Re:

Can't say i got it, why "understanding"

why is transfinite recursion chttps://en.wikipedia.org/wiki/Transfinit=_recursion> a good=model for understanding — the proof that the result is well-define= uses transfinite induction. Let F denote a (class) function = to be defined on the ordinals. The idea now is that, in defining (a) for an unspecified ordinal a, one may assume that =03) is already defined for all R < a and thus give a formula for =(a) in terms of these F(f3). It then follows by tra=sfinite induction that there is one and only one function satisfying the r=cursion formula up to and including a.

(more will be given later): define function F by letting F(α) be the smallest ordinal not in the set {F(β) | β < α}= that is, the set consisting of all F(β) for β < α. This definiti=n assumes the F(β) known in the very process of defining ; this apparent vicious circle is exactly what definition by transfi=ite recursion permits. In fact, F(0) makes sense since there is n= ordinal β < 0</=pan>, and the set {F=/em>(β) | β < 0} is empty. So F(0) is equal t= 0 (the smallest ordinal of all). Now that F(0) is known, the def=nition applied to F(1) makes sense (it is the smallest ordinal no= in the singleton set {<=m>F(0)} = {0}), and so on .