Full Idea
ZF set theory is seen as a rival to logicism as a foundational scheme. Set theory is for those who have given up the project of reducing mathematics to logic.
Gist of Idea
For those who abandon logicism, standard set theory is a rival option
Source
Bernard Linsky (Russell's Metaphysical Logic [1999], 6.1)
Book Reference
Linsky,Bernard: 'Russell's Metaphysical Logic' [CSLI 1999], p.91
A Reaction
Presumably there are other rivals. Set theory has lots of ontological commitments. One could start at the other end, and investigate the basic ontological commitments of arithmetic. I have no idea what those might be.