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Full Idea
Cantor taught that a set is 'a many, which can be thought of as one'. ...After a time the unfortunate beginner student is told that some classes - the singletons - have only a single member. Here is a just cause for student protest, if ever there was one.
Gist of Idea
If a set is 'a many thought of as one', beginners should protest against singleton sets
Source
report of George Cantor (works [1880]) by David Lewis - Parts of Classes 2.1
Book Ref
Lewis,David: 'Parts of Classes' [Blackwell 1991], p.30
A Reaction
There is a parallel question, almost lost in the mists of time, of whether 'one' is a number. 'Zero' is obviously dubious, but if numbers are for counting, that needs units, so the unit is the precondition of counting, not part of it.
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] |