Hi, I am doing a project on axiomatic set theory and i'm having a lot of trouble with transfinite recursion. It's not that the proof itself is particularly difficult to follow, only that the statement of the theorem itself is very hard to conceptualize. I have been trying to apply the theorem to show that some simple recursively defined sets - e.g. the fibbonacci sequence - exist, and I have been unable to do this. Of course, I realise that one would normally simply use recursion to show this, but my problem is that I need to understand transfinite recursion in order to, for instance, apply it in proving the equivalence of the axiom of choice and the well-ordering principle. So basically what i am looking for is examples of concrete applications of the transfinite recursion theorem. Any help is appreciated!