I’ve been doing quite a bit of work over the last couple of weeks for a major April update of the Teach Yourself Logic Guide (you can download the current March version here). And I’ve got to a subsection where I could do with advice and comments. I’m working on a new section on more advanced readings on set theory. The subsection on ZFC with all the bells and whistles (large cardinals, forcing, and other excitements) writes itself, and there’s a shorter subsection on the Axiom of Choice. But now I’m drafting a subsection on Alternative Set Theories. Here’s what I have so far written:
From earlier reading you should have picked up the idea that, although ZFC is the canonical modern set theory, there are other theories on the market. I will mention just six(!) here, which I find interesting:
SP This is by way of reminder: earlier in the Guide, we very warmly recommended
- Michael Potter, Set Theory and Its Philosophy (OUP 2004).
This presents a version of an axiomatization of set theory due to Dana Scott — hence Scott-Potter set theory. This axiomatization is consciously guided by the conception of the set theoretic universe as built up in levels (the conception that, supposedly, also warrants the axioms of ZF). What Potter’s book aims to reveal is that we can get a rich hierarchy of sets, more than enough for mathematical purposes, without committing ourselves to all of ZFC. If you haven’t read Potter’s book before, now is the time to look at it.
NBG This is also by way of a reminder if you have read Potter. We know that the universe of sets in ZFC is not itself a set — but isn’t it some sort of collection? Should we recognize, then, two sorts of collection, sets and (as they are called in the trade) proper classes which are ‘too big’ to be sets? NBG (named for von Neumann, Bernays, Gödel) is one such theory of collections. So NBG recognizes proper classes, objects having members but that cannot be members of other entities. NBG’s principle of class comprehension is predicative; quantified variables in the defining formula can range only over sets, and we get a conservative extension of ZFC (nothing in the language of sets can be proved in NBG which can’t already be proved in ZFC). For more on this and on other theories with classes as well as sets, see (briefly) Appendix C of Potter’s book. Also, for a more extended textbook presentation of NBG, see
- Elliott Mendelson, Introduction to Mathematical Logic (CRC, 4th edition 1997), Ch.4.
ZF- + AFA Here again is the familiar iterative conception of the set universe. We start with some non-sets (maybe zero of them in the case of pure set theory). We collect them into sets (as many different ways as we can). Now we collect what we’ve already formed into sets (as many as we can). Keep on going, as far as we can. On this `bottom-up’ picture, the Axiom of Foundation is compelling (any downward chain linked by set-membership will bottom out, and won’t go round in a circle).
Here, however, is another conception of the set universe. Take a set. It (as it were) points to its members. And those members point to their members. And so on and so forth. On this ‘top-down’ picture, the Axiom of Foundation is not so compelling. As we follow the pointers, can’t we come back to where we started?
It is well known that in the development of mathematics inside ZFC the Axiom of Foundation is usually non-critical. So what about considering a theory of sets which has an Anti-Foundation Axiom, which allows self-membered sets? The very readable classic here is
- Peter Aczel, Non-well-founded sets .(CSLI Lecture Notes 1988).
- Luca Incurvati, `The graph conception of set’ Journal of Philosophical Logic (published online Dec 2012), illuminatingly explores the motivation for such set theories.
NF Now for a much more substantial departure from ZF. Standard set theory lacks a universal set because, together with other standard assumptions, the idea that there is a set of all sets leads to contradiction. But by tinkering with those other assumptions, there are coherent theories with universal sets. For very readable presentations concentrating on Quine’s NF (‘New Foundations’), and explaining motivations as well as technical details, see
- T. F. Forster, Set Theory with a Universal Set Oxford Logic Guides 31 (Clarendon Press, 2nd edn. 1995).
- M. Randall Holmes, Elementary Set Theory with a Universal Set (Cahiers du Centre de Logique No. 10, Louvain, 1998). This can now be freely downloaded from the author’s website.
IST Leibniz and Newton invented infinitesimal calculus in the 1660s: a century and a half later we learnt how to rigorize the calculus without invoking infinitely small quantities. Still, the idea of infinitesimals has a strong intuitive appeal, and in the 1960s, Abraham Robinson created a theory of hyperreal numbers, based on ultrafilters: this yields a rigorous formal treatment of infinitesimal calculus. Later, a simpler and arguably more natural approach, based on so-called Internal Set Theory, was invented by Edward Nelson. As Wikipedia puts it, ”IST is an extension of Zermelo-Fraenkel set theory in that alongside the basic binary membership relation, it introduces a new unary predicate standard which can be applied to elements of the mathematical universe together with some axioms for reasoning with this new predicate.” Starting in this way we can recover features of Robinson’s theory in a simpler framework.
- Edward Nelson, ‘Internal set theory: a new approach to nonstandard analysis’ Bulletin of The American Mathematical Society 83 (1977), pp. 1165–1198. Now freely available from projecteuclid.org.
- Nader Vakin, Real Analysis through Modern Infinitesimals (CUP, 2011). A monograph developing Nelson’s ideas whose early chapters are quite approachable and may well appeal to some.
ETCS Famously, Zermelo constructed his theory of sets by gathering together some principles of set-theoretic reasoning that seemed actually to be used by working mathematicians (engaged in e.g. the rigorization of analysis or the development of point set topology), hoping to a theory strong enough for mathematical use while weak enough to avoid paradox. But does he overshoot? Could we manage with less?
- Tom Leinster, ‘Rethinking set theory‘, gives an advertising pitch for the merits of Lawvere’s Elementary Theory of the Category of Sets, and …
- E. William Lawvere and Robert Rosebrugh, Sets for Mathematicians (CUP 2003) gives a very accessible presentation which doesn’t require that you have already done any category theory.
Though perhaps to fully appreciate what’s going on, you will have to go on to dabble in category theory (see the next section of the Guide!).
More? Finally, for a brisk (and somewhat tough) overview of many other alternative set theories, including e.g. Mac Lane set theory, see
- M. Randall Holmes, ‘Alternative axiomatic set theories‘, Stanford Encyclopedia of Philosophy.
So what readings at a comparable level should I also have mentioned? What other deviant set theories should I have mentioned? Comments are open …!