On Skolem's Paradox

1. All natural numbers are in the domain. 2. The only other elements of the domain are sets of natural numbers. 3. The denotations of 0, 1, 2, ... are 0, 1, 2, ... respectively; the denotation of N is the set of all natural numbers, and the denotation of S is the set of all sets of natural numbers in the domain; the denotation of Î is the relation of membership between numbers and sets of numbers. Finally, the denotation of J is the function J extended to give some arbitrary value—say 17—for arguments that are not both numbers (that is, one or both of which are sets).