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.

This bibliography 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 )
has resulted in quite a lot of antifoundation stuff finding
its way here, and for sentimental reason we have not expunged
it. It's all good stuff, it's just that it has nothing to do with NF.

(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).

To be slightly more specific, the focus of this repository is not so
much Set theories with a Universal set (specifically the Quine
systems) as *primary research literature* thereon. Thus we do
not itemise (for example) the recent books by Incurvati and Morris
even tho' they have extended discussions of NF — because they do not
contain any original research (on those systems). There is a
gradually growing (mainly philosophical) secondary literature
.. Forster's article on the iterative
conception — which is about Church and Oswald's construction — has
attracted quite a lot of interest in some quarters. However it is
not our focus here, and we do not commit ourselves to covering it.
Sometimes that tradition throws up mathematical work:
see the article by Tim Button

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

"New Foundations", a set theory devised by W. van Orman Quine,

The

*positive set theory*originally proposed by Helen Skala and Isaac Malitz (readers of this page may be more likely to be aware of the work of Marco Forti and others on hyperuniverses and the specific theory GPK^{+}_{∞}formulated by Oliver Esser).

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 NF_{2} 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. Announcements about both print and eprint publications are welcome.

*Last revision: Southern hemisphere winter 2020 by Thomas Forster.*

**Aczel, Peter [1988]**Non-Well-Founded Sets**CSLI**

This elegant booklet is included here principally for sentimental reasons: it got put in and nobody wants to delete it. It's not really got a great deal to do with Set Theory with a Universal set.**Gian Aldo Antonelli [1998]**Extensional Quotients for Type Theory and the Consistency Problem for NF.

Journal of Symbolic Logic,**63**, n. 1, pp. 247-61, 1998.

Arruda appears to have written a Ph.D. thesis with an NF angle: Arruda A, I, Consideracões sobre os sistemas formais NF**Arruda, A. [1970a]**Sur les systèmes NF_{i}de Da Costa.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**270**, pp. 1081-1084._{n}. Tese (Cátedra em Analise Matemática e Análise Superior) Curtiba, Universidade do Paraña 1964 55pGenerally the editors have a low opinion of these contributions of Arruda and we list them here solely because of our policy of `if in doubt leave it in''

**Arruda, A. [1970b]**Sur les systèmes NF-ω.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**270**, pp. 1137-1139.**Arruda, A. [1971]**La mathématique classique dans NF-ω.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**272**, p. 1152.**Arruda, A. and Da Costa, N.C.A. [1964]**Sur une hiérarchie de systèmes formels.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**259**, pp. 2943-2945.

**Barwise, J. [1984]**Situations, sets and the axiom of foundation.

Logic Colloquium '84, ed. J. Paris, A. Wilkie, and G. Wilmers, North-Holland, pp. 21-36.**Barwise, J. and Moss, L. [1996]**Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena.

CSLI Lecture Notes (60), Stanford University, ISBN: 978-1575860084.

Note: Most of the book is concerned with ill-founded set theories without a universal set, in the manner of Forti and Honsell. Chapter 20 covers a positive set theory, SEC, with a universal set but without, for example, V — {V}.**Benes, V.E. [1954]**A partial model for NF.

Journal of Symbolic Logic**19**, pp. 197-200.**Boffa, M. [1971]**Stratified formulas in Zermelo-Fränkel set theory.

Bulletin de l'Académie Polonaise des Sciences, série Math.**19**, pp. 275-280.**Boffa, M. [1973]**Entre NF et NFU.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**277**, pp. 821-822.**Boffa, M. [1975a]**Sets equipollent to their power sets in NF.

Journal of Symbolic Logic**40**, pp. 149-150.**Boffa, M. [1975b]**On the axiomatization of NF.

Colloque international de Logique, Clermont-Ferrand 1975, pp. 157-159.**Boffa, M. [1977a]**A reduction of the theory of types.

Set theory and hierarchy theory, Springer Lecture Notes in Mathematics**619**, pp. 95-100.**Boffa, M. [1977b]**The consistency problem for NF.

Journal of Symbolic Logic**42**, pp. 215-220.**Boffa, M. [1977c]**Modèles cumulatifs de la théorie des types.

Publications du Département de Mathématiques de l'Université de Lyon**14**(fasc. 2), pp. 9-12.**Boffa, M. [1981]**La théorie des types et NF.

Bulletin de la Société Mathématique de Belgique (série A)**33**, pp. 21-31.**Boffa, M. [1982]**Algèbres de Boole atomiques et modelès de la théorie des types.

Cahiers du Centre de Logique (Louvain-la-neuve)**4**, pp. 1-5.**Boffa, M. [1984a]**Arithmetic and the theory of types.

Journal of Symbolic Logic**49**, pp. 621-624.**Boffa, M. [1984b]**The point on Quine's NF (with a bibliography).

TEORIA**4**(fasc. 2), pp. 3-13.**Boffa, M. [1988]**ZFJ and the consistency problem for NF.

Jahrbuch der Kurt Gödel Gesellschaft (Wien), pp. 102-106**Boffa, M. [1989]**A set theory with approximations.

Jahrbuch der Kurt Gödel Gesellschaft 1989, p.95-97.**Boffa, M. [1992]**Decoration ensembliste de graphes par approximations.

Cahiers du Centre de Logique (Louvain-la-Neuve),**7**(1992), p.45-50.**Boffa, M. and Casalegno, P. [1985]**The consistency of some 4-stratified subsystems of NF including NF_{3}.

Journal of Symbolic Logic**50**, pp. 407-411.**Boffa, M. and Crabbé, M. [1975]**Les théorèmes 3-stratifiés de NF_{3}.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**280**, pp. 1657-1658.**Boffa, M. and Pétry, A. [1993]**On self-membered sets in Quine's set theory NF.

Logique et Analyse**141-142**, pp. 59-60.**Bowler, Nathan and Thomas Forster [2009]**Normal Subgroups of Infinite Symmetric Groups, with an Application to Stratified Set Theory.

Journal of Symbolic Logic**74**(2009) pp 17--26.-
**Button, Tim**Boolean Level Theory

This article is struggling through a submission process, but we are confident it will appear somewhere eventually. Its relevance to us is that it establishes synonymy between ZF and a minimalist version of Church's set theory — the version that is essentially NF_{2}plus ``the wellfounded sets satisfy ZF'' and `` every set is either the same size as a wellfounded set or is the complement of such a set'' **Casalegno, P. and Boffa M. [1985]**The consistency of some 4-stratified subsystems of NF including NF_{3}.

Journal of Symbolic Logic**50**, pp. 407-411.

**Church, A. [1974]**Set theory with a Universal set.

Proceedings of the Tarski Symposium. Proceedings of Symposia in Pure Mathematics XXV, ed. L. Henkin, American Mathematical Society, pp. 297-308.

