 An ordering of type \(\omega\) has the same length as the standard natural numbers. We can prove that there is such an ordering: consider the order on the finite (i.e., standard finite) von Neumann ordinals. There must be infinite von Neumann ordinals because there is a set theoretically definable bijection between the von Neumann ordinals and the whole universe of sets: any proper semiset can be converted to a proper semiset of a set of von Neumann ordinals.