I've been always curious about Godel's incompleteness theorem and today, I decided to understand it. To understand the theorem, I started from the concepts of propositional logic and predicate logic. So to speak, I was a complete novice in mathematical logic before.
I studied through to the point that Godel's theorem states that the set of axioms cannot satisfy three conditions at the same time: 1. effective axiomatization 2. completeness 3. consistency.
I think I understand what they mean: The effective axiomatization is the existence of an algorithm that can enumerate all the theorems derived from the set of axioms. The completeness is that any statement in the language of the axioms can be proved or disproved. The consistency means that statement cannot be proved and disproved at the same time under the set of axioms.
I can connect with the concepts of completeness and consistency. Completeness is what Godel's theorem tries to disprove. Consistency seems quite natural because if it's not, the axioms are absurd as the system of axioms can't be used for proof. But I don't quite understand why we need effective axiomatization.
Can somebody please explain to me why we need effective axiomatization for Godel's theorem? And what's the implication of the condition?