Difference between revisions of "Gödel's incompleteness theorems"
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== Description == | == Description == | ||
| − | The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the [[philosophy of mathematics]]. | + | The theorems, proven by [[Kurt Gödel]] in 1931, are important both in [[mathematical logic]] and in the [[philosophy of mathematics]]. |
| − | The two results are widely, but not universally, interpreted as showing that [[David Hilbert]]'s program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem. | + | The two results are widely, but not universally, interpreted as showing that [[David Hilbert]]'s program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to [[Hilbert's second problem]]. |
== First incompleteness theorem == | == First incompleteness theorem == | ||
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* [[Axiomatic system]] | * [[Axiomatic system]] | ||
* [[Computer program]] | * [[Computer program]] | ||
| + | * [[Hilbert's second problem]] | ||
* [[Impossible Programs]] | * [[Impossible Programs]] | ||
* [[Kurt Gödel|Gödel, Kurt]] | * [[Kurt Gödel|Gödel, Kurt]] | ||
| Line 32: | Line 33: | ||
* [https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems Gödel's incompleteness theorems] @ Wikipedia | * [https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems Gödel's incompleteness theorems] @ Wikipedia | ||
| + | |||
| + | [[Category:Computation]] | ||
| + | [[Category:Computer science]] | ||
| + | [[Category:Logic]] | ||
| + | [[Category:Mathematics]] | ||
| + | [[Category:Numbers]] | ||
| + | [[Category:Paradox]] | ||
Revision as of 16:37, 20 April 2016
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic.
Contents
Description
The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics.
The two results are widely, but not universally, interpreted as showing that David Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem.
First incompleteness theorem
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic).
For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.
Second incompleteness theorem o
The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency.
See also
- Algorithm
- Axiomatic system
- Computer program
- Hilbert's second problem
- Impossible Programs
- Gödel, Kurt
- Mathematical logic
- Natural number
- Philosophy of mathematics
- Uncertainty
External links
- Gödel's incompleteness theorems @ Wikipedia
