In the Introduction to Mathematical Philosophy (1, p.26), Bertrand Russell defines natural numbers in the following way:
The “natural numbers” are the posterity of $0$ with respect to the relation “immediate predecessor” (which is the converse of “successor”).
Where:
[...] the “posterity” of a given natural number with respect to the relation “immediate predecessor” (which is the converse of “successor”) as all those terms that belong to every hereditary class to which the given number belongs.
Now, a class may be defined by extension (enumeration) or by intention (defining property). I don't see how we can use extension in the above definition, so I assume Russell means every hereditary class defined by any property to which the given number ($0$) belongs.
How do we know $0$ possesses any such property?
PS. I found similar question on SE (What does Russell mean when he defines the “Posterity… with respect to the immediate predecessor”?). However, there is no mention about defining properties.