den Berg et al. 2012, Def. 7.1) establishes that indeed \({\varPhi }(\underline{a})^{S_{\text {st}}}\equiv {\varPhi }(\underline{a})\). \(\square \)

For the rest of this paper, the notion ‘normal form’ shall refer to a formula as in (7.5), i.e. of the form \((\forall ^{\text {st}}x)(\exists ^{\text {st}}y)\varphi (x,y)\) for \(\varphi \) internal.

Finally, the previous theorems do not really depend on the presence of full Peano arithmetic. We shall study the following subsystems.

Definition 7.9

  1. 1.

    Let E-PRA \(^{\omega }\) be the system defined in (Kohlenbach 2005, §2) and let E-PRA \(^{\omega *}\) be its definitional extension with types for finite sequences as in (van den Berg et al. 2012, §2).

  2. 2.

    \(({\textsf {QF-AC}}^{\rho , \tau })\) For every quantifier-free internal formula \(\varphi (x,y)\), we have

    $$\begin{aligned} (\forall x^{\rho })(\exists y^{\tau })\varphi (x,y) \rightarrow (\exists F^{\rho \rightarrow \tau })(\forall x^{\rho })\varphi (x,F(x)) \end{aligned}$$
    (7.7)
  3. 3.

    The system \({\textsf {RCA}}_{0}^{\omega }\) is \(\textsf {E-PRA}^{\omega }+{\textsf {QF-AC}}^{1,0}\).

The system \({\textsf {RCA}}_{0}^{\omega }\) is the ‘base theory of higher-order Reverse Mathematics’ as introduced in (Kohlenbach 2005, §2). We permit ourselves a slight abuse of notation by also referring to the system \(\textsf {E-PRA}^{\omega *}+{\textsf {QF-AC}}^{1,0}\) as \({\textsf {RCA}}_{0}^{\omega }\).

Corollary 7.8

The previous theorem and corollary go through for \({\textsf {P}}\) and \(\textsf {{E-PA}}^{\omega *}\) replaced by \({\textsf {P}}_{0}\equiv \textsf {{E-PRA}}^{\omega *}+\mathcal {T}_{\text {st}}^{*} +{\textsf {HAC}}_{{\textsf {int}}} +\textsf {{I}}+{\textsf {QF-AC}}^{1,0}\) and \({\textsf {RCA}}_{0}^{\omega }\).

Proof

The proof of (van den Berg et al. 2012, Theorem 7.7) goes through for any fragment of E-PA \(^{\omega {*}}\) which includes EFA, sometimes also called \(\textsf {I}{\varDelta }_{0}+\textsf {EXP}\). In particular, the exponential function is (all what is) required to ‘easily’ manipulate finite sequences. \(\square \)

The constructive system \({\textsf {H}}\)

In this section, we define the system \({\textsf {H}}\), the constructive counterpart of \({\textsf {P}}\). The system \(\textsf {H}\) was first introduced in (van den Berg et al. 2012, §5.2), and constitutes a conservative extension of Heyting arithmetic \({\textsf {E-HA}}^{\omega } \) by (van den Berg et al. 2012, Cor. 5.6). We now study the system \({\textsf {H}}\) in more detail.

Similar to Definition 7.3, we define \( \textsf {E-HA}^{\omega *}_{\text {st}} \) as \( \textsf {E-HA}^{\omega {*}} + \mathcal {T}^{*}_{\text {st}} + \textsf {IA}^{\text {st}}\), where \(\textsf {E-HA}^{\omega *}\) is just \(\textsf {E-PA}^{\omega *}\) without the law of excluded middle. Furthermore, we define

$$\begin{aligned} {\textsf {H}}\equiv {\textsf {E-HA}}^{\omega *}_{\text {st}}+{\textsf {HAC}}+ {{\textsf {\text {I}}}}+{\textsf {NCR}}+\textsf {HIP}_{\forall ^{\text {st}}}+\textsf {HGMP}^{\text {st}}, \end{aligned}$$

where \({\textsf {HAC}}\) is \({\textsf {HAC}}_{{\textsf {int}}}\) without any restriction on the formula, and where the remaining axioms are defined in the following definition.

Definition 1.28   [Three axioms of \({\textsf {H}}\)]

  1. 1.

