Foundations of Mathematics in Polymorphic Type Theory

This essay was inspired by conversations with mathematicians who maintain in all seriousness that there is something canonical about foundations of mathematics in ZFC (Zermelo-Fraenkel set theory with the axiom of choice). (These took place on the FOM (Foundations of Mathematics) mailing list [7].) One of the parties to these conversations ventured to define mathematics in terms of ZFC. The best way to disprove such claims is to present foundations for mathematics which do not depend on ZFC. This is what is done here. The approach to foundations presented here is formalized in NFU, a theory proposed by Jensen in [17] as a “slight” modification of Quine’s set theory “New Foundations” of [22] (hereinafter NF. Actually, by NFU we will always mean at least NFU + Infinity + Choice. We say that the approach is formalized in NFU; this does not mean that it is completely embodied in NFU. Just as the underlying intuitive picture of foundations which motivates ZFC suggests further extensions of ZFC (with inaccessible cardinals and so forth), so the underlying view of foundations which motivates NFU suggests extensions of the formal theory. Some of these extensions will be discussed here. A common criticism of Quine’s “New Foundations” is that it is not motivated by any underlying picture of the world of sets; that it is merely a modification of the theory of types via a “syntactical trick”. This may be true of Quine’s original development of NF, but we will present an intuitive picture of the world of sets as well as a formal system in this essay. In [17], Jensen demonstrated the consistency of NFU + Infinity + Choice, and Jensen and later workers have described models of NFU. It is not hard to develop a clear mental picture of what a model of NFU is like. This does not fulfil our program of presenting NFU as an autonomous approach to the foundations of mathematics, because it can be objected (with considerable justice) that the whole warrant for these proceedings rests on a consistency proof carried out in ZFC, and that the models one can contemplate are structures in the world of ZFC. To achieve the philosophical program of presenting NFU as an autonomous foundational approach, it is necessary to proceed more carefully. Our development will present foundations in NFU as a natural revision of foundations in the Theory of Types of Russell (as simplified by Ramsey, and also modified by a weakening of extensionality to allow urelements in each type). We will be interested at each step not only in mathematical soundness but in intuitive appeal of the constructions. It is not the purpose of this paper to suggest that ZFC foundations are unsatisfactory or should be replaced by NFU foundations. It is not even a purpose of the paper to suggest that NFU foundations cannot be presented as a revision of ZFC foundations themselves (they can, and we indicate in the section titled “Mutual Reflections” how to do this, though we think that a development of NFU via type theory is more natural, especially since one needs to understand type theory anyway to see the point). No change in the practice of mathematics is being suggested here; our aim is to suggest that other approaches to the foundations of mathematics than the one now in fashion could have been taken, by exhibiting a different approach which is technically feasible and has a natural motivation of its own.