ABSTRACT
In this chapter, the author refers in various different ways to what Skolem, but since in fact he was embellishing a result that had earlier been established by the mathematician Lowenheim, his theorem is usually referred to as the Lowenheim–Skolem theorem. The sense in which the Lowenheim–Skolem theorem establishes that the truths of set theory do not fix what 'Set' and 'member' mean is a special technical sense that involves relations between the language of ZF and its various possible interpretations, when these are themselves being construed as elements of mathematical reality. There are unintended interpretations that make the same sentences come out true. The Lowenheim–Skolem theorem has been defused, in the sense that any special threats that it posed have been averted. To that extent, we can indeed afford to be self-confident about our set-theoretical practice; to that extent. The problem is that self-confidence and self-consciousness make notoriously bad bedfellows.
