Bill Vallicella has posted sporadically on the philosophy of set theory. What exactly are mathematical sets? Do they exist at all? I had commented occasionally without having a properly developed position of my own. By the time of Sets, Pluralities, and the Axiom of Pair, to which I attached a brief comment, I'd reached some firmer conclusions. Bill's more recent Does Potential Infinity Presuppose Actual Infinity? revisiting the question of the existence of infinite sets has prompted me to say a little more. I hope it's interesting and relevant.

I start with the idea that all reference can be pictured as pointing. A set is therefore a bunch of arrows, each arrow pointing to an object. Like proper names, being elements of language they are not in the world in the same way as ordinary objects. I have tried to indicate this in the accompanying diagram. I appreciate that philosophers are dubious of the place of drawings in philosophy but since I think the essence of reference is pointing, it seems appropriate in this case.

Relations such as membership and inclusion can be understood in terms of arrows: x is a member of S iff S has an arrow pointing to x (ie, S refers to x). S is included in T iff every thing pointed to by some arrow in S is pointed to by some arrow in T. Two sets are equal iff their arrows point to exactly the same things. Unlike the view of sets as collections there is no sense here that an element is a mereological part of a set. The extension of a set is clearly distinct from the set itself. On the view of sets as collections, equal sets are identical and this gives rise to sense/reference confusions. Here I can label a set with an intension and happily discover that it has the same extension as a set defined from and labeled with a different intension. This more accurately reflects how sets are used in proofs and constructions.

The set operations of union, intersection, and difference have simple definitions in terms of arrows.

A set of sets is a bunch of arrows with each arrow pointing to another bunch of arrows. The power set construction follows naturally. Sets of sets of sets to any required level can be constructed and have a precise meaning. This is the principal way in which sets extend the plural referencing features of natural languages. The latter are not adequate to the task of tracking the pluralities of objects that can arise in mathematical proof. They are limited in the complexity they can express and they suffer from ambiguities as evidenced by the discussion in the first of the above posts. The 'iterative hierarchy' of sets arises in a natural way as tiers of quivers bearing upward pointing arrows. In the axiomatic presentation of set theory the iterative hierarchy looks a little like a device deliberately introduced to avoid the antinomies. Here it develops as the need arises in a proof or construction to refer back to pluralities of objects previously defined, or to construct new pluralities from old. Enderton,

*Elements of Set Theory*, says that Cantor was led to set theory through his study of Fourier series:

Cantor's work on Fourier series led him to consider more and more general sets of real numbers. In 1871 he realised that a certain operation on sets of real numbers (the operation of forming the set of limit points) could be iterated more than a finite number of times; starting with a set PThis back referencing means that all arrows point upwards. That there can be no 'set of all sets' has an obvious geometric interpretation: any putative set of all sets would require an arrow pointing to itself which would thus fail to point upward._{0}one could form P_{0}, P_{1}, P_{2},...,P_{w}, P_{w+1},...,P_{w+w}...

This view of sets changes the complexion of the question of the existence of sets. If they are artificial linguistic elements akin to proper names then we need not expect to find them in the world itself nor in some Platonic heaven. Furthermore, merely referring to a plurality of objects is tantamount to their forming a set. I agree with Bill's remark that 'that from the fact that there ARE many Fs it does not straightaway follow that there IS a single thing comprising these many Fs'. But on the reference conception of set there is only a weak sense in which its members

*comprise*a set. Rather it would be a performative inconsistency to refer to the Fs yet to deny the set of the Fs. Their cardinality is of no consequence.