Peano's successor function

Question

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?

Answer 1

It is harder to work with. For example, we can "check" whether an arbitrary set $x$ is a von Neumann natural:

$x$ is finite and transitive and well-ordered by $\in$.

where finite can be formalized as, e.g., "every injective map $x\to x$ is onto" and transitive is defined as "every element is also a subset". Try to find a similar characterization for the alternative (without using "$\ldots$" anywhere). Or try to find a simple way to express the order relation between natural numbers in a simple way for the sets representing the numbers (for von Neumann we have $x<y$ iff $x\in y$ iff $x \subsetneq y$).

Answer 2

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.