Gödel's Incompleteness Theorems and Their Philosophical Implications

Kurt Gödel, then a twenty-five-year-old at the University of Vienna, published "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" in the journal Monatshefte für Mathematik und Physik in 1931, proving two theorems that permanently reshaped mathematical logic and its philosophical self-understanding. The first incompleteness theorem states that any consistent, effectively axiomatized formal system F rich enough to express elementary, Peano arithmetic contains a well-formed statement G(F) that is true but not provable within F — informally, a statement that in effect asserts "I am not provable in F," constructed via the technique now called Gödel numbering, which encodes the formulas and proofs of F as natural numbers so that syntactic properties of F become arithmetical properties expressible within F itself.