When we cannot prove a theory consistent, we can often show that it is consistent provided some other theory, e.g. Peano arithmetic or ZF is consistent.
In his [Gödel, 1940], Gödel proved that if Gödel-Bernays set theory
is consistent, then it remains consistent when the axiom of choice and
the continuum hypothesis are added to the axioms. He did this by
supposing that set theory has a model, i.e. there is a domain and an
predicate satisfying GB. He then showed that a subset of this
domain, the constructible sets, provided a model of set theory in
which the axiom of choice and the continuum hypothesis are also true.
Paul Cohen proved in 1963 that if set theory has any models it has
models in which the axiom of choice and the continuum hypothesis are
false.