The major difference is that nothing requires that your set of beliefs be consistent: however, if a given set of axioms allows you to prove and disprove some as

The major difference is that nothing requires that your set of beliefs be consistent: however, if a given set of axioms allows you to prove and disprove some assertion P, you pretty much can't use that set of axioms anymore! Well, you can, but rather than the normal huge castles made of sand that follow from any reasonably strong set of axioms, you will end up with a small puddle of mud that's useless. In this sense, a simple discovery (of an axiomatic system's inconsistency) would essentially cause that set of axioms never to be used again. Can you imagine that a simple logical proof would immediately cause the Pope to abandon making certain statements together? For example check out this post: http://www.religiousforums.com/forum/1993062-post3.html http://www.religiousforums.com/forum/1993062-post3.html Suppose a religious scholar produces an argument that the universe described by, "Jesus Christ, the only Son of God" + "who was born of the Virgin Mary" + "his kingdom will have no end" is an inconsistent Universe. It won't suddenly be the case that nobody will mention these three things in the same axiomatic set again, just because they're (through some rigorous proof) inconsistent. That's a huge, huge difference. It's like the difference between chess and Shakespearean criticism!