Computability and Non-monotone induction

Normann, Dag

doi:10.48550/arxiv.2006.03389

Cited by 1 publication

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References 22 publications

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“…Example 6.22 (Non-monotone inductive definitions). The first author has studied the computational properties of the Heine-Borel theorem and the Lindel öf lemma in relation to nonmonotone inductive definitions in [82,83]. The latter notion expresses the iteration of functionals along countable ordinals, which is not easily expressed in weak systems like RCA 0 .…”

Section: Computing De Dicto and De Rementioning

confidence: 99%

On the Uncountability Of

Normann¹,

Sanders²

2022

J. symb. log.

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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.

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Section: Computing De Dicto and De Rementioning

confidence: 99%