I have taken no formal mathematics logic course yet, I'm sorry for unclear parts of this question.
I've learned about Presburger Arithmetics few days ago, it seemed really interesting. But since then, I keep wondering, how strong can we make arithmetic while still keeping in decidable and complete?
What are useful extentions of Presburger Arithmetic, that still remain decidable and complete? Do we know any limit? Is it possible some of these theories will be "usefull" to work with?