  We show that for $\Pi_2$-properties of second or third order arithmetic as
formalized in appropriate natural signatures the apparently weaker notion of
forcibility overlaps with the standard notion of consistency (assuming large
cardinal axioms).
  Among such $\Pi_2$-properties we mention: the negation of the Continuum
hypothesis, Souslin Hypothesis, the negation of Whitehead's conjecture on free
groups, the non-existence of outer automorphisms for the Calkin algebra, etc...
In particular this gives an a posteriori explanation of the success forcing
(and forcing axioms) met in producing models of such properties.
  Our main results relate generic absoluteness theorems for second order
arithmetic, Woodin's axiom $(*)$ and forcing axioms to Robinson's notion of
model companionship (as applied to set theory). We also briefly outline in
which ways these results provide an argument to refute the Continuum
hypothesis.