Reprinted in International Logic Review**15**, pp. 11-23.The managers have a photocopy of a manuscript of Church's pertaining to this material, supplied to us by the late Herb Enderton. Its copyright status is unclear so we are not posting the scan of it. Readers who want to come to grips with Church's theory should probably start with Forster's survey article. Forster's recent work on Church's construction is in a document called ``COmodels.pdf'' which is maintained here, but is not yet ready for publication.

There is a further stash of Church's ruminations on NF lodged in the Seeley G. Mudd library at Princeton; they have never had the attentions of an NFiste. We are in touch with Princeton and hope to secure copies of the digitised version of these mss.

**Cocchiarella, N.B. [1976]**A note on the definition of identity in Quine's New Foundations.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**22**, pp. 195-197.**Cocchiarella, N.B. [1985]**Frege's double-correlation thesis and Quine's set theories NF and ML

Journal of Philosophical Logic,**14**, no. 4: 253-326.**Cocchiarella, N.B. [1992a]**Cantor's power-set theorem versus Frege's double-correlation thesis

History and Philosophy of Logic,**13**: 179-201.**Cocchiarella, N.B. [1992b]**Conceptual realism versus Quine on classes and higher-order logic,

Synthese,**90**: 379-436.**Collins, G.E. [1955]**The Modelling of Zermelo Set Theories in New Foundations

Ph.D. Thesis, Cornell University, 1955.

The managers of this document are very interested in finding out about this thesis, which — despite not being recent work — has come to our attention only recently.**Coret, J. [1964]**Formules stratifiées et axiome de fondation.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**264**, pp. 809-812 and 837-839.**Coret, J. [1970]**Sur les cas stratifiés du schema de remplacement.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**271**, pp. 57-60. Annotated English translation by Thomas Forster**Crabbé, M. [1975]**Non-normalisation de ZF

unpublished (Kiel 1974). Download from http://logoi.be/crabbe/textes/default.html.

Note: Marcel says "My old unpublished counterexample to normalisation of ZF might also be of interest..."**Crabbé, M. [1973]**NF en un nombre fini d'axiomes.

Unpublished. Downloadable from here**Crabbé, M. [1975]**Types ambigus.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**280**, pp. 1-2. also in [BROKEN] minutes of the meeting of the Groupe de Contacte: Algebre et Logique**Crabbé, M. [1976]**La prédicativité dans les théories élémentaires.

Logique et Analyse**74-75-76**, pp. 255-266.**Crabbé, M. [1978a]**Ramification et prédicativité.

Logique et Analyse**84**, pp. 399-419.**Crabbé, M. [1978b]**Ambiguity and stratification.

Fundamenta Mathematicæ**CI**, pp. 11-17.**Crabbé, M. [1982a]**On the consistency of an impredicative subsystem of Quine's NF.

Journal of Symbolic Logic**47**, pp. 131-136.**Crabbé, M. [1982b]**À propos de 2^{α}.

Cahiers du Centre de Logique (Louvain-la-neuve)**4**, pp. 17-22.**Crabbé, M. [1983]**On the reduction of type theory.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**29**, pp. 235-237.**Crabbé, M. [1984]**Typical ambiguity and the axiom of choice.

Journal of Symbolic Logic**49**, pp. 1074-1078.**Crabbé, M. [1986]**Le schéma d'ambiguïté en théorie des types.

Bulletin de la Société Mathématique de Belgique (série B)**38**, pp. 46-57.**Crabbé, M. [1991]**Stratification and cut-elimination.

Journal of Symbolic Logic**56**, pp. 213-226**Crabbé, M. [1992a]**On NFU.

Notre Dame Journal of Formal Logic**33**, pp 112-119.**Crabbé, M. [1992b]**Soyons positifs: la complétude de la théorie naïve des ensembles.

Cahiers du Centre de Logique 1992**7**, pp.51-68.**Crabbé, M. [1994]**The Hauptsatz for stratified comprehension: a semantic proof.

Mathematical Logic Quarterly**40**, pp, 481-489.**Crabbé, M. [1999]**L'axiome de l'infini dans NFU.

C. R. Acad. Sci. Paris,**329**, Série I, p. 1033-1035, 1999.**Crabbé, M. [2000]**On the set of atoms.

L. J. of the IGPL,**8****6**, pp. 751-759.**Crabbé, M. [2000]**The Rise and Fall of typed Sentences

Journal of Symbolic Logic,**65**, no. 4, pp. 1858-1862.**Crabbé, M. [2004]**Cuts and Gluts.

To appear in the Journal of Applied Non-Classical Logics. Still downloadable at http://logoi.be/crabbe/textes/default.html**Crabbé, M. [2004]**L'égalité et l'extensionnalité.

To appear in Logique et Analyse. Still downloadable at http://logoi.be/crabbe/textes/default.html**Crabbé, M. [2004]**Une élimination des coupures ne tolérant pas l'extensionnalité.

To appear in Logique et Analyse. Still downloadable at http://logoi.be/crabbe/textes/default.html

Note:Marcel says "Though not yet published, the [above] are connected with stratification and positive stuff"

There is a lot of information about Crabbé's publications on: http://logoi.be/crabbe/textes/default.html**Crabbé, M. and Boffa, M [1975]**Les théorèmes 3-stratifiés de NF_{3}.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**280**, pp. 1657-1658.**Curry, H. B. [1954]**

Review of Rosser [1953a].

Bulletin of the American Mathematical Society**60**, pp. 266-272

**Da Costa, N.C.A. [1964]**Sur une système inconsistent de théorie des ensembles.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**258**, pp. 3144-3147.**Da Costa, N.C.A. [1965a]**Sur les systèmes formels C_{i}, C_{i}*, C_{i}=, D_{i}et NF.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**260**, pp. 5427-5430.**Da Costa, N.C.A. [1969]**On a set theory suggested by Dedecker and Ehresmann I and II.

Proceedings of the Japan Academy**45**, pp. 880-888.**Da Costa, N.C.A. [1971]**Remarques sur le système NF1.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**272**, pp. 1149-1151.**Da Costa, N.C.A. [1965b]**On two systems of set theory.

Proc. Koninkl. Nederl. Ak. v. Wetens. (serie A)**68**, pp 95-99. available at https://core.ac.uk/download/pdf/82597133.pdf**Da Costa, N.C.A. [1974]**Remarques sur les Calculs C_{n}, C_{n}*, C_{n}=, et Dn.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**278**, pp. 818-821.**Dawar, A; Zachiri McKenzie and Thomas Forster**Decidable Fragments of the Simple Theory of Types with Infinity and NF

Notre Dame J. Formal Logic**58**Number 3 (2017), 433--451.**G. Dowek. [2001]**[BROKEN] The Stratified Foundations as a theory modulo.[sic]

S. Abramsky (Ed.)*Typed Lambda Calculi and Applications*, Lecture Notes in Computer Science**2044**, Springer-Verlag, 2001.**Dzierzgowski, Daniel. [1991]**Intuitionistic typical ambiguity.

Archive for Mathematical Logic**31**, pp. 171-182.**Dzierzgowski, Daniel. [1993a]**Typical ambiguity and elementary equivalence.

