Semantics of First-Order Languages

Hi,

I am reading in Ebbinghaus, Flum, Thomas (Mathematical logic), chapter
III Semantics of First Order Languages, and I think there must be
something misleading me.

In section 3, The satisfaction relation, they define this relation as
a kind of translation of formulas from any language of first order
logic to the language of set theory via some interpretation.

However, I don't see how this allows anyone to recognize the truth or
falsity of any statement, since the language of set theory is just an
instance of a first order logic language, which is precisely the kind
of thing one is trying to give sense.

I would be grateful if references to modern literature were given.

Thank you in advance for your help.
.

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