> That's circular logic; why would that be an interesting question to investigate absent the incompleteness theorem?
Because the amount of logical infrastructure that you need is surprisingly small. All you need is to be able to do basic grade-school arithmetic: add, subtract, multiple, divide, test for equality. And you don't even need all that. You can build everything you need from much simpler primitives.
The reason this is not circular is that before Goedel's theorem mathematicians were trying to construct a formal system that would allow you to derive all true statements about not just arithmetic, but all of mathematics. It turns out that this is impossible. As soon as you can do arithmetic, Goedel's theorem applies and you have lost.
> that implicitly requires understanding that small theories like Presburger Arithmetic are complete
I don't see why. Goedel's theorem matters in the context of trying to reduce all of mathematics to a formal system. Presburger arithmetic fails not because it is syntactically incomplete but because it cannot (for example) do division, so it cannot formulate (for example) the concept of a prime number and so it cannot (for example) prove the fundamental theorem of arithmetic. So yes, PB is decidable, but it is not a plausible candidate for a complete (in the informal sense) formulation of all of mathematics. Goedel's theorem shows that if you can add, subtract, multiply and divide and test for equality, then you have lost. And if you can't do those things, then you have also lost, but you don't need a theorem to tell you that.
Because the amount of logical infrastructure that you need is surprisingly small. All you need is to be able to do basic grade-school arithmetic: add, subtract, multiple, divide, test for equality. And you don't even need all that. You can build everything you need from much simpler primitives.
The reason this is not circular is that before Goedel's theorem mathematicians were trying to construct a formal system that would allow you to derive all true statements about not just arithmetic, but all of mathematics. It turns out that this is impossible. As soon as you can do arithmetic, Goedel's theorem applies and you have lost.
> that implicitly requires understanding that small theories like Presburger Arithmetic are complete
I don't see why. Goedel's theorem matters in the context of trying to reduce all of mathematics to a formal system. Presburger arithmetic fails not because it is syntactically incomplete but because it cannot (for example) do division, so it cannot formulate (for example) the concept of a prime number and so it cannot (for example) prove the fundamental theorem of arithmetic. So yes, PB is decidable, but it is not a plausible candidate for a complete (in the informal sense) formulation of all of mathematics. Goedel's theorem shows that if you can add, subtract, multiply and divide and test for equality, then you have lost. And if you can't do those things, then you have also lost, but you don't need a theorem to tell you that.