McLarty on ‘The roles of set theories in mathematics’

A couple of years back, I commented here on most of the papers in Categories for the Working Philosopher, edited by Elaine Landry (OUP). I’ve had occasion to revisit the first piece in the book, ‘The roles of set theories in mathematics’ by Colin McLarty. This is not the sort of wide-ranging discussion that we get in e.g. Penelope Maddy or John Burgess writing about the role of set theory. McLarty focuses on one familiar claim — namely that ZFC radically overshoots as an account of what ‘ordinary mathematics’ requires by way of background assumptions about sets. I have somewhat revised my comments.

McLarty in particular takes a look at the set-theoretic preliminaries expounded in two classic texts, by Munkres on topology and Lang on algebra, and comments on their relatively modest character. Though I’m not sure that McLarty is a particularly reliable close-reader of their texts. For example, he writes “Of course in ZFC everything is a set, including that the elements of sets are sets. Munkres hardly denies that all objects are sets.” But Munkres does deny just that — his informal set theory explicitly allows urelements which aren’t sets:  “The objects belonging to a set may be of any sort. One can consider the set of all even integers, and the set of all blue-eyed people in Nebraska, and the set of all decks of playing cards in the world.” 

Let that pass.  Neither Munkres or Lang gives a regimented summary of his set-theoretic assumptions in axiomatic form: if they did so, what would it look like? Something like the original Zermelo set theory with urelements plus choice, I guess. But McLarty suggests it could look like ETCS, the ‘Elementary Theory of the Category of Sets’ developed by Lawvere. Thus he writes “[Lang’s] account simply says nothing that would distinguish between ZFC and ETCS, because nothing in his book depends on those differences.” 

But recall: 

In ETCS, no element is an element of two distinct sets.And a subset of X cannot be a subset of Y unless X = Y.In standard set theory a set can be an element of X and a subset of Y: in ETCS this can’t happen unless X = Y.In standard set theory any two sets X and Y have an intersection and union: not so in ETCS.In standard set theory an element of a set X can itself have elements: not so in ETCS.

And on it goes. Yes, nothing that Munkres or Lang says explicitly rejects the deviant ETCS line. But the plausible explanation is that it just doesn’t cross their minds as an option that needs to be countenanced in an introduction to mainstream set-theoretic ideas to beginning graduate students of topology or algebra. (Any more than they countenance NF-style stratification, say.) I’d bet that Munkres and Lang would vote the traditional ticket at least when it comes to those sorts of issues on which ETCS and standard set theories disagree.

Now true enough, ETCS can be neatly re-packaged in the manner of Tom Leinster in his well-known ‘Rethinking set theory’. Leinster’s aim is, as he puts it, to show that “simply by writing down a few mundane, uncontroversial statements about sets and functions, we arrive at an axiomatization that reflects how sets are used in everyday mathematics.” And Leinster is at pains to point out that the axioms of this theory, in his presentation, do not overtly involve any essentially categorial notions — they just talk about “sets” and “functions” (without reducing the “functions” to “sets”, by the way). He is indeed emphatic that this ETCS-style story about “sets” is, unqualifiedly, a theory about sets as ordinarily thought of by mathematicians. But given the discrepancies such as those noted above between ETCS and routine assumptions about sets, that is tendentious to say the least.

McLarty himself goes on to highlight some of the differences between ETCS and standard ZFC-style theories (with or without urlements): but unlike e.g. Mike Shulman, this doesn’t make him hesitate to call ETCS a set theory. Maybe that’s impolitic. Rather than that the distracting implication (in Lawvere too) that we’ve being doing our set-theory-for-ordinary-applications wrong, I’d say it is better to spin a positive message that a significantly different way of handling pluralities may serve some purposes better (more economically, with less redundancy).  

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