Point taken, but the systems that do not address the topic of whole numbers--for that same reason--can't overturn Gödel's theorem, while the theorem applies to all systems that do address whole numbers. So the universality of the truth remains. If a system covers whole numbers, I can say before I see the axiom set that it is not both complete and consistent.
(I am laying aside the issue of whether Gödel may have been wrong, as I am not qualified to form my own opinion on the matter. There are always dissenters to any conclusion, certainly, but as an outsider I have to follow the strong consensus.)
The trade of mathematics is like cartography. Mapmakers make maps, and they use their choice of coordinate systems. Presumably, the properly made maps will all be correct according to their coordinate systems, but they rarely will look anything like each other when you compare them. Some cover different parts of the territory. Some cover the same territory, but use different projections (the shape of Greenland is very different in a Mercator projection than it is on a globe). Some use wildly different scales and rotations.
But here's the key: there is an objective territory to which the maps refer.
None of the arguments made by the formalists are wrong. It's just that they are arguments about maps. It is not possible to conclude on the basis of the maps that the maps are all that exist.
Not quite. Not only can't the systems be too small (i.e. don't emcompass arithmetic) but they can't be too large. I think the technical lingo is that the methods of proof must be finitistic.