Peano's successor function [duplicate]

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• The history of set-theoretic definitions of $\mathbb N$ 2 answers

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?

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