A really simple axiomatic system to consider:
Rule of inference: We can append | to the end of every statement.
From the only axiom, we infer |||.
From |||, we infer ||||.
From ||||, we infer |||||.
And so on.
Obviously no contradictions are possible, but we cannot derive |.
A little more complicated system with something like true and false statements:
Alphabet: ~ |
Axiom 1: |
Axiom 2: ~||
Rule of inference: You can append || to the end of each statement.
Note that we cannot derive both statement x (not beginning with '~') and statement ~x. So, no "contradictions." You cannot derive ~|. You also cannot derive ||.
So, I'm guessing, you will need at least one axiom and one rule of inference in any axiomatic system. If you want to model "true" and "false" statements, you will probably need at least two axioms.