Gödel's Second Incompleteness Theorem and Foundationalism

The Dream of Absolute Certainty: Hilbert's Programme

To understand the impact of G2, we first need to appreciate the problem it was responding to. In the early 20th century, the discovery of paradoxes in set theory (like Russell's paradox) had triggered a "foundational crisis." David Hilbert, a leading mathematician of the era, proposed a bold programme to restore certainty to mathematics. His goal was to save infinitary mathematics (like analysis and set theory), which he saw as a "paradise" from which mathematicians should not be expelled.

The following reading from Neil Barton's course notes clearly outlines Hilbert's strategy.

As the notes explain, Hilbert's Programme had two main objectives:

1. Completeness: To formalize all of mathematics into a single axiomatic system that could, in principle, prove or disprove any mathematical statement.
2. Finitary Consistency Proof: To prove that this powerful system was consistent (i.e., would never lead to a contradiction) using only elementary, 'finitary' methods (roughly, what we can formalize in a weak system like Primitive Recursive Arithmetic, or PRA).

The first goal was a hope for total knowledge. The second was the crucial "insurance policy." It would show that even though infinitary methods talk about abstract objects we can't directly inspect, they would never lead us astray in the concrete, finitary part of mathematics. As we saw last time, Gödel's first theorem shattered the goal of completeness. Now, let's see what the second theorem did to the insurance policy.

The Final Blow: The Second Incompleteness Theorem (G2)

The second incompleteness theorem is a direct and profound consequence of the first. In essence, it states:

Gödel's Second Incompleteness Theorem (G2): Any consistent formal system $T$ that is strong enough to carry out basic arithmetic cannot prove its own consistency.

The statement "T is consistent" can be expressed as a sentence in the language of arithmetic, often written as $\mathrm{Con}(T)$. G2 shows that if $T$ is indeed consistent, $\mathrm{Con}(T)$ is another example of a true but unprovable statement within $T$.

How does this follow from G1? The key is that the entire proof of the first incompleteness theorem can itself be formalized within arithmetic. Let's watch a short video that walks through this elegant and devastating argument.

Let's summarize the steps of the reasoning:

1. From our last lesson, we know that the Gödel sentence $G_T$ is equivalent to "I am not provable in $T$".
2. The proof of G1 establishes the conditional: If $T$ is consistent, then $G_T$ is unprovable (and therefore true).
3. This entire line of reasoning can be formalized and proven within the system $T$ itself. This gives us the theorem: $T⊢\mathrm{Con}(T)\to G_T$.
4. Now, suppose for the sake of argument that $T$ could prove its own consistency. That is, suppose $T⊢\mathrm{Con}(T)$.
5. By the rule of modus ponens, from $T⊢\mathrm{Con}(T)$ and $T⊢\mathrm{Con}(T)\to G_T$, it would follow that $T⊢G_T$.
6. But G1 showed that if $T$ is consistent, it cannot prove $G_T$.
7. Therefore, our initial supposition must be false. A consistent system $T$ cannot prove $\mathrm{Con}(T)$.

This result directly demolishes the second, and arguably more critical, part of Hilbert's Programme. Hilbert wanted to use a simple, finitary theory $F$ (like PRA) to prove the consistency of a much more powerful, infinitary theory $I$ (like set theory). But G2 shows that the powerful theory $I$ cannot even prove its own consistency, let alone have its consistency proven by a weaker theory like $F$. The dream of an absolute, internally-guaranteed foundation was gone.

Assessing the Aftermath: The Fate of Foundationalism

The philosophical impact of G2 was immense. It forced a re-evaluation of what a "foundation" for mathematics could possibly be.

The Refutation of Hilbert's Programme

It is widely agreed that G2 refuted Hilbert's Programme in its original, ambitious form. The goal of securing infinitary mathematics with a risk-free, finitary proof was shown to be mathematically impossible.

The "Relativized" Programme

However, this was not the end of foundational studies. Instead, Hilbert's Programme was transformed. The quest for absolute certainty was replaced by a more nuanced, "relativized" programme.

• Relative Consistency Proofs: While we cannot prove PA ⊢ Con(PA), we can prove the consistency of weaker theories from stronger ones. For example, standard set theory (ZFC) is strong enough to prove that Peano Arithmetic (PA) is consistent. We can write this as ZFC ⊢ Con(PA).
• A Hierarchy of Confidence: This creates a hierarchy of formal systems, ordered by "consistency strength." A proof of Con(PA) from ZFC doesn't give us absolute certainty in PA; it tells us that PA is consistent if ZFC is consistent. Our confidence in arithmetic is thus transferred to our confidence in set theory.
• The Role of Philosophy: This new situation brings philosophy back to the forefront. Since we can never have an ultimate, self-verifying foundational theory, the choice of which axioms to accept (e.g., the axioms of ZFC, or even stronger axioms like those for large cardinals) cannot be justified on purely mathematical grounds. As Professor Hamkins notes in the Q&A portion of his video (LINK, 01:16:58-01:18:30), mathematicians are forced to engage in what are essentially philosophical arguments about the intuitive evidence, fruitfulness, and justification for new axioms.

Impact on Other Foundationalisms

The impact of G2 extends beyond Hilbert's specific programme. Consider Logicism, the thesis that mathematics is reducible to logic. As Raatikainen discusses (LINK, section 3), G2 poses a serious epistemic problem for this view. Even if a logicist could formulate a formal system of "logic" powerful enough to derive arithmetic, G2 implies that we could never know with mathematical certainty that this system is consistent. This undermines the logicist's goal of showing mathematics is grounded in the supposedly self-evident and secure truths of logic.

Conclusion

Today, we have explored the philosophical shockwave created by Gödel's second incompleteness theorem. Where the first theorem showed that any single formal system is incomplete, the second theorem showed that no system can even vouch for its own coherence.

Key Takeaways:

• G2 states that a sufficiently strong, consistent formal system cannot prove the arithmetical statement corresponding to its own consistency.
• Impact on Hilbert's Programme: G2 delivered a decisive refutation to the programme's central goal of providing a finitary consistency proof for infinitary mathematics. It showed that the quest for absolute, self-verifying foundations is impossible.
• The Transformation of Foundationalism: The failure of Hilbert's Programme led to a "relativized" approach. The goal is no longer absolute certainty, but calibrating the consistency of different theories against each other.
• The Return of Philosophy: By closing the door on purely formal justification, G2 forces mathematicians and philosophers to debate the grounds for accepting foundational axioms—a debate that is necessarily philosophical in nature.

Preview of the Next Lesson:

Gödel's theorems revealed the limits of the proof-theoretic approach to logic and foundations, which conceives of logic as a system of formal deduction. This opened the door for a different perspective to gain prominence: the model-theoretic or semantic approach. In our next lesson, we will begin our fifth module, "The Nature of Logical Consequence," by examining Tarski's semantic theory of truth, a landmark attempt to give a rigorous definition of truth for formal languages and, from it, a new definition of logical consequence itself.