Difference between revisions of "Russell's paradox"
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| − | In the [[foundations of mathematics]], '''Russell's paradox''' (also known as '''Russell's antinomy'''), discovered by [[Bertrand Russell]] in 1901, showed that some attempted formalizations of the [[naive set theory]] created by [[Georg Cantor]] led to a | + | In the [[foundations of mathematics]], '''Russell's paradox''' (also known as '''Russell's antinomy'''), discovered by [[Bertrand Russell]] in 1901, showed that some attempted formalizations of the [[naive set theory]] created by [[Georg Cantor]] led to a [[paradox]]. |
== Description == | == Description == | ||
| Line 11: | Line 11: | ||
Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical [[Zermelo–Fraenkel set theory]] (ZFC). | Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical [[Zermelo–Fraenkel set theory]] (ZFC). | ||
| − | The essential difference between Russell's and Zermelo's solution to the paradox is that Zermelo altered the axioms of set theory while preserving the logical language in which they are expressed (the language of ZFC, with the help of Skolem, turned out to be first-order logic) while Russell altered the logical language itself. | + | The essential difference between Russell's and Zermelo's solution to the paradox is that Zermelo altered the axioms of set theory while preserving the logical language in which they are expressed (the language of ZFC, with the help of Skolem, turned out to be [[first-order logic]]) while Russell altered the logical language itself. |
== See also == | == See also == | ||
* [[Bertrand Russell|Russell, Bertrand]] | * [[Bertrand Russell|Russell, Bertrand]] | ||
| + | * [[First-order logic]] | ||
* [[Foundations of mathematics]] | * [[Foundations of mathematics]] | ||
* [[Mathematics]] | * [[Mathematics]] | ||
* [[Naive set theory]] | * [[Naive set theory]] | ||
* [[Paradox]] | * [[Paradox]] | ||
| + | * [[Zermelo–Fraenkel set theory]] | ||
== External links == | == External links == | ||
| Line 28: | Line 30: | ||
[[Category:Logic]] | [[Category:Logic]] | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
| + | [[Category:Paradox]] | ||
[[Category:Set theory]] | [[Category:Set theory]] | ||
Latest revision as of 11:30, 17 August 2016
In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that some attempted formalizations of the naive set theory created by Georg Cantor led to a paradox.
Description
The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to David Hilbert, Husserl and other members of the University of Göttingen.
According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox.
In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory.
Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory (ZFC).
The essential difference between Russell's and Zermelo's solution to the paradox is that Zermelo altered the axioms of set theory while preserving the logical language in which they are expressed (the language of ZFC, with the help of Skolem, turned out to be first-order logic) while Russell altered the logical language itself.
See also
- Russell, Bertrand
- First-order logic
- Foundations of mathematics
- Mathematics
- Naive set theory
- Paradox
- Zermelo–Fraenkel set theory
External links
- Russell's paradox @ Wikipedia
