14.3 Second Incompleteness Theorem

14.3 Second Incompleteness Theorem

The first incompleteness theorem shows that some true statements cannot be proved. The second goes further. It shows that a system strong enough to express arithmetic cannot prove its own consistency.

Formalizing Consistency

A system is consistent if it does not prove a contradiction. For example, it does not prove both $A$ and $¬A$.

Using arithmetization, we express this inside arithmetic. Let:

mean “the formula with code $y$ is provable”.

A contradiction can be represented by a fixed formula such as:

Let $\ulcorner 0 = 1 \urcorner$ be its Gödel number.

Then the statement “the system is consistent” can be written as:

This says: there is no proof of a contradiction.

What We Want to Show

We want to understand whether the system can prove:

that is, its own consistency.

At first glance, this may seem reasonable. If a system is correct, why can it not prove that it is consistent?

Gödel’s theorem shows that this expectation fails.

Link with the Gödel Sentence

Recall the Gödel sentence $G$:

The key observation is that $G$ is closely related to consistency.

If the system were inconsistent, it could prove every formula, including $G$. So inconsistency implies:

On the other hand, if the system is consistent, then $G$ is not provable. So consistency implies:

which is exactly what $G$ states.

So informally, we have:

This implication can be formalized inside the system.

The Key Argument

Assume the system proves its own consistency:

We also have, within the system:

By modus ponens, this yields:

But from the first incompleteness theorem, we know:

if the system is consistent.

This is a contradiction. Therefore:

Statement of the Theorem

Let $T$ be a formal system that:

• is consistent
• can represent arithmetic
• can formalize its own proof relation

Then:

So $T$ cannot prove its own consistency.

Interpretation

The result does not say that the system is inconsistent. It says that the system cannot establish its own consistency using only its internal methods.

To prove consistency, one must move to a stronger system.

For example:

• arithmetic cannot prove its own consistency
• but a stronger system may prove the consistency of arithmetic

This leads to a hierarchy of systems, each capable of proving the consistency of weaker ones.

Why This Happens

The limitation arises from self-reference.

If a system could prove its own consistency, then it could prove that the Gödel sentence is unprovable. But that would indirectly prove the Gödel sentence itself, which is impossible in a consistent system.

So the system must stop short. It cannot certify its own reliability from within.

Broader View

The second incompleteness theorem shows that no sufficiently strong formal system can fully justify itself.

Any such justification requires stepping outside the system.

This has important consequences for foundations:

• no single system can serve as a final, self-verifying foundation
• consistency proofs always rely on stronger assumptions
• mathematical certainty is layered rather than absolute

These ideas shape modern proof theory and the study of formal systems.