  The paper is a first of two and aims to show that (assuming large cardinals)
set theory is a tractable (and we dare to say tame) first order theory when
formalized in a first order signature with natural predicate symbols for the
basic definable concepts of second and third order arithmetic, and appealing to
the model-theoretic notions of model completeness and model companionship.
  Specifically we develop a general framework linking generic absoluteness
results to model companionship and show that (with the required care in
details) a $\Pi_2$-property formalized in an appropriate language for second or
third order number theory is forcible from some $T\supseteq\mathsf{ZFC}+$large
cardinals if and only if it is consistent with the universal fragment of $T$ if
and only if it is realized in the model companion of $T$.
  The paper is accessible to any person who has a fair acquaintance with set
theory and first order logic at the level of an under-graduate course in both
topics; however bizarre this may appear (given the results we aim to prove) no
knowledge of forcing or large cardinals is required to get the proofs of its
main results (if one accepts as black-boxes the relevant generic absoluteness
results). On the other hand familiarity with the notions of model completeness
and model companionship is essential. All the necessary model-theoretic
background will be given in full detail.
  The present work expands and systematize previous results obtained with
Venturi.
