Hello, I would like to ask you if there is a mathematical theory, that is complete (in the sense of Goedel's theorem) but practically applicable. I know about Robinson arithmetic that is very limited but incomplete already. So, I would like to know if there is some mathematics that could be practically used (expressiveness) and reduced to logics (completeness).


I'm very new to the site and to the maths as well, so please tell me if that's a silly question.



Complete mathematics
You probably intended to restrict the question to effectively axiomatizable theories. Otherwise, for example, the first-order theory of the standard model of arithmetic is a complete theory, as is the theory of the standard model of ZFC. 


Gödel's incompleteness theorem establishes some limitations on which effective theories can be complete. It shows that no effective, complete, consistent theory can interpret even weak theories of arithmetic such as Robinson arithmetic. However, there are many mathematically interesting theories that do not interpret the natural numbers.


Examples of complete, consistent, effectively axiomatizable theories include:


* For any prime $p$, the theory of algebraically closed fields of characteristic $p$
* The theory of real closed ordered fields, mentioned by Ricky Demer
* The theory of dense linear orderings without endpoints
* Many axiomatizations of Euclidean geometry


