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- Title
Coding of real‐valued continuous functions under WKL$\mathsf {WKL}$.
- Authors
Kawai, Tatsuji
- Abstract
In the context of constructive reverse mathematics, we show that weak Kőnig's lemma (WKL$\mathsf {WKL}$) implies that every pointwise continuous function f:[0,1]→R$f : [0,1]\rightarrow \mathbb {R}$ is induced by a code in the sense of reverse mathematics. This, combined with the fact that WKL$\mathsf {WKL}$ implies the Fan theorem, shows that WKL$\mathsf {WKL}$ implies the uniform continuity theorem: every pointwise continuous function f:[0,1]→R$f : [0,1]\rightarrow \mathbb {R}$ has a modulus of uniform continuity. Our results are obtained in Heyting arithmetic in all finite types with quantifier‐free axiom of choice.
- Subjects
CONTINUOUS functions; CONSTRUCTIVE mathematics; REVERSE mathematics; MODULAR arithmetic; CONTINUITY
- Publication
Mathematical Logic Quarterly, 2023, Vol 69, Issue 3, p370
- ISSN
0942-5616
- Publication type
Article
- DOI
10.1002/malq.202200031