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Gödel published his remarkable proof in 1931. He showed that in any consistent (first-order) axiomatic mathematical system there are always propositions that cannot be proved or disproved using the axioms of the system. He additionally showed that it is impossible to prove the consistency of the axioms from those same axioms.  This was the famous incompleteness theorem: any axiomatic system powerful enough to describe arithmetic on natural numbers cannot be both consistent and complete.  Moreover, the consistency of the axioms cannot be proven within the system.

Gödel published his remarkable proof in 1931. He showed that in any consistent (first-order) axiomatic mathematical system there are always propositions that cannot be proved or disproved using the axioms of the system. He additionally showed that it is impossible to prove the consistency of the axioms from those same axioms.  This was the famous incompleteness theorem: any axiomatic system powerful enough to describe arithmetic on natural numbers cannot be both consistent and complete.  Moreover, the consistency of the axioms cannot be proven within the system.

Gödel's work abruptly ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of first-order axioms for all of mathematics that is both provably consistent as well as complete.  [[Bertrand Russell]] had already published, in Principia Mathematica (1910-13), a massive attempt to axiomatize mathematics in a consistent way.  Gödel's proof also showed that the formalist approach of [[David Hilbert]] was bound to fail to prove consistency.

Gödel's work abruptly ended a half-century of attempts, beginning with the work of Frege and culminating in ''Principia Mathematica'' and Hilbert's formalism, to find a set of first-order axioms for all of mathematics that is both provably consistent as well as complete.  [[Bertrand Russell]] had already published, in ''Principia Mathematica'' (1910-13), a massive attempt to axiomatize mathematics in a consistent way.  Gödel's proof also showed that the formalist approach of [[David Hilbert]] was bound to fail to prove consistency.

The [[Gödel's incompleteness theorems|incompleteness theorems]] also imply that there is no mechanical procedure which would determine, for all sentences of mathematics S, whether or not S was a theorem of the axioms for mathematics.  

The [[Gödel's incompleteness theorems|incompleteness theorems]] also imply that there is no mechanical procedure which would determine, for all sentences of mathematics S, whether or not S was a theorem of the axioms for mathematics.