Mathematical Logic Quarterly 39, pp. 436-446.**Dzierzgowski, Daniel. [1993b]**Le théorème d'ambiguïté et son extension à la logique intuitionniste.

Dissertation doctorale. Université catholique de Louvain, Institut de mathématique pure et appliquée.**Dzierzgowski, Daniel. [1995]**Models of intuitionistic TT and NF.

Journal of Symbolic Logic**60,**pp. 640-653.**Dzierzgowski, Daniel [1996]**Finite sets and natural numbers in intuitionistic TT,

Notre Dame Journal of Formal Logic.**37**, no. 4 (1996), pp. 585-601.**Dzierzgowski, Daniel [1998]**Finite sets and natural numbers in intuitionistic TT without extensionality.

Studia Logica,**61**, no. 3 (November 1998), pp. 417-428.

**Enayat, Ali [2004]**Automorphisms, Mahlo Cardinals, and NFU

in Nonstandard Models of Arithmetic and Set Theory, (Enayat, A. and Kossak, R., eds.), Contemporary Mathematics,**361**, American Mathematical Society. Also available here.**Enayat, Ali. [2006]**From Bounded Arithmetic to Second Order Arithmetic via Automorphisms

Logic in Tehran, pp. 87--113, Lect. Notes Logic, 26, Assoc. Symbol. Logic, La Jolla, CA

Note:the author says "includes the core results about automorphisms relevant to NFU + "the universe is finite". The results about NFU are announced in section 5.1 (but see also the introduction)."**Engeler, E. and Röhrli, H. [1969]**On the problem of foundations of category theory.

Dialectica**23**, pp. 58-66.**Esser, Olivier [1996]**Inconsistency of GPK + AFA.

Mathematical Logic Quarterly,**42**, pp. 104-108.**Esser, Olivier. [1997]**An interpretation of ZF and KM in a positive set theory.

Mathematical Logic Quarterly**43**, pp. 369-377.**Esser, Olivier [1999]**On the consistency of a positive theory.

Mathematical Logic Quarterly,**45**, no. 1, pp. 105-116.**Esser, Olivier [2000]**Inconsistency of the axiom of choice with the positive set theory GPK + infinity.

Journal of Symbolic Logic,**65**, pp. 1911-1916.**Esser, Olivier [2003]**On the axiom of extensionality in the positive set theory.

Mathematical Logic Quarterly,**49**, pp. 97-100.**Esser, Olivier [2003]**A strong model of paraconsistent logic.

Notre Dame Journal of Formal Logic,**44**.**Esser, Olivier [2004]**Une theorie positive des ensembles.

Cahiers du Centre de Logique,**13**, Academia-Bruylant, Louvain-la-Neuve (Belgium), ISBN 2-8729-687-6.**Esser, Olivier and Libert, Thierry [2005]**On topological set theory

Mathematical Logic Quarterly,**51**,pp. 263-273.

**Feferman, S. [2006]**Enriched stratified systems for the foundations of category theory,

in*What is Category Theory?*(G. Sica, ed.) Polimetrica, Milano (2006), 185-203.

Dr. Feferman says "A pdf file is available on my home page at

http://math.stanford.edu/~feferman/papers.html, item #62,with publication data.

Also, my unpublished 1972 MS on which this is based, can be found there at item #58"

(item #57 also looks interesting --MRH).**Feferman, S. [2013]**Foundations of Unlimited Category Theory: What remains to be done

in The Review of Symbolic Logic**6**, Number 1, March 2013

**Forster, Thomas E. [1976]**"N.F."

Ph.D. thesis, University of Cambridge. The shortest Ph.D. thesis title ever!**Forster, Thomas E. [1982]**Axiomatising set theory with a universal set.

Cahiers du Centre de Logique (Louvain-la-neuve)**4**, pp. 61-76.**Forster, Thomas E. [1983a]**Quine's New Foundations, an introduction.

Cahiers du Centre de Logique (Louvain-la-neuve)**5**. 100 pp.**Forster, Thomas E. [1983b]**Further consistency and independence results in NF obtained by the permutation method.

Journal of Symbolic Logic**48**, pp. 236-238.**Forster, Thomas E. [1985]**The status of the axiom of choice in set theory with a universal set.

Journal of Symbolic Logic**50**, pp. 701-707.

(The author reports that the definition of "φ-hat" in this paper is faulty.)**Forster, Thomas E. [1987a]**Permutation models in the sense of Rieger-Bernays.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**33**, pp. 201-210.

(Theorem 2.3 is misstated. The correct version is theorem 3.1.30 of Forster [1992b] and [1995].)**Forster, Thomas E. [1987b]**Term models for weak set theories with a universal set.

Journal of Symbolic Logic**52**, pp. 374-387.**Forster, Thomas E. [1989]**A second-order theory without a (second-order) model.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**35**, pp. 285-286.**Forster, Thomas E. [1990]**Permutation Models and Stratified Formulæ, a Preservation Theorem.

Zeitschrift für Mathematische Logic und Grundlagen der Mathematik,**36**(1990) pp 385-388.**Forster, Thomas E. [1992a]**On a problem of Dzierzgowski.

Bulletin de la Société Mathématique de Belgique (série B)**44**, pp. 207-214.**Forster, Thomas E. [1992b]**Set Theory with a Universal Set; exploring an untyped Universe

Oxford Logic Guides Clarendon Press, Oxford.**Forster, Thomas E. [1993]**A semantic characterisation of the well-typed formulæ of lambda calculus

Theoretical Computer Science**110**, pp 405-408.**Forster, Thomas E. [1994]**Why Set theory without the axiom of foundation?

Journal of Logic and Computation, 4, number 4 (August 1994) pp. 333-335.**Forster, Thomas E. [1994]**Weak Systems of Set Theory Related to HOL.

(Invited talk given at the 1994 meeting of HUG) HUG94, Springer lecture notes in Computer Science**859**pp 193-204.**Forster, Thomas E. [1995]**Set Theory with a Universal Set, exploring an untyped Universe

Second edition. Oxford Logic Guides, Oxford University Press, Clarendon Press, Oxford.**Forster, Thomas E. [1997]**60 years of NF

The American Mathematical Monthly**104**, no. 9 (November 1997), pp. 838-845. (Reprinted in Follesdal, ed: Philosophy of Quine,**IV**Logic, Modality and Philosophy of Mathematics) Taylor-and-Francis 2001.**Forster, Thomas E. [2001]**Church's Set Theory with a Universal Set

in: Logic, Meaning and Computation: essays in memory of Alonzo Church, Synthese library 305, Kluwer, Dordrecht, Boston and London 2001. (The linked version is to be preferred to the version in print, as I remove typos and mathematical errors from it as they come to my notice. Readers who want to learn about Church's set theory with a universal set should probably start with this article. I have some more cutting-edge stuff concerning*inter alia*the possibility of iterating this construction which i am writing up. I want to do some more work on it before I publish it but I am happy to show it to interested parties if asked.)**Forster, Thomas E. [2001]**Games played on an illfounded membership relation.

in A Tribute to Maurice Boffa ed Crabbé, Point, and Michaux. (Supplement to the December 2001 number of the Bulletin of the Belgian Mathematical Society)

These games were independently rediscovered later by Denis Saveliev.**Forster, Thomas E. [2003]**Reasoning about Theoretical Entities.

