Given some first-order language and a set of axioms therefrom one still needs to specify a deductive system to turn it into a full-fledged first-order theory.
Currently I'm under the following impression. There are weaker and stronger deductive systems but there is actually a formalization of what it means for a system to be "good", i.e. "of the same strength as unformalized deductive reasoning". The definition calls for the system to be "sound" and "complete". Naturally, for any system with this quality the set of derivable formulas is the same. Furthermore, the most widely considered formal deductive systems are indeed such.
Now, a couple of citations from Wikipedia.
"A deductive system is sound if any formula that can be derived in the system is logically valid. Conversely, a deductive system is complete if every logically valid formula is derivable."
"A formula is logically valid (or simply valid) if it is true in every interpretation."
Are those really the correct definitions? I just can't wrap my head around the resulting definition of completeness. I mean: "every interpretation"?! That's one inconceivable proper class! How on earth could one then prove the completeness of some system?