The point must surely be that one can add the these true-but-unprovable statements to the original axioms without contradiction. They fit just fine: they're all

The point must surely be that one can add the these true-but-unprovable statements to the original axioms without contradiction. They fit just fine: they're all elements of Th(N). It's a poor analogy because the natural way of thinking of a jigsaw puzzle is of a set of elements (pieces) that are consistent (every piece has a place), so if a piece doesn't fit then it's not consistent with the others. But this is false if the pieces are statements in the language of arithmetic that are true of the natural numbers.