    \(\textsf {HIP}_{\forall ^{\text {st}}}\)

    $$\begin{aligned}{}[(\forall ^{\text {st}}x)\phi (x)\rightarrow (\exists ^{\text {st}}y){\varPsi }(y)]\rightarrow (\exists ^{\text {st}}y')[(\forall ^{\text {st}}x)\phi (x)\rightarrow (\exists y\in y'){\varPsi }(y)], \end{aligned}$$

    where \({\varPsi }(y)\) is any formula and \(\phi (x)\) is an internal formula of E-HA \(^{\omega *}\).

  2. 2.

    \(\textsf {HGMP}^{\text {st}}\)

    $$\begin{aligned}{}[(\forall ^{\text {st}}x)\phi (x)\rightarrow \psi ] \rightarrow (\exists ^{\text {st}}x')[(\forall x\in x')\phi (x)\rightarrow \psi ] \end{aligned}$$

    where \(\phi (x)\) and \(\psi \) are internal formulas in the language of E-HA \(^{\omega *}\).

  3. 3.

    NCR

    $$\begin{aligned} (\forall y^{\tau })(\exists ^{\text {st}} x^{\rho } ){\varPhi }(x, y) \rightarrow (\exists ^{\text {st}} x^{\rho ^{*}})(\forall y^{\tau })(\exists x'\in x ){\varPhi }(x', y), \end{aligned}$$

    where \({\varPhi }\) is any formula of E-HA \(^{\omega *}.\)

Intuitively speaking, the first two axioms of Definition 7.9 allow us to perform a number of non-constructive operations (namely Markov’s principle and independence of premises) on the standard objects of the system \({\textsf {H}}\), provided we introduce a ‘Herbrandisation’ as in the consequent of \({\textsf {HAC}}\), i.e. a finite list of possible witnesses rather than one single witness. Furthermore, while \({\textsf {H}}\) includes idealisation I, one often uses the latter’s classical contraposition, explaining why NCR is useful (and even essential) in the context of intuitionistic logic.

Surprisingly, the axioms from Definition 7.9 are exactly what is needed to convert nonstandard definitions (of continuity, integrability, convergence, et cetera) into the normal form \((\forall ^{\text {st}}x)(\exists ^{\text {st}}y)\varphi (x, y)\) for internal \(\varphi \), as is clear from e.g. Sect. 5.3. The latter normal form plays an equally important role in the constructive case as in the classical case by the following theorem.

Theorem 7.10

If \({\varDelta }_{{{\textsf {int}}}}\) is a collection of internal formulas, \(\varphi \) is internal, and

$$\begin{aligned} \mathrm{\mathsf{H}} + {\varDelta }_{{{\textsf {int}}}} \vdash \forall ^{\text {st}{}}\underline{x} \, \exists ^{\text {st}{}}\underline{y} \, \varphi (\underline{x}, \underline{y}, \underline{a}), \end{aligned}$$
(7.8)

then one can extract from the proof a sequence of closed terms t in \(\mathcal {T}^{*}\) such that

$$\begin{aligned} \mathrm{\mathsf{E}}\hbox {-}\mathrm{\mathsf{HA}}^{\omega *} + {\varDelta }_{{{\textsf {int}}}} \vdash \ \forall \underline{x} \, \exists \underline{y}\in \underline{t}(\underline{x})\ \varphi (\underline{x},\underline{y}, \underline{a}). \end{aligned}$$
(7.9)

Proof

Immediate by (van den Berg et al. 2012, Theorem 5.9). \(\square \)

The proofs of the soundness theorems in (van den Berg et al. 2012, §5-7) provide an algorithm \(\mathcal {B}\) to obtain the term t from the theorem. Finally, we point out one very useful principle to which we have access.

Theorem 7.11

The systems P, H, and P \(_{0}\) prove overspill, i.e.

figureu

for any internal formula \(\varphi \).

Proof

See (van den Berg et al. 2012, Prop. 3.3). \(\square \)

In conclusion, we have introduced the systems \({\textsf {H}}\), \({\textsf {P}}\), which are conservative extensions of Peano and Heyting arithmetic with fragments of Nelson’s internal set theory. We have observed that central to the conservation results (Corollary 7.6 and Theorem 7.5) is the normal form \((\forall ^{\text {st}}x)(\exists ^{\text {st}}y)\varphi (x, y)\) for internal \(\varphi \).

