Full Idea
The Von Neumann numbers have a structural isomorphism to the natural numbers - each number is the set of all its predecessors, so 2 is the set of 0 and 1. This helps proofs, but is unacceptable. 2 is not a set with two members, or a member of 3.
Gist of Idea
Von Neumann numbers are helpful, but don't correctly describe numbers
Source
Stephen Read (Thinking About Logic [1995], Ch.4)
Book Reference
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.106