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 …!

Morse-Kelly set theory, as in Kelly’s General Topology.

http://en.wikipedia.org/wiki/Morse-Kelley_set_theory

Grr. Misspelled “Kelley, though not in the URL. (Is there a way to edit comments?)

There’s a book on internal set theories and related matters,

Nonstandard Analysis, Axiomatically, by Vladimir Kanovei and Michael Reeken. (Unfortunately, I can’t find my copy at the moment, and it’s been ages since I looked at it, but I can remember thinking it was interesting.)There’s a different approach to infinitesimals, smooth infinitesimal analysis. I’m not sure it counts as a different set theory, exactly, but it’s related to ETCS and topos theory, uses a nonclassical logic, and is also (imo) interesting philosophically. So it might be worth mentioning somewhere. John Bell has written some accessible things about it, such as the books

A Primer of Infinitesimal AnalysisandThe Continuous and the Infinitesimal in Mathematics and Philosophyand an article in the Stanford Encyclopedia of Philosophy, Continuity and Infinitesimals which says:His book

Toposes and Local Set Theoriesmay also be relevant.Well, Morse-Kelley is hardly an “alternative set theory”. After all, it’s just second-order ZFC. But this is just quibbling over terminology.

Ackermann set theory might be worth a mention in this context. Considerations relating to a natural way of extending Ackermann set theory led Reinhardt to formulate the famous “ultimate large cardinal axiom” j: V –> V shown inconsistent by Kunen.

And of course there are also constructive (and intuitionistic) theories of sets.

Another interesting theory, from a technical point of view, and not really very “alternative”, is the system once proposed by Feferman for formalizing category theory, where we add to ZFC a constant V and a schema asserting any formula reflects down from the universe to V. This theory is a conservative extension of ZFC, as easily established using the reflection schema and compactness, but the constant V functions very much like a universe without bringing with it the proof theoretic strength of an inaccessible.

Then there are any number of fragments of ZFC that are mainly of technical interest, such as Kripke-Platek set theory. One fragment that arises naturally is Zermelo set theory + O = “every well-ordering is isomorphic to a von Neumann ordinal”. The principle O is a somewhat technical way to say that the powerset operation can be iterated along any well-ordering, nicely capturing (in a minimal sort of way) the idea that the cumulative hierarchy should extend as far as possible.

Surely Morse-Kelley is more of an alternative than NBG (which was already in the list).

If we count NBG as an alternative set theory, then as you say MK is surely also (even more) alternative. In any case, I think it’s odd to call NBG an alternative set theory. It would be more natural, to my mind at least, to call it an alternative formalization of standard set theory.

Why do you think that is a distinction that should be made? Also, what is the distinction, exactly? (I’m not saying you’re wrong, just wanting to know more.)

I am suprised no-one has mention another set-theory in which Peter Aczel has had a hand, namely CZF. This is a completely predicative set-theory, without for example a power-set operation or unbounded separation. In its basic form, it is no stronger than KPomega. With various `large set’ axioms, it provides a foundation or setting for a surprising amount of constructive mathematics. It can be modelled in various forms of Martin-Lof type theory.

As far as I know there is no book published about it, but a draft exists:

http://www.maths.manchester.ac.uk/logic/mathlogaps/workshop/CST-book-June-08.pdf

Thanks for this!

You could also consider the set theory SEAR (with or without AC) by Mike Shulman. The only account is here: http://ncatlab.org/nlab/show/SEAR