Appendix 2: Nonstandard analysis and qualitative information

We list examples of applications of Nonstandard Analysis in which the latter is explicitly used to model qualitative phenomena.

  1. 1.

    Raiman (1986, 1991) introduces a formal language FOG for reasoning with qualitative notions ‘close’, ‘negligible’, and ‘same order of magnitude’. Raiman proves FOG to be validated in Robinson’s Non-standard Analysis Robinson (1966). The system FOG has been used in economics and circuit design (Dague et al. 1987; Bourgine and Raiman 1986).

  2. 2.

    Weld studies the perturbation technique exaggeration in Weld (1990) by means of Nonstandard Analysis. In particular, he uses unlimited and infinitesimal values to study the limit behaviour for ‘large’ and ‘small’ parameter values.

  3. 3.

    Davis presents a system based on Nonstandard Analysis for reasoning with qualitative notions like ‘small’, ‘large’, and ‘medium’. Results in dynamical systems and differential equations are obtained Davis (1990).

  4. 4.

    Suenaga et al. provide deductive verification framework of signals based on Nonstandard Analysis in (Suenaga et al. 2013). Rather than using approximations up to some large finite precision, they use a correct-up-to-infinitesimals approximation using unlimited precision.

  5. 5.

    Vopenka (1991) proposes the use of various nonstandard structures (inside his alternative set theory AST) to model vague phenomena.

  6. 6.

    Tzouvaras (1998) proposes the use of Nonstandard Analysis to model vague notions like ‘similarity’ and ‘small’.

In light of the previous list, we hope the reader is convinced that Nonstandard Analysis can be used to model qualitative notions. The usual caveat applies: We do not claim that this modelling is the best or even accurate; we merely point out that people have used Nonstandard Analysis for this purpose in practice.

Appendix 3: Some theorems relating to term extraction

Theorem 9.1

The system P proves that a normal form can be derived from an implication between normal forms.

Proof

Let \(\varphi , \psi \) be internal and consider the following implication between normal forms:

$$\begin{aligned} (\forall ^{\text {st}}x)(\exists ^{\text {st}}y)\varphi (x, y)\rightarrow (\forall ^{\text {st}}z)(\exists ^{\text {st}}w)\psi (z, w). \end{aligned}$$
(9.1)

Since standard functionals have standard output for standard input by Definition 7.3, (9.1) implies

$$\begin{aligned} (\forall ^{\text {st}}\zeta )\big [(\forall ^{\text {st}}x)\varphi (x, \zeta (x))\rightarrow (\forall ^{\text {st}}z)(\exists ^{\text {st}}w)\psi (z, w)\big ]. \end{aligned}$$
(9.2)

Bringing all standard quantifiers outside, we obtain the following normal form:

$$\begin{aligned} (\forall ^{\text {st}}\zeta , z)(\exists ^{\text {st}} w, x)\big [\varphi (x, \zeta (x))\rightarrow \psi (z, w)\big ], \end{aligned}$$
(9.3)

as the formula in square brackets is internal. \(\square \)

It is an interesting exercise to establish the previous theorem for \({\textsf {H}}\) in the stead of \({\textsf {P}}\).

Theorem 9.2

For internal \(\varphi \), the system P proves that \((\forall \varepsilon \approx 0)(\forall ^{\text {st}}x)(\exists ^{\text {st}}y^{\tau })\varphi (x, y, \varepsilon )\) is equivalent to a normal form.

Proof

Written out in full, the initial formula from the theorem is:

$$\begin{aligned} \textstyle (\forall \varepsilon )\big [(\forall ^{\text {st}} k^{0})(|\varepsilon |<\frac{1}{k})\rightarrow (\forall ^{\text {st}}x)(\exists ^{\text {st}}y^{\tau })\varphi (x, y, \varepsilon )\big ], \end{aligned}$$

and bringing outside all standard quantifiers as far as possible:

$$\begin{aligned} \textstyle (\forall ^{\text {st}}x)\underline{(\forall \varepsilon )(\exists ^{\text {st}}y^{\tau }, k^{0})\big [|\varepsilon |<\frac{1}{k}\rightarrow \varphi (x, y, \varepsilon )\big ]}, \end{aligned}$$

