.. I have some trouble understanding this though -

Having painstakingly built this apparatus of mathematical self-reference, Gödel came up with an astonishing twist: he produced a formula that, while ostensibly saying something about numbers, also says, “I am not provable.” At first, this looks like a paradox, recalling as it does the proverbial Cretan who announces, “All Cretans are liars.” Having painstakingly built this apparatus of mathematical self-reference, Gödel came up with an astonishing twist: he produced a formula that, while ostensibly saying something about numbers, also says, “I am not provable.” At first, this looks like a paradox, recalling as it does the proverbial Cretan who announces, “All Cretans are liars.” But Gödel’s self-referential formula comments on its provability, not on its truthfulness. Could it be lying? No, because if it were, that would mean it could be proved, which would make it true. So, in asserting that it cannot be proved, it has to be telling the truth. But the truth of this proposition can be seen only from outside the logical system. Inside the system, it is neither provable nor disprovable. The system, then, is incomplete. Could it be lying? No, because if it were, that would mean it could be proved, which would make it true. So, in asserting that it cannot be proved, it has to be telling the truth. But the truth of this proposition can be seen only from outside the logical system. Inside the system, it is neither provable nor disprovable. The system, then, is incomplete.

So, having implied that proof and truth are not necessarily the same thing:

But Gödel’s self-referential formula comments on its provability, not on its truthfulness.

the writer then says:
Could it be lying? No, because if it were, that would mean it could be proved, which would make it true.

I just can't get my mind around that .. somewhere I must be missing the point ..

I found this page helpful though -

There are two very common but fallacious conclusions people make from this, and an immense number of uncommon but equally fallacious errors I shan't bother with. The first is that Gödel's theorem imposes some some of profound limitation on knowledge, science, mathematics. Now, as to science, this ignores in the first place that Gödel's theorem applies to deduction from axioms, a useful and important sort of reasoning, but one so far from being our only source of knowledge it's not even funny. It's not even a very common mode of reasoning in the sciences, though there are axiomatic formulations of some parts of physics. Even within this comparatively small circle, we have at most established that there are some propositions about numbers which we can't prove formally. As Hintikka says, ``Gödel's incompleteness result does not touch directly on the most important sense of completeness and incompleteness, namely, descriptive completeness and incompleteness,'' the sense in which an axiom systems describes a given field. In particular, the result ``casts absolutely no shadow on the notion of truth. All that it says is that the whole set of arithmetical truths cannot be listed, one by one, by a Turing machine.'' Equivalently, there is no algorithm which can decide the truth of all arithmetical propositions. And that is all.

Pretty hard-headed conclusions I suppose .. but his description of the theorem (in the link) is clearer to me

Anyway thanks for posting that.