A better example is the well-ordering of the reals. From wikipedia: From the [axioms of math] one can show that there is a well-order of the reals; it is also

A better example is the well-ordering of the reals. From wikipedia: From the [axioms of math] one can show that there is a well-order of the reals; it is also possible to show that [those axioms] alone are not sufficient to prove the existence of a definable (by a formula) well-order of the reals. http://en.wikipedia.org/wiki/Well-order http://en.wikipedia.org/wiki/Well-order So you can prove that set of real numbers can be well-ordered, but you can also prove that an order cannot be defined. Kind of how God exists, but cannot be perceived or comprehended ;) (or maybe not)