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ON THE UNCOUNTABILITY OF ${\mathbb R}$

Part of: Proof theory and constructive mathematics General logic Computability and recursion theory

Published online by Cambridge University Press:  07 April 2022

DAG NORMANN  and
SAM SANDERS
DAG NORMANN
Affiliation:
DEPARTMENT OF MATHEMATICS THE UNIVERSITY OF OSLO P.O. BOX 1053, BLINDERN N-0316 OSLO, NORWAY E-mail: dnormann@math.uio.no
SAM SANDERS*
Affiliation:
DEPARTMENT OF PHILOSOPHY II RUHR-UNIVERSITÄT BOCHUM (RUB) BOCHUM, GERMANY
*
E-mail: sasander@me.com
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Abstract

Cantor’s first set theory paper (1874) establishes the uncountability of ${\mathbb R}$. We study this most basic mathematical fact formulated in the language of higher-order arithmetic. In particular, we investigate the logical and computational properties of ${\mathsf {NIN}}$ (resp. ${\mathsf {NBI}}$), i.e., the third-order statement there is no injection resp. bijection from $[0,1]$ to ${\mathbb N}$. Working in Kohlenbach’s higher-order Reverse Mathematics, we show that ${\mathsf {NIN}}$ and ${\mathsf {NBI}}$ are hard to prove in terms of (conventional) comprehension axioms, while many basic theorems, like Arzelà’s convergence theorem for the Riemann integral (1885), are shown to imply ${\mathsf {NIN}}$ and/or ${\mathsf {NBI}}$. Working in Kleene’s higher-order computability theory based on S1–S9, we show that the following fourth-order process based on ${\mathsf {NIN}}$ is similarly hard to compute: for a given $[0,1]\rightarrow {\mathbb N}$-function, find reals in the unit interval that map to the same natural number.

Keywords

Uncountability of ℝReverse MathematicsKleene S1–S9higher-order computability theory

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics)
Secondary: 03F35: Second- and higher-order arithmetic and fragments 03D55: Hierarchies 03D30: Other degrees and reducibilities

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Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 4 , December 2022 , pp. 1474 - 1521
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© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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