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A Local-Global Principle for the Real Continuum

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Contradictions, from Consistency to Inconsistency

Part of the book series: Trends in Logic ((TREN,volume 47))

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Abstract

We discuss the implications of a local-global (or global-limit) principle for proving the basic theorems of real analysis. The aim is to improve the set of available tools in real analysis, where the local-global principle is used as a unifying principle from which the other completeness axioms and several classical theorems are proved in a fairly direct way. As a consequence, the study of the local-global concept can help establish better pedagogical approaches for teaching classical analysis.

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Notes

  1. 1.

    Precise definitions will be given in Sect. 4. Some of the statements require the Archimedean property: Any real number is upper bounded by a natural number.

  2. 2.

    Possibly infinite, e.g., \(\sup \;\mathbb {R} =\inf \;\varnothing =+\infty \), \(\sup \;\varnothing =\inf \;\mathbb {R} =-\infty \), \(\lim \;\pm n=\pm \infty \).

  3. 3.

    This is Borel’s statement, also (somewhat wrongly) attributed to Heine, and later generalized by Lebesgue and others [1].

  4. 4.

    Although proposed at the same time as Borel’s BL, Cousin has been largely overlooked since. It was only recently re-exhumed as a fundamental lemma for deriving the gauge (Kurzweil–Henstock) integral (e.g. [13]). Ind is much more recent and in fact inspired from LG (see [10, 19]).

  5. 5.

    A non-degenerate interval. Hence two adjacent intervals [u, v] and [v, w] (where \(u<v<w\)) are not overlapping since their intersection is reduced to a point.

  6. 6.

    Even though the proposed axioms appear to be second-order statements since they are quantified over properties of sets, in fact any property is simply identified to a family of subintervals. Therefore, the axioms only require the basic (first order) ZF theory (with or without the axiom of choice), as is usual when teaching real analysis at an elementary level.

  7. 7.

    We found it convenient for later developments that functions assume only finite values in order to leverage on the complete metric space property of the set of function values.

  8. 8.

    A literal translation of the French “inégalité des accroissements finis”, which advantageous replaces the “théorème des accroissements finis”, which is the mean value theorem.

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Acknowledgements

The authors would like to thank Pete Clark for his detailed review of the paper and his many useful comments and suggestions.

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Authors and Affiliations

  1. Télécom ParisTech, Université Paris-Saclay, Paris, France

    Olivier Rioul

  2. Faculdade de Tecnologia, Unicamp, Limeira, Brasil

    José Carlos Magossi

Authors
  1. Olivier Rioul
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  2. José Carlos Magossi
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Correspondence to Olivier Rioul .

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Editors and Affiliations

  1. Department of Philosophy, Centre for Logic, Epistemology and the History of Science, University of Campinas-Unicamp, Campinas, São Paulo, Brazil

    Walter Carnielli

  2. Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland

    Jacek Malinowski

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Rioul, O., Magossi, J.C. (2018). A Local-Global Principle for the Real Continuum. In: Carnielli, W., Malinowski, J. (eds) Contradictions, from Consistency to Inconsistency. Trends in Logic, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-319-98797-2_11

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