About an erroneous recursive definition

[Guest]

Hi, I'm reading Introduction to set theory of Monk, and on page 87 in the paragraph of recursive definitions, it starts with a wrong proof about the existence and uniqueness of such a recursive function. The example function is that of addition. The argument is by induction: [tex]m+n[/tex] is defined for all [tex]m,n \in \omega[/tex]. Proof: [tex]m+0=m[/tex] and so [tex]m+0[/tex] is defined; assuming [tex]m+n[/tex] is defined, [tex]m+Sn=S(m+n)[/tex], and so [tex]m+Sn[/tex] is defined ([tex]S[/tex]=the successor).

I can see that this proof is wrong in the set theory formal system.
What I cannot understand, however, is what the author states immediately after this "proof" about the nature of the error; he says: "It mixes language and metalanguage, since the argument talks about an expression's being defined on the same level as the integers themselves."

My questions are:

(1) Where's the metalanguage in the above proof? Is peraphs the "+" symbol?

(2) Saying "on the same level as the integers themselves" does he mean peraphs that when the definition of the function is still up in the air, the function to be proven is already considered existing (and this obviously is wrong)? If not, what does he really mean?