Bibliography: Set Theory with a Universal Set


This is a comprehensive bibliography on axiomatic set theories which have a universal set. The policy has been to put in pointers to anything that anyone doing a literature search on set theory with a universal set might hope to find. (``Completeness is all''!) This policy has been adhered to even at the cost of including articles that are obsolete, erroneous or just plain worthless. Less discouragingly, it also means that we have included numerous papers that contain results that are of interest to people who study set theories with a universal set (even when those papers do not bear on the subject directly) simply because this is a place where people who might want them could be foraging.

As we update the bibliography we gradually enlarge the set of items that have linked text. The two constraints on this of course are effort and copyright, and there is a large archive of NF-related manuscripts that are in various stages of becoming public. Some are itemised here and available publicly (linked) in electronic form; some are itemised here and are scanned but not publicly available because of copyright etc concerns; some are itemised but not even scanned. There is even a body of material not listed here - yet! Feel free to contact the managers if there is a document in the penumbra that you desire or whose existence you suspect.

At present the field includes two main areas of study:

The model construction of Alonzo Church and Urs Oswald probably really belongs with NF: the models created by their technique are really best uderstood as fancy models of NF2 or NF0. If the NF bloc is to be divided into two the natural division one would reach for is the division between NF (and fragments thereof) with full extensionality, and the systems that allow distinct empty sets or urelemente. These systems arise from Jensen's consistency proof for NFU (actually of NFU + Infinity + Choice). Specker's disproof of Choice in NF shows that NF is quite different from NFU+Choice. The latter theory is perhaps best understood as a cunning way of describing a model of ZF (or rather KF) with an automorphism.

Do these areas exhaust set theory with a universal set? Perhaps not. Recent papers by Holmes (and the original papers of Andrzej Kisielewicz) on "double extension set theory" are referenced in the main body of the bibliography but not under "recent work"; the jury is still out on this system (two versions of which have been shown to be inconsistent) but it must be admitted that if the surviving version is consistent it is a set theory with a universal set. - MRH

For those unfamiliar with the field, two places to start are the New Foundations Home Page and Thomas Forster's book Set Theory with a Universal Set. A new option is afforded by the recent appearance of Holmes's elementary text.

Comments, corrections, and information about new publications should be sent to Randall Holmes. Announcements about both print and eprint publications are welcome.

This bibliography is a lineal descendent of the bibliography in Forster's thesis, which was the first attempt at a comprehensive NF bibliography, and thanks are due to Paul West, who did the first round of virtual typesetting.

Last revision: late January (by Randall Holmes), in progress (and we don't always remember to update this date when we change it).

Recent Work

Comprehensive Bibliography