Advances in Logic,**3**World Scientific (UK), Imperial College Press 2003.**Forster, Thomas E. [2003]**ZF + Every set is the same size as a wellfounded set

Journal of Symbolic Logic,**58**, (2003) pp 1-4.**Forster, Thomas E. [2004]**AC fails in the natural analogues of V and L that model the stratified fragment of ZF

in Nonstandard Models of Arithmetic and Set Theory, (Enayat, Ali. and Kossak, Roman., eds.), Contemporary Mathematics,**361**, American Mathematical Society.**Forster, Thomas E. [2006]**Permutations and Wellfoundedness: the True Meaning of the Bizarre Arithmetic of Quine's NF

Journal of Symbolic Logic,**71**(2006) pp 227-240.**Forster, Thomas E. [2007]**Implementing Mathematical Objects in Set theory

Logique et Analyse,**50**No.197 (2007)**Forster, Thomas E. [2008]**The Iterative Conception of Set.

Review of Symbolic Logic,**1**(2008) pp 97-110. (This was voted one of the ten best philosophy articles of 2008 by The Philosophers' Annual!!)**Forster, Thomas E. [2009]**A Tutorial on Constructive NF

in the NF 70th anniversary volume, Cahiers du Centre de Logique,**16**2009 pp 137--171.

*This is work-in-progress; a later version of this document is available here*.**Forster, Thomas E. [2009]**Paris-Harrington in an NF context

in One hundred Years of Axiomatic Set Theory Cahiers du Centre de Logique,**17**This paper takes up Harvey Friedman's

*aperçu*that the statement of Paris-Harrington is unstratified, and runs with it. Unfortunately it turns out that the strength of Paris-Harrington does not, after all, have anything to do with failure of stratification. This is explained in a sequel available at https://www.dpmms.cam.ac.uk/~tf/parisharringtonredux.pdf**Forster, Thomas E. [2010]**NF at (nearly) 75. In the Special Quine number of Logique et Analyse, (212):483 (2010) edited by Cresswell and Rini. 2010.**Forster, Thomas E.[2014],**

“ Mathematical entities arising from equivalence relations, and their implementation in Quine's NF,”*Philosophia Mathematica.***24**2016**Forster, Thomas E. and Nathan Bowler, [2009]**Normal Subgroups of Infinite Symmetric Groups, with an Application to Stratified Set Theory.

Journal of Symbolic Logic**74**(2009) pp 17--26.

**Forster, Thomas E. and Esser, Olivier E. [2007]**Relaxing Stratification

Bull. Belg. Math. Soc. Simon Stevin, vol 14, (2007), pp. 247-258.

**Forster, Thomas E. and Randall Holmes [2009]**Permutation methods in NF and NFU.

NF 70th anniversary volume, Cahiers du Centre de Logique, vol 16 2009 pp 33--76.**Forster, Thomas E. and Kaye, Richard. [1991]**End-extensions preserving power set.

Journal of Symbolic Logic**56**, pp. 323-328.

(Errata in Forster [1992b], p. 139; reiterated in Forster [1995], p. 152.)**Forster, Thomas E. and Thierry Libert [2011]**An Order-Theoretic account of some set-theoretical paradoxes

Notre Dame Journal of Formal Logic

**Forster, Thomas E. and Rood, C.M. [1996]**Sethood and situations.

Computational Linguistics. 22 (1996) pp 405-408.

Online at https://www.aclweb.org/anthology/J96-3005.pdf**Forster, Thomas E., Holmes, M.R., and Libert, Thierry.**``Alternative Set Theories"

in volume 6 (``Sets in the Twentieth Century'') of the Handbook of the History of Logic, Elsevier/North-Holland.**Forster, Thomas E, Zachiri McKenzie and Anuj Dawar**Decidable Fragments of the Simple Theory of Types with Infinity and NF

Notre Dame J. Formal Logic**58**Number 3 (2017), 433--451.**Forti, M. [1987]**Models of the generalized positive comprehension principle.

Preprint, Università di Pisa.**Forti, M. and Hinnion, R. [1989]**The consistency problem for positive comprehension principles.

Journal of Symbolic Logic**54**, pp. 1401-1418.**Forti, M. and Honsell, F. [1983]**Set theory with free construction principles.

Annali della Scuola Normale Superiore di Pisa, Scienze fisiche e matematiche 10, pp. 493-522.**Forti, Marco and F. Honsell.[1989]**Models of Selfdescriptive Set Theories.

in Partial Differential equations and the calculus of Variations, Essays in Honor of Ennio De Giorgi, I. (F. Colombini et al., eds), Birkhäuser, Boston (1989), pp. 473-518.**Forti, M. and Honsell, F. [1992a]**Weak foundation and anti-foundation properties of positively comprehensive hyperuniverses.

Cahiers du Centre de Logique (Louvain-la-Neuve)**7**, pp. 31-43.**Forti, M. and Honsell, F. [1992b]**A general construction of hyperuniverses.

Preprint, Università di Pisa.**Forti, Marco and F. Honsell.[1996]**Choice Principles in Hyperuniverses.

Annals of Pure and Applied Logic**77**(1996), pp. 35-52.**Forti, Marco and F. Honsell.[1998]**Addendum and Corrigendum to "Choice Principles in Hyperuniverses".

Annals of Pure and Applied Logic**92**(1998), pp. 211-214.**Gilmore, Paul C [1974]**The Consistency of partial Set Theory without Extensionality.

Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics,**13**, part 2, AMS, Providence RI, pp.147-153.**Gilmore, Paul C [1986]**Natural Deduction Based Set Theories: A New Resolution of the Old Paradoxes.

JSL,**51**, pp.393-411.**Grishin, V.N. [1969]**Consistency of a fragment of Quine's NF system

Soviet Mathematics Doklady**10**, pp. 1387-1390.**Grishin, V.N. [1972a]**The equivalence of Quine's NF system to one of its fragments (in Russian).

Nauchno-tekhnicheskaya Informatsiya (series 2)**1**, pp. 22-24.**Grishin, V.N. [1972b]**Concerning some fragments of Quine's NF system (in Russian).

Issledovania po matematicheskoy lingvistike, matematicheskoy logike i informatsionym jazykam (Moscow), pp. 200-212.**Grishin, V.N. [1972c]**The method of stratification in set theory (in Russian).

Ph.D. thesis, Moscow University.**Grishin, V.N. [1973a]**The method of stratification in set theory (Abstract of Ph.D. thesis, in Russian).

Academy of Sciences of the USSR (Moscow). 9pp.**Grishin, V.N. [1973b]**An investigation of some versions of Quine's systems.

Nauchno-tekhnicheskaya Informatsiya (series 2)**5**, pp. 34-37.**Hailperin, T. [1944]**A set of axioms for logic.

