  Of course, as Jané notices, there are a number of strategies that we could use to evade this application of Skolem's Paradox: we could use uncountably many axioms to force our models to have uncountable domains, we could appeal to the Upward Löwenheim-Skolem theorem to show that Zermelo's axioms also have uncountable models (see section  1), or we could move to a second-order version of Zermelo's axioms and then prove that these axioms can only be satisfied by models with uncountable domains (see section  2.5). Unfortunately, each of these strategies presupposes that we already have a prior grip on the notion of an uncountable set—e.g., to initially characterize an uncountable set of axioms, to formulate the Upward Löwenheim-Skolem theorem, or to prove that second-order ZFC has only uncountable models. So, none of these strategies can be used to introduce uncountable sets into mathematics in the first place. Or so, at any rate, Jané takes Skolem to be arguing.