Kurt Gödel (nonfiction): Difference between revisions
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Considered along with [[Aristotle (nonfiction)|Aristotle]], [[Alfred Tarski (nonfiction)|Alfred Tarski]], and [[Gottlob Frege (nonfiction)|Gottlob Frege]] to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as [[Bertrand Russell (nonfiction)|Bertrand Russell]], [[Alfred North Whitehead (nonfiction)|Alfred North Whitehead]], and [[David Hilbert (nonfiction)|David Hilbert]] were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by [[Georg Cantor (nonfiction)|Georg Cantor]]. | Considered along with [[Aristotle (nonfiction)|Aristotle]], [[Alfred Tarski (nonfiction)|Alfred Tarski]], and [[Gottlob Frege (nonfiction)|Gottlob Frege]] to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as [[Bertrand Russell (nonfiction)|Bertrand Russell]], [[Alfred North Whitehead (nonfiction)|Alfred North Whitehead]], and [[David Hilbert (nonfiction)|David Hilbert]] were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by [[Georg Cantor (nonfiction)|Georg Cantor]]. | ||
Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers. | Gödel published his two [[Gödel's incompleteness theorems (nonfiction)|incompleteness theorems]] in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers. | ||
He also showed that neither the [[Axiom of choice (nonfiction)|axiom of choice]] nor the continuum hypothesis can be disproved from the accepted axioms of [[Set theory (nonfiction)|set theory]], assuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. | He also showed that neither the [[Axiom of choice (nonfiction)|axiom of choice]] nor the continuum hypothesis can be disproved from the accepted axioms of [[Set theory (nonfiction)|set theory]], assuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. | ||
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* [[Gödel's incompleteness theorems (nonfiction)]] | |||
* [[Mathematician (nonfiction)]] | * [[Mathematician (nonfiction)]] | ||
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* [https://en.wikipedia.org/wiki/Kurt_G%C3%B6del Kurt Gödel] @ Wikipedia | * [https://en.wikipedia.org/wiki/Kurt_G%C3%B6del Kurt Gödel] @ Wikipedia | ||
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[[Category:People (nonfiction)]] | [[Category:People (nonfiction)]] | ||
[[Category:Philosophers (nonfiction)]] | [[Category:Philosophers (nonfiction)]] | ||
Revision as of 10:37, 15 November 2017
Kurt Friedrich Gödel (/ˈɡɜːrdəl/, US: /ˈɡoʊ-/; German: [ˈkʊɐ̯t ˈɡøːdl̩] (About this sound listen); April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher.
Considered along with Aristotle, Alfred Tarski, and Gottlob Frege to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell, Alfred North Whitehead, and David Hilbert were analyzing the use of logic and set theory to understand the foundations of mathematics pioneered by Georg Cantor.
Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.
He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs.
He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.
In the News
Fiction cross-reference
Nonfiction cross-reference
- Axiom of choice (nonfiction)
- Gödel's incompleteness theorems (nonfiction)
- Mathematician (nonfiction)
External links:
- Kurt Gödel @ Wikipedia