"To work with a set of axioms, we must believe that that set is both logically consistent ... The first question cannot be answered for the interesting sets of

"To work with a set of axioms, we must believe that that set is both logically consistent ... The first question cannot be answered for the interesting sets of axioms" Consistency of a theory may be proven or disproven in a different set of axioms. For instance, peano or presburger arithmetic are consistent in ZF. In the field of program verification for instance, people use all kinds of theories that they prove to be consistent. But I agree that at some point, one has to believe in an axiomatic system.