Journal of Symbolic Logic**9**, pp. 1-19.*A Classic; the first (of several) finite axiomatisations of NF***Hatcher, W.S. [1963]**La notion d'équivalence entre systèmes formels et une généralisation du système dit "New Foundations" de Quine.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**256**, pp. 563-566.**Henson, C.W. [1969]**Finite sets in Quine's New Foundations.

Journal of Symbolic Logic**34**, pp. 589-596.**Henson, C.W. [1973a]**Type-raising operations in NF.

Journal of Symbolic Logic**38**, pp. 59-68.**Henson, C.W. [1973b]**Permutation methods applied to NF.

Journal of Symbolic Logic**38**, pp. 69-76.**Hiller, A.P. and Zimbarg, J.P. [1984]**Self-reference with negative types.

Journal of Symbolic Logic**49**, pp. 754-773.**Hinnion, Roland. [1972]**Sur les modèles de NF.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**275**, p. 567.**Hinnion, Roland. [1974]**Trois résultats concernant les ensembles fortement cantoriens dans les "New Foundations" de Quine.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**279**, pp. 41-44.**Hinnion, Roland. [1975]**Sur la théorie des ensembles de Quine.

Ph.D. thesis, ULB Brussels.

Annotated English translation by Thomas Forster.**Hinnion, Roland. [1976]**Modèles de fragments de la théorie des ensembles de Zermelo-Fraenkel dans les "New Foundations" de Quine.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**282**, pp. 1-3. Also in [BROKEN] minutes of the meeting of the Groupe de Contacte: Algebre et Logique**Hinnion, Roland. [1979]**Modèle constructible de la théorie des ensembles de Zermelo dans la théorie des types.

Bulletin de la Société Mathématique de Belgique (série B)**31**, pp. 3-11.**Hinnion, Roland. [1980]**Contraction de structures et application à NFU: Définition du "degré de non-extensionalité" d'une relation quelconque.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**290**, pp. 677-680.**Hinnion, Roland. [1981]**Extensional quotients of structures and applications to the study of the axiom of extensionality.

Bulletin de la Société Mathématique de Belgique (série B)**33**, pp. 173-206.**Hinnion, Roland. [1982]**NF et l'axiome d'universalité.

Cahiers du Centre de Logique (Louvain-la-neuve)**4**, pp. 45-59.**Hinnion, Roland. [1986]**Extensionality in Zermelo-Fraenkel set theory.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**32**, pp. 51-60.**Hinnion, Roland. [1987]**Le paradoxe de Russell dans des versions positives de la theorie naïve des ensembles

Comptes Rendus de l'Academie des Science de Paris,**304**, pp. 307-310.**Hinnion, Roland. [1989]**Embedding properties and anti-foundation in set theory.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**35**, pp. 63-70.**Hinnion, Roland. [1990]**Stratified and positive comprehension seen as superclass rules over ordinary set theory.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**36**, pp. 519-534.**Hinnion, Roland. [1994]**Naïve set theory with extensionality in partial logic and in paradoxical logic.

Notre Dame Journal of Formal Logic,**35**, pp. 15-40..**Hinnion, Roland. [2003]**About the coexistence of classical sets with non-classical ones: a survey

Logic and Logical Philosophy,**11**, pp. 79-90.**Hinnion, Roland. [2006]**Intensional positive set theory

Reports on Mathematical Logic,**40**.**Hinnion, Roland and Libert, Thierry [2003]**Positive abstraction and extensionality

Journal of Symbolic Logic,**68**, pp. 828-836.**Hinnion, Roland and Thierry Libert [2008].**Topological Models for Extensional Partial Set Theory

Notre Dame J. Formal Logic,**49**, Number 1 (2008), 39-53.**Holmes, Randall [1991a]**Systems of combinatory logic related to Quine's 'New Foundations.'

Annals of Pure and Applied Logic**53**, pp. 103-133.**Holmes, Randall [1991b]**The Axiom of Anti-Foundation in Jensen's 'New Foundations with Ur-Elements.'

Bulletin de la Société Mathématique de Belgique (série B)**43**, pp. 167-179.**Holmes, Randall [1992]**Modelling fragments of Quine's 'New Foundations.'

Cahiers du Centre de Logique (Louvain-la-Neuve)**7**, pp. 97-112.**Holmes, Randall [1993]**Systems of combinatory logic related to predicative and 'mildly impredicative' fragments of Quine's 'New Foundations.'

Annals of Pure and Applied Logic**59**, pp 45-53.**Holmes, Randall [1994]**The set theoretical program of Quine succeeded (but nobody noticed).

Modern Logic**4**, pp. 1-47.**Holmes, Randall [1995a]**The equivalence of NF-style set theories with "tangled" type theories; the construction of ω-models of predicative NF (and more).

Journal of Symbolic Logic**60**, pp. 178-189.**Holmes, Randall [1995b]**Untyped lambda-calculus with relative typing.

Typed Lambda-Calculi and Applications (Proceedings of TLCA '95), Springer, pp. 235-248.**Holmes, Randall. [1998]**Elementary set theory with a universal set.

volume**10**of the Cahiers du Centre de logique, Academia, Louvain-la-Neuve (Belgium), 241 pages, ISBN 2-87209-488-1.

See http://www.cahiersdelogique.be/cahiersangl.html

here for an on-line errata slip. By permission of the publishers, a corrected text is published online here; an official second edition will appear online eventually.

Marcel reminds us that the official online version is at http://www.logic-center.be/cahiersdelogique/cahiersangl.html; this should be free of the errors in the original errata slip, but will not reflect more recent revisions found in the version on my site.**Holmes, Randall [1999]**Subsystems of Quine's ``New Foundations'' with Predicativity Restrictions

Notre Dame Journal of Formal Logic,**40**, no. 2, pp. 183-196.

appeared physically in 2001.**Holmes, Randall [2002]**Forcing in NFU and NF

in M. Crabbe, C. Michaux, and F. Point, eds., A tribute to Maurice Boffa, Belgian Mathematical Society, 2002.**Holmes, Randall [2001]**Foundations of mathematics in polymorphic type theory.

Topoi,**20**, pp. 29-52.

NOTE: this is my official answer to the claim by certain parties on the FOM list that mathematics must be defined in terms of what we can do in ZFC...**Holmes, Randall [2001]**Strong axioms of infinity in NFU.

Journal of Symbolic Logic,**66**, no. 1, pp. 87-116.

(brief notice of errata with corrections to appear in a future issue).

**Holmes, Randall [2001]**Foundations of mathematics in polymorphic type theory.

Topoi,**20**, pp. 29-52.

NOTE: this is Holmes' official answer to the claim by certain parties on the FOM list that mathematics must be defined in terms of what we can do in ZFC**Holmes, Randall [2001]**Strong axioms of infinity in NFU.

Journal of Symbolic Logic,**66**, no. 1, pp. 87-116.

(brief notice of errata with corrections to appear in a future issue).**Holmes, Randall [2001]**The Watson theorem prover.

