# Changes

Gödel took [[set theory]] originally developed by [[Georg Cantor]] to new heights

[[Image:Goel.jpg|thumb|right|Kurt Gödel at Institute for Advanced Study]]

'''Kurt Gödel''' (1906-1978) was an Austrian mathematician who did pioneering work in logic and the foundations of mathematics. His Incompleteness Theorem demonstrated some limitations of the program that would have placed all of mathematics on a complete axiomatic basis. He immigrated to the [[United States]] and worked at the Institute for Advanced Study at Princeton, New Jersey~~. He died insane after starving himself to death~~.

Gödel took [[set theory]] originally developed by [[Georg Cantor]] to new heights, as explained by Boston University Professors Juliet Floyd and Akihiro Kanamori in "How Gödel Transformed Set Theory": {{cquote|How Gödel transformed set theory can be broadly cast as follows: On the larger stage, from the time of Cantor, sets began making their way into topology, algebra, and analysis so that by the time of Gödel, they were fairly entrenched in the structure and language of mathematics. But how were sets viewed among set theorists, those investigating sets as such? Before Gödel, the main concerns were what sets are and how sets and their axioms can serve as a reductive basis for mathematics. Even today, those preoccupied with ontology, questions of mathematical existence, focus mostly upon the set theory of the early period. After Gödel, the main concerns became what sets do and how set theory is to advance as an autonomous field of mathematics.<ref>www.ams.org/notices/200604/fea-kanamori.pdf</ref>}} Gödel published his remarkable proof of the Incompleteness Theorem 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.