the underlined formula is suitable for Idealisation. Applying the latter yields

$$\begin{aligned} \textstyle (\forall ^{\text {st}}x)(\exists ^{\text {st}}w^{0^{*}}, z^{\tau ^{*}}){(\forall \varepsilon )(\exists y^{\tau } \in z, k^{0}\in w)\big [|\varepsilon |<\frac{1}{k}\rightarrow \varphi (x, y, \varepsilon )\big ]}, \end{aligned}$$

and let \(N^{0}\) be the maximum of all w(i) for \(i<|w|\). We obtain:

$$\begin{aligned} \textstyle (\forall ^{\text {st}}x)(\exists ^{\text {st}}z^{\tau ^{*}}, N){(\forall \varepsilon )(\exists y^{\tau }\in z)\big [|\varepsilon |<\frac{1}{N}\rightarrow \varphi (x, y, \varepsilon )\big ]}. \end{aligned}$$
(9.4)

Clearly, (9.4) implies the initial formula from the theorem. \(\square \)

Appendix 4: Examples in reverse mathematics of the computational content of nonstandard analysis

Theorems equivalent to \({\textsf {ACA}}_{0}\)

In this section, we study the monotone convergence theorem MCT, i.e. the statement that every bounded increasing sequence of reals is convergent, which is equivalent to arithmetical comprehension \({\textsf {ACA}}_{0}\) by (Simpson 2009, III.2.2). We prove an equivalence between a nonstandard version of \(\mathrm {\textsf {MCT}}\) and a fragment of Transfer. From this nonstandard equivalence, we obtain an effective RM equivalence involving \(\mathrm {\textsf {MCT}}\) and arithmetical comprehension.

Firstly, the nonstandard version of \(\mathrm {\textsf {MCT}}\) (involving nonstandard convergence) is:

figurev

where ‘\((\forall K\in {\varOmega })(\dots )\)’ is short for \((\forall K^{0}))(\lnot \text {st}(K)\rightarrow \dots )\). The effective version MCT \(_{\textsf {ef}}(t)\):

$$\begin{aligned} \textstyle (\forall c_{(\cdot )}^{0\rightarrow 1},k^{0})\big [(\forall n^{0})(c_{n}\le c_{n+1}\le 1)\rightarrow (\forall N,M\ge t(c_{(\cdot )})(k))[|c_{M}- c_{N}|\le \frac{1}{k} ] \big ]. \end{aligned}$$
(10.1)

We require two equivalent (Kohlenbach 2005, Prop. 3.9) versions of arithmetical comprehension:

figurew
figurex

Clearly, \((\exists ^{2})\) (and therefore \((\mu ^{2})\)) is the functional version of \({\textsf {ACA}}_{0}\). We also recall the restriction of Nelson’s axiom Transfer as follows:

figurey

Denote by \(\textsf {MU}(\mu )\) the formula in square brackets in \(\mu ^{2}\). We have the following theorem which establishes the explicit equivalence between \((\mu ^{2})\) and uniform \(\mathrm {\textsf {MCT}}\).

Theorem 10.1

We have P \(\vdash \) MCT \(_{{\textsf {ns}}}\leftrightarrow {\varPi }_{1}^{0}\)-TRANS. From this proof, terms su can be extracted such that E-PA \(^{\omega *}\) proves:

(10.2)

Proof

See (Sanders 2016a, §4.1). \(\square \)

We point out (10.2) is the ‘effective’ version of the equivalence \({\textsf {ACA}}_{0}\leftrightarrow \mathrm {\textsf {MCT}}\); the former is obtained from the corresponding ‘nonstandard’ equivalence \({\varPi }_{1}^{0}{-\textsf {TRANS}}\leftrightarrow \mathrm {\textsf {MCT}}_{{\textsf {ns}}}\). Note that the latter proof proceeds by contradiction.

Finally, while we did not emphasise this in Sect. 2, Reverse Mathematics usually studies mathematical theorems formalised in second-order arithmetic. The latter only involves natural numbers and sets thereof, i.e. continuous functions on the real numbers are indirectly present in the form of codes (See Kohlenbach 2002, §4). Now, (10.2) is clearly not part of second-order arithmetic (as it involves objects of type two), but it is possible to obtain results in second-order arithmetic from (10.2), as explored in Sanders (2017).

