At this point, we can hypothesize that the operators that create fermion states do not commute.
In fact, if we assume that the operators creating fermion states anti-commute (as do the Pauli matrices),
then we can show that fermion states are antisymmetric under interchange.
are the creation and annihilation operators for fermions and that they anti-commute.
The states are then antisymmetric under interchange of pairs of fermions.
Its not hard to show that the occupation number for fermion states is either zero or one.