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In his book on set theory, Kunen often emphasizes how important it is to distinguish between statements in the theory and the meta-theory. I have two questions:

a) When we are talking about set theory, isn't this distinction superfluous? For example when we formalize logic in set theory, there exists an enumeration of all formulas and you can make recursive definitions with them. This makes some definitions easier and somehow more natural, I think. Or is it possible that we lose something with this approach, perhaps just the philosophical strength of the statements in the meta-theory? Or is it even possible that we can "prove" wrong statements?

b) As I said, Kunen seems to make a distinction between the natural numbers in the meta-theory and the natural numbers in models of set theory. Now, for example at the beginning of chapter V, there is the following lemma:

Let $\phi(x_0,...,x_{n-1})$ be any formula whose free variables are among $x_0,...,x_{n-1}$; then

$\forall A ( \{s \in A^n : \phi^A(s(0),...,s(n-1))\} \in Df(A,n))$.