ionism has enjoyed some preliminary success with Hume’s Principle, which says that the number of F s is identical to the number of Gs just in case the F s and the Gs can be one-to-one correlated. This abstraction principle can be formalized as (HP) #F = #G ↔ F ≈ G where #F denotes the number of F s and F ≈ G abbreviates the second-order claim that there is a relation that one-to-one correlates F s and the Gs. Burgess describes (with unsurpassed attention to historical detail) how Hume’s Principle has been found to have two very desirable properties. Let Frege Arithmetic (or FA for short) be the second-order theory whose sole nonlogical axiom is (HP). Then firstly, FA is consistent. And secondly, Frege’s own definitions and derivations show how FA gives rise to all the axioms of ordinary Peano arithmetic—a result known as Frege’s theorem. The previous section alluded to the question how strong concept comprehension axioms are needed for Frege’s theorem to go through. Linnebo, 2004 proves that, given Frege’s own definitions of the primitive signs of arithmetic, predicative comprehension is insufficient to prove that every number has a successor. In Section 2.3 Burgess cleverly bypasses this limitative result by adopting an alternative, non-Fregean definition of natural number predicate, which allows him to interpret Robinson arithmetic Q (which of course says that every number has a successor) in FA with only predicative comprehension. If one imposes no requirements (beyond the obvious technical ones) on one’s definitions of the primitive signs of arithmetic, then Burgess’s result will dispel any worries caused by Linnebo, 2004. If, on the other hand, one requires that these definitions should correspond to our ordinary arithmetical thought and practice, and if one thinks that Frege’s definitions are implicit in such thought and practice, then the result will be of no immediate philosophical significance.10 Setting aside all questions about impredicative comprehension, it is clear that any viable abstractionism must be able to delineate a much wider class of acceptable abstraction principles than just Hume’s Principle. Burgess summarizes some of the efforts towards this goal Heck, gives a different predicative proof of Frege’s theorem, using Frege’s own definitions transposed to the un-Fregean setting of ramified second-order logic.