[Verse 1] In nineteen-thirty-one, a young logician dreamed Of systems talking back about themselves, it seemed He took the statements, gave each one a code Numbers became the language that provability showed Gödel built a sentence, twisted and clever Said "I cannot be proven" — a paradox forever [Chorus] Self-reference breaks the chain Truth outrunning proof's domain What the system cannot see Lives inside its own machinery Incomplete by its design Every formal paradigm [Verse 2] If you prove the statement, then you've proven false The system contradicts itself, logic somersaults If you cannot prove it, then the statement's true But unprovable forever — incompleteness breaks through Diagonal construction, Cantor's ghost returns In the halls of mathematics, certainty burns [Chorus] Self-reference breaks the chain Truth outrunning proof's domain What the system cannot see Lives inside its own machinery Incomplete by its design Every formal paradigm [Bridge] Hilbert's program crumbles down Complete foundations cannot be found From arithmetic's simple rules Spring the limits of our tools Turing machines will halt or spin Rice's theorem echoes from within [Verse 3] Every system strong enough to count and add Contains these hidden statements, neither good nor bad Just true but unprovable, blind spots in the light Self-awareness costs us mathematical sight The template spreads through logic's vast expanse Undecidability's eternal dance [Chorus] Self-reference breaks the chain Truth outrunning proof's domain What the system cannot see Lives inside its own machinery Incomplete by its design Every formal paradigm [Outro] What we cannot capture In our nets of proof Gödel's ghost still whispers Mathematical truth
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