Brief Summary
This video explains Gödel's incompleteness theorems, starting with an introduction to formal reasoning and interactive proof systems. It covers the first incompleteness theorem, which states that within any consistent formal system capable of expressing basic arithmetic, there exist statements that cannot be proven true or false within the system itself. The video also touches on the second incompleteness theorem, which asserts that such a system cannot prove its own consistency.
- Gödel's incompleteness theorems demonstrate fundamental limitations in formal systems.
- The first theorem shows the existence of unprovable statements within a system.
- The second theorem proves that a system cannot prove its own consistency.
- The concept relies on encoding the system within itself, similar to writing a compiler in its own language.
Introduction to Formal Reasoning and Completeness
The discussion begins with an introduction to formal reasoning using the interactive proof system Lean. Two propositions, P1 (for all natural numbers n, n + 0 = n) and P2 (for all natural numbers n, n + n = n), are examined. P1 can be proven in Lean, while the negation of P2 can also be proven. This leads to the central question: for any proposition P, can either P or its negation be proven within a given proof system? This question of completeness was originally posed by Hilbert.
The P vs. NP Problem and Hilbert's Approach
The video introduces the famous P versus NP problem in computer science, which asks whether every problem whose solution can be quickly verified (NP) can also be quickly solved (P). While most believe P is not equal to NP, there is no proof. Hilbert believed that if a logical system is incomplete, axioms could be added to make it complete, suggesting that adding "P is not equal to NP" as an axiom might solve the problem.
Gödel's Incompleteness Theorem
Gödel demonstrated that within the theory of arithmetic (natural numbers with basic axioms), it's possible to define a proposition G such that neither G nor its negation can be proven. This extends beyond arithmetic to any system that includes arithmetic, such as Lean. The key is that a system can talk about itself.
Encoding Propositions and Provability
Gödel's approach involves representing propositions as natural numbers. Any proposition P can be represented by a unique natural number denoted as "code of P". A predicate "proof" is defined as the set of natural numbers corresponding to provable propositions. The statement "I can prove P" is equivalent to "I can prove that the code of P is in the set proof," effectively encoding provability within the system. This is analogous to writing a compiler for C in C.
Predicates and Diagonalization
The concept is extended to predicates, which are sets of natural numbers. A predicate Q can be encoded such that proving Q(n) is equivalent to proving that the pair (code of Q, n) is in a "proof two" predicate. A set V is defined as the set of codes n such that it cannot be proven that (n, n) is not in proof two. This uses a diagonalization technique, similar to proving undecidability or the existence of different sizes of infinity.
Constructing the Gödel Sentence
A Gödel sentence G is constructed, asking whether the code of V is an element of V. By definition, this is true if and only if it's not provable that the code of V is an element of V. This leads to a contradiction: G is provable if and only if not G is provable. If G could be proven, then not G could also be proven, leading to a contradiction and making the system inconsistent. The same occurs if not G could be proven. Therefore, G is a sentence that can neither be proven nor disproven within the system, illustrating the incompleteness theorem.
The Second Incompleteness Theorem
The second incompleteness theorem states that a system cannot prove its own consistency. Specifically, it cannot prove that "false is not provable." Proving consistency within the system is akin to "pulling yourself out of the swamp." This theorem is proven by taking the first incompleteness theorem and encoding its proof within arithmetic.
Implications and Limitations
The incompleteness theorem applies to any system that includes arithmetic but can be avoided by making the system weaker. For example, Presburger arithmetic (addition without multiplication) is complete but cannot talk about itself. Systems like the logic of real numbers or Euclidean geometry are complete because they lack the ability to encode themselves.
