I was reading about Godel's incompleteness theorem which is true for any formal theory that satisfies certain properties. One of these properties is the following: "The theory is assumed to be effective, which means that the set of axioms must be recursively enumerable.". I have also been told that Godel's incompleteness theorem hold for any axiomatic system large enough to be capable of expressing elementary arithmetic. (Is this because we assume the theory to have a signature that specifies the non-logical symbols in the language of such theory, such as '0' or '+'? )
But, clearly, if our assumption is false, such that our theory is not effective, Godel's theorem would not apply. Then, can't we create an axiomatic system which defines Arithmetic but is not effective? If so, it could be complete and consistent at the same time, right?. Why do we assume that the theory is effective? Why is that so important? In this case, being effective equivalent to having a finite list of axioms?