effective procedures in general

Question

Any Godel numbering must be effective. Given a number, there should be an effective procedure to determine whether it numbers a formula or proof, and, if it does, what it numbers. Similarly, given a formula or proof, there should be an effective procedure to determine its Godel number. However, effectiveness is normally defined as a property of functions on the natural numbers. How, then, does one define effectiveness in general such that a Godel numbering counts? Thanks!