  Already in 1931 Gödel pointed out a natural way to justify new axioms.  He pointed out that if one moves beyond the natural numbers and climbs the hierarchy of types (the sets of natural numbers, the sets of sets of natural numbers, etc.) one arrives at axioms (the axioms of second-order arithmetic PA2, the axioms of third-order arithmetic PA3, etc.) that settle the undecided statements that he discovered.  The axiom system for the second level, PA2, settles the statement left undecided at the first level, namely Con(PA); in fact, PA2 proves Con(PA), which is the desired result.  But now we have a problem at the second level.  For the second incompleteness theorem shows that (under similar background assumptions to those above) PA2 does not settle Con(PA2). Fortunately, the axiom system for the third level, PA3, settles the statement left undecided at the second level, namely Con(PA2).  This pattern continues.  For every problem there is a solution and for every solution there is a new problem.  In this way, by climbing the hierarchy of types one arrives at systems that successively settle the consistency statements that arise along the way.