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Full Idea
With the ordinary view of classes you would say that a class that has only one member was the same as that one member; that will land you in terrible difficulties, because in that case that one member is a member of that class, namely, itself.
Clarification
(the problem is classes that turn out to be members of themselves)
Gist of Idea
Normally a class with only one member is a problem, because the class and the member are identical
Source
Bertrand Russell (The Philosophy of Logical Atomism [1918], §VII)
Book Ref
Russell,Bertrand: 'Russell's Logical Atomism', ed/tr. Pears,David [Fontana 1972], p.126
A Reaction
The problem (I think) is that classes (sets) were defined by Frege as being identical with their members (their extension). With hindsight this may have been a mistake. The question is always 'why is that particular a member of that set?'
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| 6103 | Normally a class with only one member is a problem, because the class and the member are identical [Russell] |
| 13203 | The singleton is defined using the pairing axiom (as {x,x}) [Enderton] |
| 10813 | What on earth is the relationship between a singleton and an element? [Lewis] |
| 10814 | Are all singletons exact intrinsic duplicates? [Lewis] |
| 15497 | We can replace the membership relation with the member-singleton relation (plus mereology) [Lewis] |
| 15506 | If we don't understand the singleton, then we don't understand classes [Lewis] |
| 15511 | If singleton membership is external, why is an object a member of one rather than another? [Lewis] |
| 15513 | Maybe singletons have a structure, of a thing and a lasso? [Lewis] |
| 9551 | What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara] |
| 8956 | What is a singleton set, if a set is meant to be a collection of objects? [Szabó] |
| 14243 | The unit set may be needed to express intersections that leave a single member [Oliver/Smiley] |