Full Idea
Ernie's number progression is [φ],[φ,[φ]],[φ,[φ],[φ,[φ,[φ]]],..., whereas Johnny's is [φ],[[φ]],[[[φ]]],... For Ernie 3 belongs to 17, not for Johnny. For Ernie 17 has 17 members; for Johnny it has one.
Clarification
See also Idea 9899
Gist of Idea
For Zermelo 3 belongs to 17, but for Von Neumann it does not
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
Benacerraf's point is that there is no proof-theoretic way to choose between them, though I am willing to offer my intuition that Ernie (Zermelo) gives the right account. Seventeen pebbles 'contains' three pebbles; you must pass 3 to count to 17.
Related Ideas
Idea 9899 The successor of x is either x and all its members, or just the unit set of x [Benacerraf]
Idea 8762 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]