Work · Analytic Philosophy
On Formally Undecidable Propositions of Principia Mathematica and Related Systems I
The incompleteness theorems: no consistent system of this kind can prove all arithmetical truths, nor its own consistency.
Introduction
Kurt Gödel, aged 24 and working in Vienna, announced the result at a conference in Königsberg in September 1930; the paper appeared in the Monatshefte für Mathematik und Physik in 1931. It concerns formal systems strong enough to express elementary arithmetic, such as the system of Russell and Whitehead’s Principia Mathematica. Gödel encodes formulas and proofs as numbers (“Gödel numbering”) and constructs a sentence that in effect says of itself that it cannot be proved; if the system is consistent, this sentence is true but unprovable within it. The second theorem follows from this. The result is generally taken to have defeated the original aim of David Hilbert’s programme, to secure all of mathematics by a finitary consistency proof. The planned Part II never appeared.
Connections
Responds to
- Principia Mathematica1910 · Bertrand Russell
Taken up in
- On Computable Numbers, with an Application to the Entscheidungsproblem1936 · Alan Turing
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