Firstly, let's establish what exactly I mean by these symbols. Let $\omega_0 = \{ 0, 1, 2, \ldots \}$, where $0, 1, 2, \ldots$ are the usual von Neumann representations of the natural numbers. Let $n$ be a finite natural number. For each $n$, define $\omega_{n+1} = \sup S$, where $S$ the image of the set $R$ under the function-class mapping well-orders to their von Neumann ordinal, and $R \subset \mathcal{P}(\omega_n \times \omega_n)$ is the set of all well-orders on $\omega_n$. We define $\aleph_n = \omega_n$.
Unless I'm mistaken, this establishes the existence of $\omega_n$ for all $n \in \mathbb{N}$ as sets under the axioms of ZF. It's straightforward to see that there is no injection $\omega_{n+1} \to \omega_n$, as that would establish (via pullback) that $|\omega_{n+1}| \le \aleph_n$, and this is a contradiction as $\omega_{n+1}$ is strictly greater than all ordinals in $S$. This in turn implies, by the axiom of choice, that there is no surjection $\omega_n \to \omega_{n+1}$, and the conclusion that there is no surjection $\aleph_n \to \aleph_{n+1}$ follows.
My question is: Can this be done, using my definitions above, without the axiom of choice? I'm willing to accept reasonable alternative definitions, provided that they don't render the conclusion tautological.
(This is a self-imposed extension to a homework problem: Should I tag with homework?)