Can we simplify the classification of infinite Dedekind-finite subsets of $\mathbb{N}$?

Question

In ZF, does there exist a bijection between the infinite subsets X of $\mathbb{N}$ for which there exists no injection $\mathbb{N}\rightarrow X$ and the set of infinite Dedekind-finite subsets of $\mathbb{N}$? If so, can we construct one and what would it look like?

Answer 1

The answer is yes, for the disappointing reason that both families are empty.

There are no infinite Dedekind-finite subsets of $\mathbb{N}$, since it is well ordered. If $X\subseteq\mathbb{N}$ is infinite, then $X$ is equinumerous with $\mathbb{N}$.