| Résume | We 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
$T\supseteq\ZFC+$\emph{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 $T$. |
