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On the Computational Properties of the Uncountability of the Real Numbers

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Logic, Language, Information, and Computation (WoLLIC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13468))

Abstract

The uncountability of the real numbers is one of their most basic properties, known (far) outside of mathematics. Cantor’s 1874 proof of the uncountability of the real numbers even appears in the very first paper on set theory, i.e. a historical milestone. Despite this famous status and history, the computational properties of the uncountability of the real numbers have not been studied much. In this paper, we study the following computational operation that witnesses that the real numbers not countable:

$$ on\; input\; a \;countable \;set\; of\; reals,\; output\; a\; real\; not\; in\; that\; set. $$

In particular, we formulate a considerable number of operations that are computationally equivalent to the centred operation, working in Kleene’s higher-order computability theory based on his S1-S9 computation schemes. Perhaps surprisingly, our equivalent operations involve most basic properties of the Riemann integral and Volterra’s early work circa 1881.

This research was supported by the Deutsche Forschungsgemeinschaft (DFG) via the grant Reverse Mathematics beyond the Gödel hierarchy (SA3418/1-1).

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Notes

  1. 1.

    The functional \(\textsf {S} _{k}^{2}\) from Sect. 1.3.2 can decide \(\Pi _{k}^{1}\)-formulas, but the centred operation is not computable in \(\textsf {S} _{k}^{2}\) (or their union). Kleene’s \(\exists ^{3}\) from Sect. 1.3.2 computes the centred operation, but the former also yields full second-order arithmetic..

  2. 2.

    If \(A_{n}\) were infinite, the Bolzano-Weierstrass theorem implies the existence of a limit point \(y\in [0,1]\) for \(A_{n}\). One readily shows that \(f(y+)\) or \(f(y-)\) does not exist, a contradiction as f is assumed to be regulated.

  3. 3.

    A point \(x\in [0,1]\) is a strict local maximum of \(f:[0,1]\rightarrow \mathbb {R}\) in case \((\exists N\in \mathbb {N})( \forall y \in B(x, \frac{1}{2^{N}}))(y\ne x\rightarrow f(y)<f(x))\).

  4. 4.

    If \(g\in C([0,1])\), then \(x\in [0,1]\) is a strict local maximum iff for some \(\epsilon \in \mathbb {Q}^{+}\):

    • \(g(y) < g(x)\) whenever \(|x-y] < \epsilon \) for any \(q\in [0,1]\cap \mathbb {Q}\), and:

    • \(\sup _{y\in [a,b]}g(y) < g(x)\) whenever \(x \not \in [a,b]\), \(a,b\in \mathbb {Q}\) and \([a,b] \subset [x - \epsilon ,x + \epsilon ]\).

    Note that \(\mu ^{2}\) readily yields \(N\in \mathbb {N}\) such that \((\forall y\in B(x, \frac{1}{2^{N}}))( g(y)<g(x))\)..

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Acknowledgement

We thank Anil Nerode for his valuable advice. Our research was supported by the Deutsche Forschungsgemeinschaft via the DFG grant SA3418/1-1. Initial results were obtained during the stimulating MFO workshop (ID 2046) on proof theory and constructive mathematics in Oberwolfach in early Nov. 2020. We express our gratitude towards the aforementioned institutions.

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  1. Department of Philosophy II, RUB Bochum, Bochum, Germany

    Sam Sanders

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  1. Sam Sanders
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  1. Technische Universität Wien, Vienna, Austria

    Agata Ciabattoni

  2. University College London, London, UK

    Elaine Pimentel

  3. Universidade Federal de Pernambuco, Recife, Pernambuco, Brazil

    Ruy J. G. B. de Queiroz

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Sanders, S. (2022). On the Computational Properties of the Uncountability of the Real Numbers. In: Ciabattoni, A., Pimentel, E., de Queiroz, R.J.G.B. (eds) Logic, Language, Information, and Computation. WoLLIC 2022. Lecture Notes in Computer Science, vol 13468. Springer, Cham. https://doi.org/10.1007/978-3-031-15298-6_23

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  • DOI: https://doi.org/10.1007/978-3-031-15298-6_23

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