I want to describe a system with two non-interacting and definitely different fermions, say an electron neutrino, $\nu_e$, and an electron, $e^-$.
The state describing a single electron is given using the Dirac creation operator $$c^\dagger_e|0\rangle$$ where $|0\rangle$ is the vacuum state.
Similarly the state describing a single neutrino is $$c_\nu^\dagger|0\rangle$$
My question is what is the state with both fermions present? Is it simply $$c^\dagger_ec^\dagger_\nu|0\rangle$$ If so, does it matter which way around the two creation operators are? If not, surely that means different fermion operators commute, since I could have just as well have chosen to write $$c^\dagger_\nu c^\dagger_e|0\rangle$$ Other questions on this topic (e.g. this question) suggest to me that they should anticommute, and in general I've come to believe that the whole point of these operators and what makes them so special is they anticommute with just about anything.