First, here are some definition:
$A$ is finite set - there exists bijective function between $[n]=\{0,\ldots,n-1\}$ and $A$
$A$ is infinite set - $A$ not finite
$A$ is Dedekind infinite - there exists bijective function between $A$ and a proper subset of $A$
$A$ is Dedekind finite - $A$ is not Dedekind infinite
I learned that without assuming choice(or some weak version of choice) the order of cardinality(exists injective function from $A$ to $B$) is not connex; exists $A,B$ such that no $|A|\le |B|$ and $|B|\le |A|$.
When I asked, I was told that such $A,B$ are sets such that $|A|\ge|\Bbb N|$ and $B$ is Dedekind finite and infinite, this made me think for example of such $A,B$, but I couldn't think on set in the form of $B$.
Is there a simple example for such $A,B$? And how to prove that there is no one to one function from either direction?