Tameness in Set Theory

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 extending 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.

Part (but not all) of our results are conditional to the proof of Schindler and Asperò that Woodin’s axiom (*) can be forced by a stationary set preserving forcing