I think that if you allow some axioms, you can describe it. But I'm not sure if it will be called complete. I think because "(next axiom)" is not defined, it w

I think that if you allow some axioms, you can describe it. But I'm not sure if it will be called complete. I think because "(next axiom)" is not defined, it will make your language not mappable to integer numbers and you avoid Godel proof.