, it also belongs to
, the successor of
. Similarly a class is said to be "hereditary" if, whenever
is a member of the class, so is
. It is easy to see, though we are not yet supposed to know, that to say a property is hereditary is equivalent to saying that it belongs to all the natural numbers not less than some one of them, e.g. it must belong to all that are not less than 100, or all that are less than 1000, or it may be that it belongs to all that are not less than 0, i.e. to all without exception.
A property is said to be "inductive" when it is a hereditary property which belongs to 0. Similarly a class is "inductive" when it is a hereditary class of which 0 is a member.