Elements of an Alternative Philosophy of Set Theory

The two favoured solutions of the class paradoxes have been the 'limitation of size' view of sets and the iterative view of sets. A rival is offered: a view of sets based on an idea of definite pluralities, augmented to allow for sets of just one member and an empty set. An advantage of this alternative is that it flows from an ordinary non-mathematical idea, but with a slightly refined understanding of indefiniteness. Some consequences for axioms of set theory are presented. The axioms of Separation and Replacement are natural outcomes of this view. Another consequence is that the Foundation axiom is not universally true. So, we should take seriously certain non-well-founded set theories, and let the context of investigation determine when to use a set theory with an anti-foundation axiom and when to stay with well-founded set theory.