Bibliography: Set Theory with a Universal Set

Introduction

This is a comprehensive bibliography of the primary literature on axiomatic set theories which have a universal set. It is maintained by Thomas Forster and Randall Holmes. Thanks are due to Paul West, who did the first round of virtual typesetting. It is a lineal descendent of the bibliography in Forster's thesis, which was the first attempt at a comprehensive NF bibliography. Its primary focus remains NF and the other Quine systems.

It aims to be comprehensive and fault-tolerant. It is fault-tolerant because it is intended as an aid to scholarship: an article on set theory with a universal set that a researcher might expect to find here should be here. Erroneous and useless articles are not debarred; they are part of history too. Even a connection with a universal set is not an absolute requirement. Amicable relations between the NFistes and the people who study antifoundation axioms—specifically the anti-foundation axiom of Forti-and-Honsell (popularly misattributed to Aczel because of his excellent monograph below ) go back a long way and have resulted in quite a lot of antifoundation stuff finding its way here, and for sentimental reasons we have not expunged it. It's all good stuff, and the fact that it has nothing to do with NF is not a compelling reason for leaving it out. (There is also the consideration that scholars interested in antifoundation axioms but not attuned to the difference between antifoundation axioms and existence of a universal set will come looking for quarry here, and we do not wish to disappoint them. This is an aid to scholarship after all).

Because the idea of stratification is so important in NF studies almost any paper treating it is likely to end up being listed here. Ryan-Smith's paper, and Max Newman's 1943 paper for example, and the recent paper of Tin Adlešić and Vedran Čačić.

To be slightly more specific, the focus of this repository is not so much Set theories with a Universal set but rather primary mathematical research literature thereon. We do not aim to cover the gradually growing philosophical literature; our contribution is to curate and make available to its authors the mathematical sources that they will want to consult.
Thus we do not itemise (for example) the recent books by Incurvati and Morris (even though they have extended discussions of NF) nor the recent Birkbeck Ph.D. thesis of Kriener (which has an extended discussion of Forster's article about Church-and-Oswald's construction in Church [1974])—because they do not contain any original mathematical research (on those systems).
However, sometimes that tradition throws up mathematical work— see the article by Tim Button — and material of that kind we definitely do want to know about.

At present the field of Set Theory with a Universal Set includes two main areas of study:

Recently a third area has presented itself for development. The articles of Holmes and Andrzej Kisielewicz below concern Kisielewicz's interesting idea of double extension set theory. The idea looks bizarre initially but may be fruitful.

It might appear that there are four, but the putative fourth, the model construction of Alonzo Church and Urs Oswald in the 1970s belongs with NF: the models created by their technique are really best understood as fancy models of NF2 or NF0, and is not really separate from NF at all. If the NF bloc is to be divided into two then the natural division one would reach for is the division between — on the one hand — NF (and fragments thereof) with full extensionality, and — on the other — the systems (NFU plus modifications) 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 the Kaye-Forster system KF) with an automorphism.


The work of Specker has revealed deep and important connections between NF and various typed Set Theories going back ultimately to Russell and Whitehead, and some of the articles itemised below concern those type theories: TST (theory of simple types) and TZT (theory of (positive and) negative types).

For those unfamiliar with the field, possible places to start are the New Foundations Home Page, Thomas Forster's book Set Theory with a Universal Set, Holmes's elementary text (which treats NFU rather than NF). There is also the Stanford Encyclopædia of Philosophy article and of course Wikipædia.

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 some material not yet listed here. Feel free to contact the managers if there is a document in the penumbra that you desire or whose existence you suspect.

Items are listed alphabetically by author; items with multiple authors are entered separately under each author. Two-author papers for a given author come below single-author papers, and three-author papers come below two-author papers.

Comments, corrections, and information about new publications should be sent to Randall Holmes or Thomas Forster. Electronic copy of relevant publications will be gratefully received. Announcements about both print and eprint publications are welcome.

Last revision: Southern hemisphere summer 2023-4 by Thomas Forster.

Comprehensive Bibliography