Full Idea
What Russell tried to show [at this time] was that Peano's Postulates (based on 'zero', 'number' and 'successor') could in turn be dispensed with, and the whole edifice built upon nothing more than the notion of 'class'.
Gist of Idea
Russell tried to replace Peano's Postulates with the simple idea of 'class'
Source
report of Bertrand Russell (The Principles of Mathematics [1903]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.4
Book Reference
Monk,Ray: 'Bertrand Russell: Spirit of Solitude' [Vintage 1997], p.132
A Reaction
(See Idea 5897 for Peano) Presumably you can't afford to lose the notion of 'successor' in the account. If you build any theory on the idea of classes, you are still required to explain why a particular is a member of that class, and not another.
Related Idea
Idea 5897 0 is a non-successor number, all successors are numbers, successors can't duplicate, if P(n) and P(n+1) then P(all-n) [Peano, by Flew]