  We give a brief survey on the interplay between forcing axioms and various
other non-constructive principles widely used in many fields of abstract
mathematics, such as the axiom of choice and Baire's category theorem.
  First of all we outline how, using basic partial order theory, it is possible
to reformulate the axiom of choice, Baire's category theorem, and many large
cardinal axioms as specific instances of forcing axioms. We then address
forcing axioms with a model-theoretic perspective and outline a deep analogy
existing between the standard {\L}o\'s Theorem for ultraproducts of first order
structures and Shoenfield's absoluteness for $\Sigma^1_2$-properties. Finally
we address the question of whether and to what extent forcing axioms can
provide a "complete" semantics for set theory. We argue that to a large extent
this is possible for certain initial fragments of the universe of sets: The
pioneering work of Woodin on generic absoluteness show that this is the case
for the Chang model $L(\text{Ord}^\omega)$ in the presence of large cardinals,
and recent works by the author show that this can also be the case for the
Chang model $L(\text{Ord}^{\omega_1})$ in the presence of large cardinals and
maximal strengthenings of Martin's maximum or of the proper forcing axiom. The
major open question we leave open is whether this situation is peculiar to
these Chang models or can be lifted up also to $L(\text{Ord}^\kappa)$ for
cardinals $\kappa>\omega_1$.
