>>13289000Well one crucial consequence is that any system with a list of axioms {A1, A2, An}, say, that can at least represent arithmetic, contains some sentences G such that you can neither prove G nor it's negation not(G). Provable here means with respect to the system we set up.
Consequently, you can also not complete the system. You might say well let's consider the system which has all the axioms but also G taken to be true by definition, i.e. {A1, A2, An, G}, but by the theorem there will now be a new sentence G' which now is neither provable nor it's negation not(G).
As described so far, this concerns (un)provability.
(You can mingle "true but unprovable" into the mix. But it that makes it significantly more fragile to talk about informally, as it opens significantly more avenues to misread the thing - which imho little gain)
