Can Any Formal System Prove Its Own Consistency?

Since an inconsistent system can prove its own consistency...

• Say a proof for the consistency of a formal system (proved within the formal system) is known. There are two possible cases: 1. the formal system is consistent and it can be and has been proven to be, or 2. the formal system is inconsistent (i.e. contains a contradiction), thus anything is provable, hence the proof of its consistency. Is there a way to determine whether 1 or 2 is the case?

• Answer:

  Gödel's Incompleteness Theorem says that if a system is consistent, recursively axiomatised and adequate for arithmetics then it is cannot prove its own consistency. If your formal system can prove its own consistency, it must either be able to prove anything, eg that $0=1$, or be (consistent,) recursively axiomatised and adequate for arithmetic. So the question comes down to whether you can prove $0=1$ in it. In general this is undecidable. Indeed, this is exactly the Entcheidungsproblem, if I recall correctly.