[H8crilA]

The model of natural numbers with addition and multiplication that we intuitively understand as the correct one. As Godel showed you can't actually describe this model in first order logic (Skolem-Lowenheim further shows how hopeless we are in describing the model). "Every child knows what natural numbers actually are".

[lisper]

More concretely: you can add an axiom to standard PA that says, "There exists a number that is the Godel number of a proof of G." The resulting system is consistent, and includes a new kind of number that is not a natural number (because you can prove that N is not a proof of G for any N that is a natural number). It's analogous to adding an axiom that says, "There exists a number whose successor is 0", which introduces a new kind of number that we call "negative numbers." Math is extended in familiar interesting directions like this all the time by adding axioms like "There exists a number whose square is 2" (which gives you irrational numbers) or "there exists a number such that the successor of its square is 0" (which gives you imaginary numbers).

[Y_Y]

You didn't say otherwise, but it's interesting to note that the irrationals is quite "big" in the sense that you can do a lot without them. Not only can you make a nice consistent extension of usual structures on N, Z, Q etc by just adding sqrt(2), you can even add all square roots, or all n-th roots, or the solutions to all polynomials without getting plenty of good irrationals. Sets between Q and R are often neglected, but there's a wealth of good maths in there.