Example of Non-axiomatic theory?

Question

Im fairly sure that non-axiomatic theories exist (in a mathematical logic sense, aka not considering Creationism and all that...), but I cant think of any examples. Im wondering if I have a theory $T$ which extends First Order Arithmetic and is complete, can that theory be non-axiomatizable? Since $T$ extends FOA and is complete, it must be inconsistent by Godel's thms, but what does this say about its axiomatization? I imagine that it can be axiomatic with inconsistent axioms, but is there some way for it to be non-axiomatic as well? Thanks

Answer 1

You forgot a hypothesis: a theory extending FOA can be both complete and consistent, if it is not axiomatizable by a recursively enumerable set of axioms. (roughly speaking, this includes fintie sets of axioms, as well as infinite sets of axioms so long as there is a "computer" program that can tell whether any given statement is one of the axioms or not)

All theories have axiomatizations: for example, we can take every theorem of the theory to be one of the axioms. This is even a useful thing to do, on occasion.

Answer 2

You seem to have misunderstood Godel's theorem. By Godel's theorems we know that $\operatorname{Th}(\mathbb{N},+,.,0,S)$ is not recursively axiomatizable. But this does not at all imply that it is inconsistent. In fact it is consistent, since the theory has a model, namely $(\mathbb{N},+,.,0,S)$.