I'm learning some about Set Theory of my own and I'm following the appendix of Kelley's General Topology. In his development, he constructs first the ordinals, and then the naturals and the cardinals. He never mentions Transfinite recursion theorem nor Transfinite induction principle for that construction (as far as I know).
On the other hand, Enderton's book Elemntary of set theory starts with natural numbers and then, in chapter 7, he introduces ordinal numbers. I think the background both constructions is the same, but I think Enderton takes more care. He presents the Transfinite Recursion Theorem and Transfinite induction principle before to start with the ordinal's construction.
I know that this is always a difficult question but... what approach do you follow to learn about the constrction of naturals and ordinals, Enderton or Kelley? Remember Kelley's theory is allowed to discuss classes, while Enderton's (ZFC) not. Does it matter for the construction?
Thanks