Believing that a mathematical system has a model usually corresponds to believing that a certain computable ordinal is well-ordered (the proof-theoretic ordinal of that system), and large ordinals imply the well-orderedness of all smaller ordinals.

I'm no expert in this -- my comment is based just on reading the post, but I take the above to mean that there's some large ordinal for ZFC whose existence implies that ZFC has a model. And if ZFC has a model, it's consistent.