Full Idea
The problem for logicism was to find definitions of the primitive notions of Peano's theory, number, successor and 0, in terms of logical notions, so that the postulates could then be derived by logic alone.
Gist of Idea
The task of logicism was to define by logic the concepts 'number', 'successor' and '0'
Source
Bernard Linsky (Russell's Metaphysical Logic [1999], 7)
Book Reference
Linsky,Bernard: 'Russell's Metaphysical Logic' [CSLI 1999], p.115
A Reaction
Both Frege and Russell defined numbers as equivalence classes. Successor is easily defined (in various ways) in set theory. An impossible set can exemplify zero. The trouble for logicism is this all relies on sets.
Related Ideas
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]
Idea 18179 For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy]
Idea 18178 For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy]