I am currently studying ZF set theory in terms of first-order logic and I am having trouble understanding the motivation behind this axiomatic formulation of set theory.
ZF set theory is a first-order language with the proper axioms of extensionality, foundation, specification, union, pairing, replacement, power set, well- ordering and infinity.
Is ZF set theory just a model for universal set theory in the same manner that Lagrangian Classical Mechanics is just a model for the universe. I.e neither are perfect?
Otherwise, what is the justification in choosing these axioms, what 'logic' are we basing our choices on?
For when we extend the theory to include the axiom of choice, we have added a new axiom.
What is the justification of adding the axiom of choice, and not other statements which we cannot prove (have been proved to be unprovable in set theory); such as the continuum hypotheses, provided they are consistent with the current axioms?
Why can't we add every statement that has been proved to be unprovable within the system, why do we only add the axiom of choice?
In this light, what is the scope of ZF(C) set theory?