Second Incompleteness Theorem

Mathematical Logic · Axiom Academy

LESSON Second Incompleteness Theorem No consistent system can prove its own consistency Mathematical Logic • Unit 5 - Gödel's Theorems 1. Statement of the Second Incompleteness Theorem Gödel's Second Incompleteness Theorem is perhaps even more devastating than the first. It states that no sufficiently powerful consistent formal system can prove its own consistency. Gödel's Second Incompleteness Theorem If PA is consistent, then PA cannot prove its own consistency. More precisely: If Peano Arithmetic is consistent, then there is no proof in PA of the statement "PA is consistent". This theorem follows from the First Incompleteness Theorem and reveals a fundamental limitation: to prove a system is consistent, you must step outside that system. 2. Con(PA): The Formal Statement of Consistency To make the theorem precise, we need to formalize the statement "PA is consistent" within the language of arithmetic itself. The Consistency Statement Con(PA) We define Con(PA) to be the formalized arithmetic statement that expresses "PA is consistent". This says: "There does not exist a proof in PA of the contradiction " Informal: "PA will never prove a contradiction" Formal: "There is no natural number that codes a proof of 0=1 in PA" Using Gödel numbering, we can express: 3. Con(PA) is Equivalent to the Gödel Sentence G This is the key insight: the consistency statement Con(PA) is logically equivalent to the Gödel sentence G we constructed in the First Incompleteness Theorem.

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