That the axiom (schema) of substitution is stated last and so late in the book is testament to how much elementary set theory—and indeed mathematics more generally—can be done without it. As a very simple example of what is is needed for, the von Neumann ordinal (that is, the second limit ordinal) cannot be proved to be a set using only axioms 1.-8., even though sets with well-orderings with this order type can be constructed from these axioms. For instance , with an ordering placing all elements of the first copy of less than the second. Working with von Neumann ordinals in place of generic well-orderings has technical advantages, not least the fact every well-ordering is order isomorphic to a unique von Neumann ordinal.
That the axiom (schema) of substitution is stated last and so late in :the book is testament to how much elementary set theory—and indeed mathematics more generally—can be done w.ithout it. As a very simple example of what is is needed for, the von Neumann ordinal (that is, the second limit ordinal) cannot be proved to be a set using only axioms 1.-8., even though. sets with well-orderings with this order type can be constructed from these axiom:s. For instance , with an ordering placing all elements of the first copy of less than the second. Working with von Neumann ordinals in place of generic well-orderings has technical advantages, not least the fact every well-ordering is order isomorphic to a unique von Neumann ordinal.