It is commonly thought that logic and sometimes mathematics can be used to prove that a god exists. Indeed, many theologians and philosophers have written great tracts that offer proofs that God exists, although, people are rarely converted by such proofs, but they are pleased to know that their beliefs are well founded. Unfortunately, it can be shown quite easily that there are contradictions in logic (Russell's paradox) and with some effort, it is possible to produce contradictions in mathematics (Kurt Godel's inconsistency). When we study the natural world, however, there cannot be contradictions. There are just facts. Logic and mathematics are human constructs and as such are fundamentally flawed. We are not gods. It follows that our logic (so far) is not an exact model of the world. The world is not logical in that sense. Suppose I tell you I have a pet dog and you said, "prove it". I wouldn't consider using a mathematical proof, I would produce physical evidence. For example, I could show you a photograph and unless are are pedantic, that would be proof enough. If you demanded absolute proof, then I have to do a lot more work, too much to bother, perhaps. So if your project is to prove that God exists and you tell me he's undetectable, then you have no hope. You certainly can't do it using human logic.