Is computability theory the only theory that doesn't introduce new axioms to develop its theory?

Question

Generally, a mathematical theory is characterized by its axioms (like group theory). Computability theory is characterized by definitions (such as those of Turing machines or general recursive functions) and doesn't introduce any new unique axioms of its own. So my question is: are there any other mathematical theories that share this feature or is it just computability theory?

Answer 1

A group is usually defined as a pair $(G, \cdot)$ where $G$ is a set, $\cdot$ an operation on $G$, which together satisfy certain conditions. In this way it does not introduce any axioms, it only says things about objects which meet the definition.

In other words, the distinction between definition and axiom is often kind of arbitrary.