An axiom can be wrong wrt the standard model for some first-order theory: http://lesswrong.com/lw/g0i/standard_and_nonstandard_numbers...

But, demonstrating an axiom is inconsistent with the standard model of peano arithmetic or ZFC is in the general case uncomputable.

For example, an axiom that claims some program halts when it does not actually halt cannot in general be proved to be inconsistent wrt the standard model, because otherwise one could decide the halting problem.