I'm having trouble with a proof of the Eliminability Theorem in Set Theory given by the book "Basic Set Theory" by Azriel Levy.
I have two questions:
I don't quite understand how they use the implicit proof by induction. Could someone explain this to me?
Also, why can we use induction here? This isn't spoken of at this stage and even if it were to be presented as an axiom, it relies on natural numbers. Isn't Set Theory supposed to define numbers their own way?
Thank you in advance.