Kurt Gödel - Revision history

Alex Kosh: /* References */

2021-12-12T04:54:58Z

‎References

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2021-12-12T04:54:24Z

‎References

Aschlafly at 21:15, September 2, 2019

2019-09-02T21:15:32Z

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2019-08-14T04:42:31Z

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2019-06-23T00:28:37Z

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2019-06-23T00:06:11Z

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Aschlafly: Politically, he supported Republican President Dwight Eisenhower and was critical of the Democrat Harry Truman.

2019-06-23T00:05:08Z

Politically, he supported Republican President Dwight Eisenhower and was critical of the Democrat Harry Truman.

Aschlafly: Gödel was offered only an unpaid lecturer position after he published his breakthrough papers.

2019-06-21T04:37:31Z

Gödel was offered only an unpaid lecturer position after he published his breakthrough papers.

Aschlafly at 04:31, June 21, 2019

2019-06-21T04:31:10Z

Aschlafly: He completely rejected the atheism of the Vienna Circle to which he was introduce, and Godel was ostracized from academia despite his immense intellectual achievements.

2019-06-21T04:30:41Z

He completely rejected the atheism of the Vienna Circle to which he was introduce, and Godel was ostracized from academia despite his immense intellectual achievements.

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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.
-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.

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 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 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.
 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.

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 [[Image:Goel.jpg|thumb|right|Kurt Gödel at Institute for Advanced Study]]
-'''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, and Gödel was ostracized from academia despite his immense intellectual achievements.<ref>Gödel was offered only an unpaid lecturer position after he published his breakthrough papers.</ref>
+'''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 took [[set theory]], as 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":