I know very little about modal logic (only some set theory) in mathematics, but I am aware that there exists a completeness theorem, incompleteness theorem, and the axiom choice, and that maths is not always decidable.
However, my interest in modal logic has been growing and I started wondering if there was a way for some system of mathematics to be all: complete, consistent, and decidable, while being most similar to current mathematics. The closest thing I've seen is Presburger arithmetic, but I believe it does not have the Axiom of Infinity.
Could someone tell me if there is or isn't and provide a reasonably clear justification for the answer? I do not mind if you use logic symbols that you think I may not understand, but it would be appreciated if you added a comment next to them to explain what they mean.