Editing Definition (section)

===Recursive definitions===
A [[recursive definition]], sometimes also called an ''inductive'' definition, is one that defines a word in terms of itself, so to speak, albeit in a useful way. Normally this consists of three steps:
# At least one thing is stated to be a member of the set being defined; this is sometimes called a "base set".
# All things bearing a certain relation to other members of the set are also to count as members of the set. It is this step that makes the definition [[Recursion|recursive]].
# All other things are excluded from the set

For instance, we could define a [[natural number]] as follows (after [[Peano axioms|Peano]]): 
# "0" is a natural number.
# Each natural number has a unique successor, such that:
#* the successor of a natural number is also a natural number;
#* distinct natural numbers have distinct successors;
#* no natural number is succeeded by "0".
# Nothing else is a natural number.

So "0" will have exactly one successor, which for convenience can be called "1". In turn, "1" will have exactly one successor, which could be called "2", and so on. The second condition in the definition itself refers to natural numbers, and hence involves [[self-reference]]. Although this sort of definition involves a form of [[Circular definition|circularity]], it is not [[Vicious circle principle|vicious]], and the definition has been quite successful.

In the same way, we can define [[ancestor]] as follows:
#A parent is an ancestor.
#A parent of an ancestor is an ancestor.
#Nothing else is an ancestor.
Or simply: an ancestor is a parent or a parent of an ancestor.