which is endless, contains no repetitions, has a beginning, and has no terms that cannot be reached from the beginning in a finite number of steps, we have a set of terms verifying Peano's axioms. This is easily seen, though the formal proof is somewhat long. Let "0" mean
, let "number" mean the whole set of terms, and let the "successor" of
mean
. Then
(1) "0 is a number," i.e.