### Single Idea 21723

#### [catalogued under 6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism]

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]**