Section on Rank in Enderton's Text.

Question

I am confused with a statement Enderton made in his text, Elements of Set Theory on page 202, Chapter 7. There were two Lemmas,

Lemma 7Q: For any ordinal number $\delta$ there is a function $F_{\delta} $ with domain $\delta$ such that $$ F_{\delta} ( \alpha )= \bigcup \{ \mathcal{P} F_{ \delta} ( \beta) \, | \, \beta \in \alpha \} $$

Proof: Apply transfinite recursion...

Lemma 7R: Let $\delta$ and $\epsilon$ be ordinal numbers; let $F_{\delta}$ and $F_\varepsilon$ be function of form in Lemma 7Q. Then $$F_\delta (\alpha) = F _ \varepsilon (\alpha)$$ for all $\alpha \in \delta \cap \varepsilon$.

And we have one statement following Lemma 7R,

The statement: In particular (by taking $\delta = \varepsilon$) we see that the function $F_{\delta}$ from lemma 7Q is unique. We can now...

My question is, the proof of 7Q uses the Transfinite Recursion Theorem, which already establishes the uniqueness of $F_\delta$. What is the point of this statement? Am I misunderstanding something here?

For reference, the recursion theorem is included below. Please tell me if not enough information is provided, or more clarification is needed. Thanks!

Transfinite Recursion Theorem Schema For any formula $\gamma(x,y)$ the following is a theorem. Assume $<$ is a well ordering on $A$. Assume that for any $f$ there is a unique $y$ such that $\gamma (f,y)$. Then there exists a unique function $F$ with domain $A$ such that $$ \gamma (F \upharpoonright \text{seg }t , F(t)) $$ for all $t \in A$.

Answer 1

I don't think you are misunderstanding anything here. The transfinite recursion theorem schema does indeed give uniqueness of the recursively defined function, and so your $F_δ$ is indeed uniquely defined.

Answer 2

I know what you are asking, because I thought the same thing, but Enderton is being precise here because he does not formally explain the tool of defining something on all of the ordinals.

For example, much later he defines $\aleph_\alpha$ for each ordinal $\alpha$. And then he will define $\beth_\alpha$ for each ordinal $\alpha$. He does this with a 'theorem' that I will post below. What he seeks to do here is define $V_\alpha$ for each ordinal $\alpha$, but because he has not taught that formal tool yet, he decides to define $V_\alpha$ in a manner that can be generalized to proving that 'theorem.'

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The Theorem: Transfinite Recursion on the Ordinals (page 210) For each formula $ \phi\left(x,y\right) $, if for each $ x $, there exists a unique $ y $ such that $ \phi\left(x,y\right) $ holds, then there exists a formula $ \Phi\left(\alpha,z\right) $ such that for each ordinal $ \alpha $, there exists a unique $ z $ such that $ \Phi\left(\alpha,z\right) $ holds, and for each ordinal $ \alpha $ and for each function $ f $ such that $$ \operatorname{dom}\left(f\right)=\alpha\qquad\text{and}\qquad \Phi\left(\beta,f\left(\beta\right)\right) $$ for every $ \beta\in\alpha $ and for each $ z $, $$ \phi\left(f,z\right) $$ if and only if $ \Phi\left(\alpha,z\right) $ holds.