Gödel's theorems do not say that we can never know our axiom systems are consistent. Not at all. What they say is that we can never prove that certain systems are consistent within those systems themselves. This leaves open the possibility that we can prove their consistency in other axiom systems, or can convince ourselves of their consistency by methods that are not completely formal.
My recommended reference on the incompleteness theorems for a general reader is "Gödel's Theorem: An Incomplete Guide to its Use and Abuse" by Torkel Franzén. This book has the rare combination of being written to be broadly accessible while still being precise enough to be satisfying.
