Theorem. Let J and C be well-ordered sets; assume that there is no surjective function map...

Theorem. Let J and C be well-ordered sets; assume that there is no surjective function mapping a section of J onto C. Then there exists a unique function h : J → C satisfying the equation

for each x ∈ J, where Sx is the section of J by x.

Proof.

(a) If h and k map sections of J, or all of J, into C and satisfy (*) for all x in their respective domains, show that h(x) = k(x) for all x in both domains.

(b) If there exists a function h : Sα → C satisfying (*), show that there exists a function k : Sα ∪ {α} → C satisfying (*).

(c) If K ⊂ J and for all α ∈ K there exists a function hα : Sα → C satisfying (*), show that there exists a function

satisfying (*).

(d) Show by transfinite induction that for every β ∈ J, there exists a function hβ : Sβ → C satisfying (*). [Hint: If β has an immediate predecessor α, then Sβ = Sα ∪ {α}. If not, Sβ is the union of all Sα with α<β.]

(e) Prove the theorem.