Journal of Automated Reasoning,**26**, no. 4, pp. 357-408.

This paper describes a theorem prover using a higher order logic based on NFU.**Holmes, Randall [2001]**Tarski's Theorem and NFU

in C. Anthony Anderson and M Zeleny (eds.), Logic, Meaning and Computation, Kluwer, 2001, pp. 469--478.**Holmes, Randall [2002]**Forcing in NFU and NF

in M. Crabbé, C. Michaux, and F. Point, eds., A tribute to Maurice Boffa, Belgian Mathematical Society, 2002.**Holmes, Randall [2004]**Paradoxes in double extension set theories

Studia Logica,**77**(2004), pp. 41-57.**Holmes, Randall [2005]**The structure of the ordinals and the interpretation of ZF in double extension set theory

Studia Logica,**79**, pp. 357-372.**Holmes, Randall [2008]**Symmetry as a criterion for comprehension motivating Quine's ``New Foundations''

Studia Logica,**88**, no. 2 (March 2008).**Holmes, Randall and Alves-Foss, J. [2001]**The Watson theorem prover.

Journal of Automated Reasoning,**26**, no. 4, pp. 357-408.**Holmes, Randall and Alves-Foss, J. [2000]**A strong and mechanizable grand logic.

in Theorem Proving in Higher Order Logics: 13th International Conference, TPHOLs 2000, Lecture Notes in Computer Science,**1869**, Springer-Verlag, pp. 283-300.

This is the theoretical paper on the foundations of the Watson theorem prover.**Honsell, Furio**All Honsell's publications in this area are joint with Marco Forti and listed under his name**Jamieson, M.W. [1994]**Set theory with a Universal Set.

Ph.D. thesis, University of Florida. 114pp.**Jech, T. [1995]**OTTER experiments in a system of combinatory logic

Journal of Automated Reasoning,**14**, pp. 413-426.**Jensen, R.B. [1969]**On the consistency of a slight(?) modification of Quine's NF.

Synthese**19**, pp. 250-263.

**Kaye, R.W. [1991]**A generalisation of Specker's theorem on typical ambiguity.

Journal of Symbolic Logic**56**, pp 458-466.**Kaye, R.W. [1996]**The quantifier complexity of NF.

Bulletin of the Belgian Mathematical Societé Simon Stevin, ISSN 1370-1444,**3**, pp 301-312.**Kemeny, J.G. [1950]**Type theory vs. set theory (abstract of Ph.D. thesis).

Journal of Symbolic Logic**15**, p. 78.**Khakhanian.**Systema NFI ravnoneprotivoretchivaya c Systema Quine NF

Online journal "Logical Studies"**8**(2002)**Kirmayer, G. [1981]**A refinement of Cantor's theorem.

Proceedings of the American Mathematical Society**83**, p. 774.**Kisielewicz, Andrzej [1989]**

Double extension set theory

Reports on Mathematical Logic**23**81--89, 1989.**Kisielewicz, Andrzej [1998]**A very strong set theory?

Studia Logica**61**171--178, 1998.

Note: as we comment above, the jury is still out on double extension set theory; but if the remaining version of the 1998 paper is consistent it is certainly appropriate here — MRH.**Körner, F. [1994]**Cofinal indiscernibles and some applications to New Foundations.

Mathematical Logic Quarterly**40**, pp. 347-356.**Körner, F. [1998]**Automorphisms moving all non-algebraic points and an application to NF.

Journal of Symbolic Logic**63**, p. 815-830.**Kreinovich, V. and Oswald, U. [1982]**A decision method for the Universal sentences of Quine's NF.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**28**, pp. 181-187.**Kühnrich, M. and Schultz, K. [1980]**A hierarchy of models for Skala's set theory.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**26**, pp. 555-559.**Kuzichev, A.C. [1981]**Arithmetic theories constructed on the basis of lambda-conversion.

Soviet Mathematics Doklady**24**, pp. 584-589. [BROKEN: Randall now has a pdf of \mendelson's translation**Kuzichev, A.C. [1983]**Nyeprotivoretchivost' Sistema NF Quine.

Doklady Akademia Nauk 270, pp. 537-541.In the 1980's Kuzichev produced an alleged consistency proof for NF using ideas from λ-calculus, and some attempt was made in the West to come to grips with this work (since the idea that techniques from λ-calculus might eventually prove Con(NF) is not crazy at all) but not — so far— with any success. Some manuscript translations from that era survive and have now been scanned (by Zachiri McKenzie): [BROKEN] here is a translation into English prepared by a professional translator (not a mathematician but a friend of tf); [BROKEN] here is another english version of (i think) the same text (possibly due to Eliot Mendelson?) and finally [BROKEN] here is a translation into French supplied by Marcel Crabbé. The matter certainly merits investigation, but these texts come with no guarantees! — tf [None of these links work, I need the files -MRH 2020]

**Lake, J. [1974]**Some topics in set theory.

Ph.D. thesis, Bedford College, London University.

(Lake's thesis is listed here because towards the end of it he considers the possibility of having an inhomogeneous ordered pair function in NF.)**Lake, J. [1975]**Comparing type theory and set theory.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**21**, pp. 355-356.**Libert, Thierry. [2004]**Semantics for naïve set theory in many-valued logics, technique and historical account

in, J. van Benthem and G. Heintzmann, eds., The age of alternative logics, Kluwer, 2004.**Libert, Thierry. [2008].**Positive Frege and its Scott-style semantics.

Math. Log. Quart.**54**, No. 4, 378 – 402**Libert, Thierry. [2008].**Positive Frege and its Scott-style semantics.

Math. Log. Quart.**54**, No. 4, 378 ? 402**Libert, Thierry. [2006]**More studies on the axiom of comprehension

Cahiers du Centre de Logique,**15**, Academia-Bruylant, Louvain-la-Neuve (Belgium).**Libert, Thierry. [2004]**Semantics for naïve set theory in many-valued logics, technique and historical account

in, J. van Benthem and G. Heintzmann, eds., The age of alternative logics, Kluwer, 2004.**Libert, Thierry. [2005]**Models for a Paraconsistent Set Theory

Journal of Applied Logic,**3**, pp. 15-41.**Libert, Thierry. [2006]**More studies on the axiom of comprehension

Cahiers du Centre de Logique,**15**, Academia-Bruylant, Louvain-la-Neuve (Belgium).**Libert, Thierry and O. Esser [2005]**On topological set theory

Mathematical Logic Quarterly,**51**, pp. 263-273.**Libert, Thierry and Roland Hinnion. [2008].**Topological Models for Extensional Partial Set Theory

Notre Dame J. Formal Logic,**49**, Number 1 (2008), 39-53.**R. Hinnion and Libert, Thierry [2003]**Positive abstraction and extensionality

Journal of Symbolic Logic,**68**, pp. 828-836.

