Linnebo's abstractionism and the bad company problem

In Thin Objects, Linnebo offers what he describes as a ‘simple and definitive’ solution to the bad company problem facing abstractionist accounts of mathematics. ‘Bad’ abstraction principles can be rendered ‘good’ by taking abstraction to have a predicative character. But the resulting predicative axioms are too weak to recover substantial portions of mathematics. Linnebo pursues two quite different strategies to overcome this weakness in the case of set theory and arithmetic. I argue that neither infinitely iterated abstraction nor abstraction on possible specifications fully resolves the bad company problem.