Found this fascinating post from phi_of_sci

One of the fascinating things about the Godel incompleteness results is that infinity itself is not the problem. In some systems of arithmetic you can believe you can count to infinity and still be complete and consistent. Some even include the infinite set of induction axioms

Yet the incompleteness problem is not even due to multiplication alone. A system with all of Peano's axioms apart from addition, but inclusive of multiplication was shown by Thorlaf Skolem to be complete. The completeness problem derives from the combination of addition and multiplication on variables. These operations together provide the ability to construct the Godel numbering by allowing the definition of the unique prime factorisation of every integer and the mechanism by which Godel's unprovable statement can be constructed.
So we have yet more demonstration of how easy it is to skirt Godelian incompleteness. The Godelian incompleteness results are really about carefully choosing the formal language concepts we use to describe systems we wish to use. The real problem faced by humans is not even Godelian-incompleteness, its computational complexity. Even solving relatively small problems of Presburger arithmetic takes impossibly large amounts of time.