TITLE: Meaning Confused " Theorem Schema"
QUESTION [5 upvotes]: I was reading Enderton's "Elements of Set Theory", and came upon, 

Transfinie Recrursion 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 $t \in A$.

I am confused what is meant by Theorem Schema here. Enderton explains this as an "infinite package of theorems". But what is the point? 
When we write out a theorem, don't we just write a statement? Why restrict ourselves to a specific form? I feel like the Theorem Schema is limiting our scope rather than helping - or maybe I completely misunderstood its purpose. 
May someone explain? Thanks!

REPLY [2 votes]: In short, what is being proven is a meta-theorem that says:

Take any $2$-parameter sentence $γ$ over ZFC such that ZFC proves "$\forall f\ \exists! y\ ( γ(f,y) )$".
Then ZFC proves $φ_γ$, which is the sentence ...

This meta-theorem can be applied (in the meta-system) to all formulae over ZFC, giving a whole schema of theorems over ZFC. Incidentally, this meta-theorem is itself provable in ZFC, but that is not what you want, because it would be one level deeper. Specifically, within ZFC you cannot from "ZFC proves $φ_γ$" derive "$φ_γ$". So when you are working within ZFC you do not want to invoke the theorem schema inside, because then you are stuck with a statement about ZFC and not $φ_γ$ alone. Instead you go outside to the meta-system, observe there that ZFC proves $φ_γ$, and then go back inside where you now have $φ_γ$.
In other words, the meta-theorem shows that we can for any particular such $γ$ write down a proof over ZFC of "$φ_γ$", and so it is valid for us to use "$φ_γ$" as an axiom anywhere in a proof over ZFC. It is then just like the ZFC axiom schemas where each schema cannot be expressed as a single axiom over ZFC.