**McKenzie, Zachiri and Vu, D.**Permutation methods yielding models of the stratified axioms of Zermelo Fraenkel set theory

NF 70th anniversary volume, Cahiers du Centre de Logique,**16**2009.**McKenzie, Zachiri, Anuj Dawar and Thomas Forster**Decidable Fragments of the Simple Theory of Types with Infinity and NF

Notre Dame J. Formal Logic**58**Number 3 (2017), 433--451.

**McLarty, C. [1992]**Failure of cartesian closedness in NF.

Journal of Symbolic Logic**57**, pp. 555-556.**McNaughton, R. [1953]**Some formal relative consistency proofs.

Journal of Symbolic Logic**18**, pp. 136-144.**Malitz, R.J. [1976]**Set theory in which the axiom of foundation fails.

Ph.D. thesis, UCLA. A central text in the study of Positive Set Theory.

Malitz was a student of Church.**Manakos, J. [1984]**On Skala's set theory.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik 30, pp. 541-546.**Mitchell, E. [1976]**A model of set theory with a universal set.

Ph.D. thesis, University of Wisconsin, Madison, Wisconsin. Mitchell was Church's student.**Moss, L and Barwise, J. [1996]**Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena.

CSLI Lecture Notes (60), Stanford University, ISBN: 978-1575860084.

Note: Most of the book is concerned with ill-founded set theories without a universal set, in the manner of Forti and Honsell. Chapter 20 covers a positive set theory, SEC, with a universal set but without, for example, V — {V}.**Oberschelp, A. [1964]**Eigentliche Klasse als Urelemente in der Mengenlehre.

Mathematische Annalen**157**, pp. 234-260.**Oberschelp, A. [1973]**Set theory over classes.

Dissertationes Mathematicæ**106**. 62 pp.**Oksanen, M. [1999]**The Russell-Kaplan Paradox and Other Modal Paradoxes: A New Solution

Nordic Journal of Philosophical Logic,**4**, No. 1, pp. 73-93, June 1999, Scandinavian University Press.

Also available on- line at http://www.hf.uio.no/filosofi/njpl/**Orey, S. [1955]**Formal development of ordinal number theory.

Journal of Symbolic Logic**20**, pp. 95-104.**Orey, S. [1956]**On the relative consistency of set theory.

Journal of Symbolic Logic**21**, pp. 280-290.

These two papers of Orey seem to be the most publicly visible record of a project Rosser had in the 1950's (and which he wished on at least three of his Ph.D. students---Firestone, Orey and Collins), namely that of the*converse consistency problem*: How much of ZF can we prove consistent in (or relative to) NF? How much do we have to add to NF to obtain a system in which we can prove con Z or con ZF? (This task is addressed in Hinnion's Ph.D. thesis, itemised here). One idea is that instead of defining Gödel's*L*as a class of sets, one defines it instead (or rather an isomorphic copy of it) as a special binary relation on the ordinals. These two papers of Orey's seem to be concerned with this possibility. So does the thesis of Firestone, and too (we think) Collins. There is also an article of Takeuti (Journal of the Mathematical Society of Japan,**6**no 2 pp 197-220) that seems to do the same thing — tho' not in an NF context. (I have copies of some of these works, tho' not the Collins or Firestone theses).**This needs to be investigated properly**— tf**Orey, S. [1964]**New Foundations and the Axiom of Counting.

Duke Mathematical Journal**31**, pp. 655-660.

This paper shows how to use Specker's tie-up between NF and Type-Theory to show that the assumption that every finite set is cantorian implies Con(NF). There are ideas in this paper that still need to be worked out. It's one of the fundamental papers in NF studies, and also a wonderful example of a proof in set theory using truth-definitions.**Oswald, U. [1974]**Sur les modeles d'un fragment de NF

in in the [BROKEN] minutes of the meeting of the Groupe de Contacte: Algebre et Logique**Oswald, U. [1976]**Fragmente von "New Foundations" und Typentheorie.

Ph.D. thesis, ETH Zürich. 46 pp.**Oswald, U. [1981]**Inequivalence of the fragments of New Foundations.

Archiv für mathematische Logik und Grundlagenforschung**21**pp. 77-82.**Oswald, U. [1982]**A decision method for the existential theorems of NF_{2}

Cahiers du Centre de Logique (Louvain-la-neuve)**4**, pp. 23-43.**Oswald, U. and Kreinovich, V. [1982]**A decision method for the Universal sentences of Quine's NF.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**28**, pp. 181-187.

**Pabion, J.F. [1980]**TT_{3}I est équivalent à l'arithmétique du second ordre.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**290**, pp. 1117-1118.**Pétry, A. [1974]**À propos des individus dans les "New Foundations" de Quine.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A)**279**, pp. 623-624.**Pétry, A. [1975]**Sur l'incomparabilité de certains cardinaux dans le "New Foundations" de Quine.

Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris (série A) 281, pp. 673-675.**Pétry, A. [1976]**Sur les cardinaux dans le "New Foundations" de Quine.

Ph.D. thesis, University of Liège. 66 pp. electronically accessible here**Pétry, A. [1976]**On cardinal numbers in Quine's NF.

Set theory and hierarchy theory, Bierutowice 1976, Springer Lecture Notes in Mathematics**619**, pp. 241-250.**Pétry, A. [1979]**On the typed properties in Quine's NF.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**25**, pp. 99-102.**Pétry, A. [1982]**Une charactérisation algébrique des structures satisfaisant les mêmes sentences stratifiées.

Cahiers du Centre de Logique (Louvain-la-neuve)**4**, pp. 7-16.**Pétry, A. [1992]**Stratified languages.

Journal of Symbolic Logic**57**, pp. 1366-1376.**Pétry, A. and Boffa, M [1993]**On self-membered sets in Quine's set theory NF.

Logique et Analyse**141-142**, pp. 59-60.**Prati, N. [1994]**A partial model of NF with E.

Journal of Symbolic Logic**59**, pp. 1245-1253.**Putnam, Hilary [1957]**Axioms of Class Existence

Summaries of talks presented at the Summer Institute for Symbolic Logic, Cornell University, 1957, pp. 271-274.**Quine, W.V. [1937a]**New Foundations for Mathematical Logic.

American Mathematical Monthly**44**, pp. 70-80.

Reprinted in Quine [1953a]**Quine, W.V. [1937b]**On Cantor's theorem.

Journal of Symbolic Logic**2**, pp. 120-124.**Quine, W.V. [1945]**On ordered pairs.

Journal of Symbolic Logic**10**, pp. 95-96.**Quine, W.V. [1951a]**Mathematical logic, revised ed.

Harvard University Press.**Quine, W.V. [1951b]**On the consistency of "New Foundations."

Proceedings of the National Academy of Sciences of the USA**37**, pp. 538-540.**Quine, W.V. [1953a]**From a logical point of view.

Harper & Row.**Quine, W.V. [1953b]**On ω-inconsistency and a so-called axiom of infinity.

Journal of Symbolic Logic**18**, pp. 119-124.

Reprinted in Quine [1966] (Selected Logic Papers).**Quine, W.V. [1963]**Set theory and its logic.

