Difference between revisions of "Kurt Gödel"
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Considered with [[Aristotle]] 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]], A. N. Whitehead, and [[David Hilbert]] were pioneering the use of [[logic]] and [[set theory]] to understand the [[foundations of mathematics]]. | Considered with [[Aristotle]] 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]], A. N. Whitehead, and [[David Hilbert]] were pioneering the use of [[logic]] and [[set theory]] to understand the [[foundations of mathematics]]. | ||
| − | 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. | + | Gödel published his [[Gödel's incompleteness theorems|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. | 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. | ||
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== See also == | == See also == | ||
| + | * [[Gödel's incompleteness theorems]] | ||
* [[Mathematician]] | * [[Mathematician]] | ||
Revision as of 05:29, 21 August 2015
Kurt Friedrich Gödel (/ˈkɜrt ɡɜrdəl/; German: [ˈkʊʁt ˈɡøːdəl] ( listen); April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher.
Considered with Aristotle 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, A. N. Whitehead, and David Hilbert were pioneering the use of logic and set theory to understand the foundations of mathematics.
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.
See also
External links
- Kurt Gödel @ Wikipedia
