Difference between revisions of "Class"
Asymptosis (talk | contribs) (New page: In set theory, a class refers to a set made up of other sets. For example, if <math>A=\{a,b,c\}</math> is a set, then <math>\mathscr{A}=\{\{a\},\{b\},\{b,c\}\}</math> is a class ma...) |
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| − | In [[set theory]], a class | + | In [[set theory]], a class is essentially a set which we do not call a set for logical or semantic reasons. For example, if <math>A=\{a,b,c\}</math> is a set, then <math>B=\{\{a\},\{b\},\{b,c\}\}</math> is a class consisting of some subsets of <math>A</math>. In this example though, <math>B</math> can also be called a set. |
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| + | To understand why one would make such a distinction, consider Russell's Paradox: | ||
| + | "Define <math>T</math> to be the set of all sets which do not contain themselves. Is it true or not that <math>T\in T</math>?" If <math>T\in T</math>, then <math>T</math> must not contain itself; that is, <math>T\not\in T</math>. If <math>T\not\in T</math>, then it must be because <math>T \in T</math>. Either way there is a contradiction. One resolution to Russell's paradox changes the language every so slightly: "Define <math>T</math> to be the class of all sets which do not contain themselves. Is it true or not that <math>T\in T</math>?" Indeed, <math>T\not\in T</math> for <math>T</math> is not a set. | ||
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| + | Compare Russell's Paradox to the Barber of Seville problem: "The barber of Seville shaves exactly those men who do not shave themselves. How can this be?" Naturally, the barber is a woman. | ||
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[[Category:Mathematics]] | [[Category:Mathematics]] | ||
Latest revision as of 14:28, 11 August 2008
In set theory, a class is essentially a set which we do not call a set for logical or semantic reasons. For example, if 
is a set, then 
is a class consisting of some subsets of 
. In this example though, 
can also be called a set.
To understand why one would make such a distinction, consider Russell's Paradox:
"Define 
to be the set of all sets which do not contain themselves. Is it true or not that 
?" If , then must not contain itself; that is, 
. If , then it must be because 
. Either way there is a contradiction. One resolution to Russell's paradox changes the language every so slightly: "Define to be the class of all sets which do not contain themselves. Is it true or not that ?" Indeed, for is not a set.
Compare Russell's Paradox to the Barber of Seville problem: "The barber of Seville shaves exactly those men who do not shave themselves. How can this be?" Naturally, the barber is a woman.
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