Zermelo-Fraenkel Set Theory

CHAPTER 4. ORDINALS 29
There are versions of the recursion theorem with F having parameters. For instance,
we might have a recursion equation of the form
F (x 1 , . . . , x n , α) = H(x 1 , . . . , x n , α, {(β, F (x 1 , . . . , x n , β)) | β < α}).
However, the same proof works.
An abstract version of the recursion theorem holds:
Theorem 4.13 Suppose that ε is a well-founded relation on the class U such that for all
a ∈ U, {b ∈ U | b ε a} is a set. Then for every operation H : V → V there is a unique
operation F : U → V such that for all a ∈ U:
Exercises
F (a) = H(F |{b ∈ U | b ε a}).
64 ♣ Prove Theorem 4.13.
Hints. First, assume that ε is transitive. Check that the proof for this special case can
be copied, word for word, replacing OR by U, from that of Theorem 4.12.
Next, using this, recursion along a possibly non-transitive ε can be reduced to recursion
along its transitive closure ε ⋆ : Given H, define an auxiliary operation H ′ by
H ′ (f) = def H(f|{y | y ε x}) if x is such that Dom(f) = {y | y ε ⋆ x}. (Its values for other
arguments are irrelevant.) Now if F satisfies the ε ⋆ -recursion equation F (x) = H ′ (F |{y |
y ε ⋆ x}), it follows that F (x) = H ′ (F |{y | y ε ⋆ x}) = H(F |{y | y ε x}).
65 ♣ Let a 0 ∈ V be a set and G : V → V an operation. Show: there exists a unique
operation F : OR → V on OR such that
• F (0) = a 0 ,
• F (α + 1) = G(F (α)) ,
• for limits γ: F (γ) = ⋃ ξ