Look up ‘transitive sets’ if you haven’t heard of them already. The primary answer to your question is that in the von Neumann definition the ordinals are tran

Look up ‘transitive sets’ if you haven’t heard of them already. The primary answer to your question is that in the von Neumann definition the ordinals are transitive and well ordered by the epsilon (membership) relation, which is a pair of stipulations that can then be used as a definition of the ordinals if you like. This in turn is nice for many other reasons! Also, you can make convenient definitions like defining the supremum of a set of ordinals to be the union of that set. The union of two of your ordinals usually won’t be an ordinal. In general, it’s nice to have an ordinal simply be the set of its predecessors, which is something that this definition implies.