What is it "that cannot be referred to in any way whatsoever"? That is obviously a reference but what does it refer to? Is there a concept which is inherently undefinable and cannot be referred to in any way whatsoever? And didn't I just refer to it?
That concept surely exists but it must non-exist without actually non-existing. Call it "Recursion" and insofar as words can be put to it, say that it "Is not what it is" and "Is what it is not".
That is about as independent and nonarbitrary as possible and thus Necessary. In a universe where there is anything, there will be existence and nonexistence and thus Recursion. In a universe where there is nothing at all and no possibility of anything at all there will be definite nonexistence and thus Recursion. And so forth for any condition or noncondition of condition. ((this probably goes down easier with a shot or two of vodka))
Thus the single axiom will be called "Recursion"