# How do logicians think of strength of proof systems?

I want to understand how logicians reason about strengths of proof systems and argue relative strengths of proof systems. I want to appreciate the validity of the reasoning by which we establish relative proof strengths. Some milestones in this area are the works by Godel, Gentzen and Turing, but I am unequipped to follow Gentzen's "Consistency Proof" or Turing's dissertation "Systems of Logic based on Ordinals".

What are the starting areas I must master to be able to understand these topics on strengths of proof systems?

• There are different ways by which logicians measure "strength", consistency strength and interpretability strength are the most common ones, some interrelations are discussed in this MathOverflow post. For foundations in the work of Hilbert, Gödel and Gentzen and pointers to further reading see SEP, Proof theory. Sep 20, 2021 at 0:46
• @Conifold Is proof theory different from Metamathematics? How exactly do they overlap?
– Ajax
Sep 20, 2021 at 13:07
• It is not entirely clear what you are asking. A proof system is normally considered to be a distinct thing from a logic. A logic might be identified with the set of its theorems, while a proof system is a formal way of determining what those theorems are. A given logic can have several proof systems. For example, propositional logic has Hilbert-style axiom systems, natural deduction systems, sequent calculus, tableau methods. These are of equal strength in the sense that they all prove the same theorems. Sep 20, 2021 at 17:29
• Logics themselves can be of different strengths, in that one logic might have as its theorems a proper subset of the theorems of another. In this sense, minimal logic is weaker than intuitionistic logic, and intuitionistic logic is weaker than classical logic. Sep 20, 2021 at 17:29
• Concerning what you must master to understand what you are interested in: What should a person interested in the philosophy of mathematics know? Concerning the strength of formal systems, see this post (which assumes you know what is listed in the other post). Oct 25, 2021 at 8:37