Full Idea
For Ernie, the successor of a number x was the set consisting of x and all the members of x, while for Johnny the successor of x was simply [x], the unit set of x - the set whose only member is x.
Clarification
Ernie actually refers to Von Neumann's set theory, and Johnny refers to Zermelo's!
Gist of Idea
The successor of x is either x and all its members, or just the unit set of x
Source
Paul Benacerraf (What Numbers Could Not Be [1965], II)
Book Reference
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.278
A Reaction
See also Idea 9900. Benacerraf's famous point is that it doesn't seem to make any difference to arithmetic which version of set theory you choose as its basis. I take this to conclusively refute the idea that numbers ARE sets.
Related Ideas
Idea 9900 For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf]
Idea 8762 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]