On the Computational Properties of the Uncountability of the Real Numbers

Sanders, Sam

doi:10.1007/978-3-031-15298-6_23

Logic, Language, Information, and Computation

2022

DOI: 10.1007/978-3-031-15298-6_23

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On the Computational Properties of the Uncountability of the Real Numbers

Sam Sanders

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Cited by 2 publications

References 21 publications

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On Robust Theorems Due to Bolzano, Weierstrass, Jordan, and Cantor

Normann¹,

Sanders²

2022

J. symb. log.

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Reverse Mathematics (RM hereafter) is a program in the foundations of mathematics where the aim is to identify the minimal axioms needed to prove a given theorem from ordinary, i.e., non-set theoretic, mathematics. This program has unveiled surprising regularities: the minimal axioms are very often equivalent to the theorem over the base theory, a weak system of ‘computable mathematics’, while most theorems are either provable in this base theory, or equivalent to one of only four logical systems. The latter plus the base theory are called the ‘Big Five’ and the associated equivalences are robust following Montalbán, i.e., stable under small variations of the theorems at hand. Working in Kohlenbach’s higher-order RM, we obtain two new and long series of equivalences based on theorems due to Bolzano, Weierstrass, Jordan, and Cantor; these equivalences are extremely robust and have no counterpart among the Big Five systems. Thus, higher-order RM is much richer than its second-order cousin, boasting at least two extra ‘Big’ systems.

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On Robust Theorems Due to Bolzano, Weierstrass, Jordan, and Cantor

Normann¹,

Sanders²

2022

J. symb. log.

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Big in Reverse Mathematics: The Uncountability Of the reals

Sanders

2023

J. symb. log.

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The uncountability of $\mathbb {R}$ is one of its most basic properties, known far outside of mathematics. Cantor’s 1874 proof of the uncountability of $\mathbb {R}$ even appears in the very first paper on set theory, i.e., a historical milestone. In this paper, we study the uncountability of ${\mathbb R}$ in Kohlenbach’s higher-order Reverse Mathematics (RM for short), in the guise of the following principle: $$\begin{align*}\mathit{for \ a \ countable \ set } \ A\subset \mathbb{R}, \mathit{\ there \ exists } \ y\in \mathbb{R}\setminus A. \end{align*}$$ An important conceptual observation is that the usual definition of countable set—based on injections or bijections to ${\mathbb N}$ —does not seem suitable for the RM-study of mainstream mathematics; we also propose a suitable (equivalent over strong systems) alternative definition of countable set, namely union over ${\mathbb N}$ of finite sets; the latter is known from the literature and closer to how countable sets occur ‘in the wild’. We identify a considerable number of theorems that are equivalent to the centred theorem based on our alternative definition. Perhaps surprisingly, our equivalent theorems involve most basic properties of the Riemann integral, regulated or bounded variation functions, Blumberg’s theorem, and Volterra’s early work circa 1881. Our equivalences are also robust, promoting the uncountability of ${\mathbb R}$ to the status of ‘big’ system in RM.

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On the Computational Properties of the Uncountability of the Real Numbers

Cited by 2 publications

References 21 publications

On Robust Theorems Due to Bolzano, Weierstrass, Jordan, and Cantor

On Robust Theorems Due to Bolzano, Weierstrass, Jordan, and Cantor

Big in Reverse Mathematics: The Uncountability Of the reals

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