I just read an article called Rethinking Set Theory by Tom Leinster in the most recent issue of the American Mathematical Monthly. I found the article very interesting and appealing, so I thought I’d mention it here.
Set theory is the basis of the vast majority classical mathematics. As such, most mathematicians use set theory every day in their work. Yet the majority of mathematicians don’t appeal to the formal properties of set theory–i.e. the axioms of set theory–in their work. Rather, mathematicians tend to use a set of working principles about sets which they constantly rely upon. There is nothing wrong with this approach, as the working principles can be rigorously justified, but it reveals a (perhaps philosophical) disconnect between the theory and practice of set theory.
Leinster’s article in the Monthly proposes an alternate set of axioms for set theory. Instead of starting for the standard ZFC axioms, he suggests formulating axioms that appear more similar to the way in which sets are used by most mathematicians. The inspiration for Leinster’s axioms (which he attributes to F. W. Lawvere) comes from category theory. They place equal emphasis on the concepts of “set” and “function,” whereas in the traditional ZFC theory, the only formal objects are sets.
I find Leinster’s approach to be very appealing. From a practical standpoint, I very much like that his axioms directly appeal to common set theoretic applications. Philosophically, I also appreciate that the axioms seem to reference the way in which sets interact (via functions) rather than describing the internal structure of sets (as does ZFC). This emphasis of external interaction over internal structure seems analogous to humankind’s position in the universe: the internal structure of the universe is a mystery, so we must learn what we can by interacting with as diverse phenomena as possible. Thus perhaps we should learn to understand sets the same way.