This is a followup to an earlier post where I talked about Gödel’s First Incompleteness Theorem. Here, I discuss the Second Incompleteness Theorem, and further implications.
Could you remind me what the theorem was?
The theorem states that a consistent formal system cannot prove its own consistency.
As previously discussed, there are a couple qualifiers. The formal system must include some amount of arithmetic, and must have a computable set of axioms.
What does consistency mean?
A system is consistent if it cannot prove any contradictions. A system is inconsistent if it can prove a contradiction.
Contradictions sound bad. Are they bad?
Yes. The Explosion Principle states that if you can prove a direct contradiction, then you can prove absolutely any statement.
Here’s how the Explosion Principle works. Suppose A and not-A are both provable. Now consider statement B. “(A implies B) or (not-A implies B)” is a tautology. Since both A and not-A, that means we can prove B. Following the same procedure we can also prove not-B.