The Myhill isomorphism theorem does not generalize much

TL;DR

This paper examines the limitations of extending the Myhill isomorphism theorem beyond natural numbers, showing it can be extended to conatural numbers under certain conditions but generally does not extend to other infinite sets.

Contribution

The paper demonstrates the extent and limitations of extending the Myhill isomorphism theorem to various infinite sets, highlighting essential restrictions and impossibility results.

Findings

Extension to conatural numbers under Markov's principle with bicomplemented sets.

Impossibility to extend the theorem to many other infinite sets.

Restriction to bicomplemented sets is essential for the extension.

Abstract

The Myhill isomorphism is a variant of the Cantor-Bernstein theorem. It states that, from two injections that reduces two subsets of $\mathbb{N}$ to each other, there exists a bijection $\mathbb{N} \to \mathbb{N}$ that preserves them. This theorem can be proven constructively. We investigate to which extent the theorem can be extended to other infinite sets other than $\mathbb{N}$. We show that, assuming Markov's principle, the theorem can be extended to the conatural numbers $\mathbb{N}_{\infty}$ provided that we only require that bicomplemented sets are preserved by the bijection. This restriction is essential. Otherwise, the picture is overall negative: among other things, it is impossible to extend that result to either $2 \times \mathbb{N}_{\infty}$, $\mathbb{N} + \mathbb{N}_{\infty}$, $\mathbb{N} \times \mathbb{N}_{\infty}$, $\mathbb{N}_{\infty}^2$, $2^{\mathbb{N}}$ or $\mathbb{N}^{\mathbb{N}}$.