 (b) An objection to the choice principle was also put forward by Peano. This objection seems to be that since the choice principle cannot be proved ‘syllogistically’ (i.e., from the principles of Peano's Formulario), then it has to be rejected (see Peano 1906). (Peano does think, however, that finite versions of the choice principle are provable, relying essentially on repeated applications of a version for classes of the basic logical principle EI mentioned above (§2.2.1). Zermelo's reply is the following. Axiom systems like Peano's are constructed so as to be adequate for mathematics; but how does one go about selecting the ‘basic principles’ required? One cannot assemble a complete list of adequate principles, says Zermelo, without careful inspection of actual mathematics and thereby a careful assessment of what principles are actually necessary to such a list, and such inspection would show that the choice principle is surely one such; in other words, a selection of principles such as Peano's is very much a post hoc procedure. The reply to Peano is of a piece with the reply to Borel, and recalls strongly the invocation in Zermelo (1908b: 261), that it is necessary to distill principles from the actual operation with sets. He supports his claim that the choice principle is necessary by a list of seven problems which ‘in my opinion, could not be dealt with at all without the principle of choice’ (Zermelo 1908a:  113).[26]  In particular he points out that the principle is indispensable for any reasonable theory of infinite cardinality, for only it guarantees the right results for infinite unions/sums, and in addition is vital for making sense of the very definition of infinite product. That Peano cannot establish the choice principle from his principles, says Zermelo, strongly suggests that his list of principles is not ‘complete’ (Zermelo 1908a: 112). 