Higher-Order Skolem’s Paradoxes and the Practice of Mathematics: a Note

We will formulate some analogous higher-order versions of Skolem’s paradox and assess the generalizability of two solutions for Skolem’s paradox to these paradoxes: the textbook approach and that of Bays (2000). We argue that the textbook approach to handle Skolem’s paradox cannot be generalized to solve the parallel higher-order paradoxes, unless it is augmented by the claim that there is no unique language within which the practice of mathematics can be formalized. Then, we argue that Bays’ solution to the original Skolem’s paradox, unlike the textbook solution, can be generalized to solve the higher-order paradoxes without any implication about the possibility or order of a language in which mathematical practice is to be formalized.