Full Idea
Field needs the notion of logical consequence in second-order logic, but (since this is not recursively axiomatizable) this is a semantical notion, which involves the idea of 'true in all models', a set-theoretic idea if there ever was one.
Gist of Idea
Field needs a semantical notion of second-order consequence, and that needs sets
Source
comment on Hartry Field (Science without Numbers [1980], Ch.4) by James Robert Brown - Philosophy of Mathematics
Book Reference
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.54
A Reaction
Brown here summarises a group of critics. Field was arguing for modern nominalism, that actual numbers could (in principle) be written out of the story, as useful fictions. Popper's attempt to dump induction seemed to need induction.