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On the uncountability of \(\mathbb{R}\). (English) Zbl 1523.03004

This article continues the authors’ exploration of higher-order reverse mathematics and computability theory. The focus here is on principles asserting the non-existence of injections (or bijections) from \([0,1]\) to the natural numbers. These principles are independent of surprisingly strong comprehension axioms. For example, \(\mathsf{NIN}\) (non-existence of an injection) is not provable in \(\mathbb{Z}^\omega_2\), a higher-order extension of full second-order arithmetic. On the other hand, \(\mathsf{NIN}\) is a consequence of higher-order formulations of well-known theorems, for example, the Baire category theorem as stated in Section 6 of [the authors, J. Log. Comput. 30, No. 8, 1639–1679 (2020; Zbl 1472.03012)]. The results here lay the foundation for analysis of other results in measure and category, as shown in the preprint of S. Sanders [“Big in reverse mathematics: measure and category”, Preprint, arXiv:2303.00493].

MSC:

03B30 Foundations of classical theories (including reverse mathematics)
03F35 Second- and higher-order arithmetic and fragments
03D65 Higher-type and set recursion theory
03D55 Hierarchies of computability and definability
03D30 Other degrees and reducibilities in computability and recursion theory

Citations:

Zbl 1472.03012

References:

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