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Full Idea
Given any x we have the singleton {x}, which is defined by the pairing axiom to be {x,x}.
Gist of Idea
The singleton is defined using the pairing axiom (as {x,x})
Source
Herbert B. Enderton (Elements of Set Theory [1977], 2:19)
Book Ref
Enderton,Herbert B.: 'Elements of Set Theory' [Posts + Telecoms 2006], p.19
A Reaction
An interesting contrivance which is obviously aimed at keeping the axioms to a minimum. If you can do it intuitively with a new axiom, or unintuitively with an existing axiom - prefer the latter!
| 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] |