Help with Kunen and recursion on wellfounded sets.

Hi everyone,

I'm a bit stuck on the proof (page 103-4 in Kunen's Set theory and independence book) of the existence of a functional class G for any class A wellfounded relation R on A, and functional class F from A x V to V, which is set-like such that:

(*) forallx in A(G(x) = F(x, G|pred(x))

where pred(x) are the predecessors of x under R and in A.

So Kunen first wants to prove that for every x of A the set {x}Ucl(x) (cl(x) is the closure of x under R) admits of a function g with domain {x}Ucl(x) and such that:

(**) forall y in {x}Ucl(x)(g(x) = F(x, g|pred(x))

Now, his way of doing this is by induction. He assumes that for each predecessor of x there is such a function g and then takes the union of these functions h, and adds to that the pair <x, F(x, h)> to get an new function h'.

The claim is then that h' satisfies (**). One way to prove this is by showing that h is satisfies (**) on cl(x) and then to show that F(x, h) = F(x, h'|pred(x)). This in turn we could do if we could show that h'|pred(x) = h.

But in general it seems that ~(h = h'|pred(x)). The reason is that h has as a domain the closure of x whereas h'|pred(x) only has x's predecessors as its domain.

Any help in sorting this out would be really great.

Thanks

Sam