Theorems equivalent to \({\textsf {ATR}}_{0}\) and \({\varPi }_{1}^{1}{\hbox {-}\textsf {CA}}_{0}\)

In this section, we study equivalences relating to \({\textsf {ATR}}_{0}\) and \({\varPi }_{1}^{1}{\hbox {-}\textsf {CA}}_{0}\), the strongest Big Five systems. We shall work with the Suslin functional \((S^{2})\), the functional version of \({\varPi }_{1}^{1}{\hbox {-}\textsf {CA}}_{0}\).

figurez

Feferman has introduced the following version of the Suslin functional (See e.g. Avigad and Feferman (1998)).

figureaa

where the formula in square brackets is \({\textsf {MUO}}(\mu _{1})\). We require another instance of Transfer:

figureab

We first consider \({\textsf {PST}}\), i.e. the statement that every tree with uncountably many paths has a non-empty perfect subtree. As proved in (Simpson (2009), V.5.5), we have \({\textsf {PST}}\leftrightarrow {\textsf {ATR}}_{0}\) and a uniform version of \({\textsf {PST}}\) is equivalent to the Suslin functional by (Sakamoto and Yamazaki 2004, Theorem 4.4). Now, \({\textsf {PST}}\) has the following nonstandard and uniform versions.

Theorem 10.2

(PST \(_{{{\mathbf {\mathsf{{ns}}}}}}\)) For all standard trees \(T^{1}\), there is standard \(P^{1}\) such that

$$\begin{aligned} (\forall f_{(\cdot )}^{0\rightarrow 1})(\exists f\in T)(\forall n)(f_{n}\ne _{1} f) \rightarrow P \text {is a non-empty perfect subtree of} T. \end{aligned}$$

Theorem 10.3

(PST \(_{{{\mathbf {\mathsf{{ef}}}}}}(t)\)) For all trees \(T^{1}\), we have

$$\begin{aligned} (\forall f_{(\cdot )}^{0\rightarrow 1})(\exists f\in T)(\forall n)(f_{n}\ne _{1} f) \rightarrow t(T)\text { is a non-empty perfect subtree of} T. \end{aligned}$$

As a technicality, we require that P as in the previous two principles consists of a pair \((P', p')\) such that \(P'\) is a perfect subtree of T such that \(p'\in P'\).

Theorem 10.4

We have P \(\vdash \) PST \(_{{\textsf {ns}}}\leftrightarrow {\varPi }_{1}^{1}\)-TRANS. From the latter, terms su can be extracted such that \({\textsf {E-PA}}^{\omega *}\) proves:

(10.3)

Proof

See (Sanders 2016a, §4.5). \(\square \)

In conclusion, (10.3) is the ‘effective’ version of (Sakamoto and Yamazaki 2004, Theorem 4.4); the former is obtained from the corresponding ‘nonstandard’ equivalence \({\varPi }_{1}^{1}\hbox {-}{\textsf {TRANS}}\leftrightarrow {\varPi }_{1}^{1}\)-TRANS. Note that the latter proof proceeds by contradiction.

Another more mathematical statement which can be treated along the same lines is every countable Abelian group is a direct sum of a divisible and a reduced group. The latter is called \({\textsf {DIV}}\) and equivalent to \({\varPi }_{1}^{1}{\hbox {-}\textsf {CA}}_{0}\) by (Simpson 2009, VI.4.1). By the proof of the latter, the reverse implication is straightforward; We shall therefore study \({\textsf {DIV}}\rightarrow {\varPi }_{1}^{1}{\hbox {-}\textsf {CA}}_{0}\).

To this end, let \({\textsf {DIV}}(G, D, E)\) be the statement that the countable Abelian group G satisfies \(G=D\oplus E \), where D is a divisible group and E a reduced group. The nonstandard version of \({\textsf {DIV}}\) is as follows:

figureac

where we used the same technicality as for PST \(_{{\textsf {ns}}}\). The effective version is:

figuread

We have the following theorem.

Theorem 10.5

We have have P \(\vdash \) DIV \(_{{\textsf {ns}}}\rightarrow {\varPi }_{1}^{1}\)-TRANS. From the latter, a term u can be extracted such that E-PA \(^{\omega *}\) proves:

(10.4)

Proof

See (Sanders 2015, §4.5). \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Sanders, S. Reverse formalism 16. Synthese (2017). https://doi.org/10.1007/s11229-017-1322-2

Download citation

Keywords

  • Abraham Robinson
  • Formalism
  • Nonstandard analysis
  • Computational content