Comment by lmm

Even if an axiom system could prove its own consistency, that wouldn't be any less circular - we could believe T because T proves that T is consistent - but if T were inconsistent then it might still prove that T was consistent.

Most of us take some axioms and believe them, ultimately because they correspond to physical experience - if you take a pile of n pebbles and add another pebble to the pile, you get a pile of n+1 pebbles, and n+1 is always a new number that doesn't =0 (that is, our pile looks different from a pile of 0 pebbles). Maybe there's some (very large) special n where this wouldn't happen - where you'd add 1 and get a pile of 0 pebbles, or where you can have n pebbles but you can't have n sheep. But so far we haven't found that. So far the universe remains simple, and the axioms of arithmetic are simpler than conceivable alternatives.