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Kurt Godel

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To every ?-consistent recursive class ? of formulae there correspond recursive class signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg (?) (where v is the free variable of r).
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Proposition VI, On Formally Undecidable Propositions in Principia Mathematica and Related Systems I (1931)
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Informally, recursive systems of axioms cannot be complete.

 
Kurt Godel

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