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'''Kurt Gödel''' (1906-1978) was an Austrian-born mathematician who is considered perhaps the greatest logician since [[Aristotle]].  His Incompleteness Theorem was a stunning proof of limitations on logic, at a time when leading mathematicians were working diligently to establish its completeness.  He immigrated to the [[United States]] and worked at the Institute for Advanced Study at Princeton, New Jersey, and is buried in that town with his wife.  Gödel was a devout [[Christian]] who believed in an afterlife and read the [[Bible]] regularly.<ref>https://www.quora.com/What-led-Kurt-G%C3%B6del-to-become-a-Christian</ref>  He completely rejected the [[atheism]] of the [[Vienna Circle]] to which he was introduced.

'''Kurt Gödel''' (1906-1978) was an Austrian-born mathematician who is considered perhaps the greatest logician since [[Aristotle]].  His Incompleteness Theorem was a stunning proof of limitations on logic, at a time when leading mathematicians were working diligently to establish its completeness.  He immigrated to the [[United States]] and worked at the Institute for Advanced Study at Princeton, New Jersey, and is buried in that town with his wife.  Gödel was a devout [[Christian]] who believed in an afterlife and read the [[Bible]] regularly.<ref>https://www.quora.com/What-led-Kurt-G%C3%B6del-to-become-a-Christian</ref>  He completely rejected the [[atheism]] of the [[Vienna Circle]] to which he was introduced.

Gödel was ostracized from academia despite his immense intellectual achievements, and was never offered a tenure-track position of professor.<ref>Gödel was offered only an unpaid lecturer position after he published his breakthrough papers.</ref>  He worked on developing a logical proof of [[God]] that build on the ontological proof by [[Saint Anselm]].  Politically, he supported Republican President [[Dwight Eisenhower]] and was critical of the [[Democrat]] [[Harry Truman]].

Gödel was ostracized from academia despite his immense intellectual achievements, and was never offered a tenure-track position of professor.<ref>Gödel was offered only an unpaid lecturer position after he published his breakthrough papers.</ref>  He worked on developing a logical proof of [[God]] that build on the ontological proof by [[Saint Anselm]].  Politically, he supported Republican President [[Dwight Eisenhower]] and was critical of the [[Democrat]] [[Harry Truman]].<ref>https://www.usna.edu/Users/math/meh/godel.html</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 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.