As the title says, I wish to know if every system that is not axiomatic, not consistent or it's theorems can not be listed by an effective procedure is not complete.
I think that this is false. That is because The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers.
It does not say anything about the negation of this statement. Therefore, my intuition is that it is false. I just wish I could find an example that shows it.