TITLE: Peano's successor function
QUESTION [7 upvotes]: I have been trying to get my head round ZF set theory and Peano's axioms, but I have hit some confusion over Peano's definition of the successor function, or more accurately von Neumann's model. 
Why did von Neumann use $S(x) := x \bigcup \{x\} $ and not just plain old $S(x) := \{x\} $? The latter seems a lot more simple and easy to work with, so am I missing some major advantage of the former, or is my function incompatible in some way?
Edit: is the only advantage just that the cardinality of the set is equal to the value it represents?

REPLY [5 votes]: The most immediately obvious benefit of the Von Neumann representation is that the set that represents the number $n$ has exactly $n$ elements. This makes it technically easy to use the representation to reason about counting.
It also provides a connection to a "naive" definition of numbers where, for example, the number two is regarded as "the property that a set may have that it has one more than one element", and represented by the set of all sets that have exactly two elements. Unfortunately such a set can't actually exist in standard set theory, but the Von Neumann definition points to a particular representative of this class that we can use to represent it instead.
A more advanced -- but technically quite important -- benefit of the Von Neumann naturals is that they generalize directly to a representation of transfinite ordinal numbers. Representing finite numbers as towers of singletons has no natural continuation beyond the finite ones, but the Von Neymann representation does.