 Moreover, Frege recognized the need to employ the Principle of Mathematical Induction in the proof that every number has a successor. One cannot prove the claim that every number has a successor simply by producing the sequence of expressions for cardinal numbers (e.g., the second of the two sequences described above). All such a sequence demonstrates is that for every expression listed in the sequence, one can define an expression of the appropriate form to follow it in the sequence. This is not the same as proving that  every natural number has a successor.