TITLE: Infinite sets with cardinality less than the natural numbers
QUESTION [16 upvotes]: Are there any infinite sets that have a lower cardinality than the natural numbers? Is there a proof of this?

REPLY [6 votes]: Every set $X$ that has cardinality at most that of $\mathbb{N}$, i.e. such that $X$ injects into $\mathbb{N}$, is either finite or has the same cardinality as $\mathbb{N}$ by the argument in Asaf's answer.
Moreover assuming the Axiom of Choice or even just a weak fragment thereof, every set $X$ that does not have cardinality at least that of $\mathbb{N}$, i.e. such that $\mathbb{N}$ does not inject into $X$, is finite by the argument in user103567's answer.
However, perhaps it is worth pointing out that it is consistent with $\mathsf{ZF}$ set theory, which is like the usual theory $\mathsf{ZFC}$ but without the Axiom of Choice, that there are infinite sets $X$ whose cardinality is incomparable with that of $\mathbb{N}$; i.e. neither $X$ nor $\mathbb{N}$ injects into the other.
Such pathological sets would be infinite and Dedekind finite: see here for the definitions.