Belknap Press.**Quine, W.V. [assorted editions]**Selected logic papers.

Random House.**Quine, W.V. [1969]**Set theory and its logic, revised edition.

Belknap Press.**Quine, W.V. [1993]**The Inception of NF.

Bulletin de la Société Mathématique de Belgique (série B)**45**, pp. 325-328.

(This paper was written for the NF 50th anniversary meeting in Oberwolfach in 1987. It can be found in all recent editions of Quine's*Selected Logic Papers*.)

**Rood, Cathy and Thomas Forster [1996]**Sethood and situations.

Computational Linguistics.**22**(1996) pp 405-408.

Rood now publishes under her married name: Wyss.**Rosser, J.B. [1939a]**On the consistency of Quine's new foundations for mathematical logic.

Journal of Symbolic Logic**4**, pp. 15-24.**Rosser, J.B. [1939b]**Definition by induction in Quine's new foundations for mathematical logic.

Journal of Symbolic Logic**4**, p. 80.**Rosser, J.B. [1942]**The Burali-Forti paradox.

Journal of Symbolic Logic**7**, pp. 11-17.**Rosser, J.B. [1952]**The axiom of infinity in Quine's New Foundations.

Journal of Symbolic Logic**17**, pp. 238-242.**Rosser, J.B. [1953a]**Logic for mathematicians.

McGraw-Hill. The first edition is electronically accessible here; only in the second edition do you get the appendices on the proof of Infinity and the negation of Choice, but it is a wonderful book anyway.**Rosser, J.B. [1953b]**Deux esquisses de logique.

Paris.**Rosser, J.B. [1954]**Review of Specker [1953].

Journal of Symbolic Logic**19**, p. 127.

This can be usefully read in conjunction with Specker [1953]**Rosser, J. B. [1956]**The relative strength of Zermelo's set theory and Quine's new foundations.

Proceedings of the International Congress of Mathematicians (Amsterdam 1954) III, pp. 289-294.**Rosser, J. B. [1978]**Logic for mathematicians, second edition.

Chelsea Publishing.**Rosser, J.B. and Wang, H. [1950]**Non-standard models for formal logic.

Journal of Symbolic Logic**15**, pp. 113-129.**Rouvelas, P**Increasing sentences in Simple Type Theory.

Ann. Pure Appl. Logic,**168**, No 10, p. 1902--1926, 2017**Rouvelas, P.**Partial type-shifting automorphisms.

Logique et Analyse,**60**, No 238, p. 167--177, 2017**Rouvelas, P.**Decreasing sentences in Simple Type Theory.

Math. Logic Quart.,**63**, Issue 5, p. 342--363, 2017**Russell, B.A.W. [1908]**Mathematical logic as based on the theory of types.

American Journal of Mathematics**30**, pp. 222-262.**Russell, B.A.W. and Whitehead, A. N.[1910]**Principia Mathematica. Cambridge University Press.

**Schultz, K. [1977]**Ein Standardmodell für Skala's Mengenlehre.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**23**, pp. 405-408.**Schultz, K. [1980]**The consistency of NF.

Unpublished. (The pdf linked here was scanned by Zachiri McKenzie from a photocopied typescript in the possession of Thomas Forster, who was probably given it by Boffa.)**Scott, D.S. [1960]**Review of Specker [1958].

Mathematical Reviews**21**, p. 1026.A very useful summary of a fundamental paper.

**Scott, D.S. [1962]**Quine's individuals.

Logic, methodology and philosophy of science, ed. E. Nagel, Stanford University Press, pp. 111-115.**Scott, D.S. [1980]**The lambda calculus: some models, some philosophy.

The Kleene Symposium, North-Holland, pp. 116-124.**Sharlow, Mark [2001]**Broadening the Iterative Conception of Set

Notre Dame J. Formal Logic**42**, Number 3 (2001), pp 149-170.(The abstract concludes with the words: ``It is suggested that this modified iterative conception of set supports the axioms of Quine's set theory NF.'')

**Sheridan, Flash [2014]**A Variant of Church's Set Theory with a Universal Set in which the Singleton Function is a Set.

Logique et Analyse**59**(233) pp. 81-131, doi:10.2143/LEA.233.0.3149532

This is an abridged version of an Oxford doctoral thesis awaiting resubmission. The full version is online at http://www.logic-center.be/Publications/Bibliotheque/SheridanVariantChurch.pdf.

Slides for a talk summarizing the results and philosophy are at http://pobox.com/~flash/Fixing_Freges_Set_Theory.pdf . **Skala, H. [1974a]**Eine neue Methode, die Paradoxien der naïven Mengenlehre zu vermeiden.

Annalen der Österreichen Akademie der Wissenschaften Math-Nat. Kl. II, pp. 15-16.**Skala, H. [1974b]**An alternative way of avoiding the set-theoretical paradoxes.

Zeitschrift für mathematische Logik und Grundlagen der Mathematik**20**, pp. 233-237.**Solovay, R. [2008]**Correspondence describing Solovay's proof that NFU* (NFU + Counting + "every definable subclass of a strongly cantorian set is a set") is equiconsistent with Zermelo + Σ_{2}Replacement. This proof was presented in a talk at Stanford in October 2008.

http://math.berkeley.edu/~solovay/NFU_star.html**Specker, E.P. [1953]**The axiom of choice in Quine's new foundations for mathematical logic.

Proceedings of the National Academy of Sciences of the USA**39**, pp. 972-975. See also Rosser's review.**Specker, E.P. [1958]**Dualität.

Dialectica**12**, pp. 451-465.

Note: There is an annotated English translation by Forster of this important and elegant article in Follesdal, ed: Philosophy of Quine,**IV**Logic, Modality and Philosophy of Mathematics pp 7-16. Taylor-and-Francis 2001.**Specker, E.P. [1962]**Typical ambiguity.

Logic, methodology and philosophy of science, ed. E. Nagel, Stanford University Press, pp. 116-123.**These three papers of Specker are absolutely fundamental to NF studies.****Stanley, R.L. [1955]**Simplified foundations for mathematical logic.

Journal of Symbolic Logic**20**, pp. 123-139.

**Tsouvaras, Athanassios**Combinatorics related to NF consistency

Presented at the NF 70th anniversary meeting, Cambridge 2007**Tsouvaras, Athanassios**A reduction of the NF consistency problem

Presented at the NF 70th anniversary meeting, Cambridge 2007**Tupailo, S. [2010]**Consistency of strictly impredicative NF and a little more...

Journal of Symbolic Logic**75**(4) pp. 1326-1338.**Vayl, V.**Gentzen systems of postulates for set theory.

AMS translations series 2**135**pp. 23-37.**Vayl, V.**On Models of Quine's NF.

Logique et Analyse**131-132**pp. 287-293.**Vu, D. [2010].**Symmetric Sets and Graph Models of Set and Multiset Theories.

Ph.D Thesis, University of Cambridge**Vu, D. and Zachiri McKenzie**Permutation methods yielding models of the stratified axioms of Zermelo Fraenkel set theory

NF 70th anniversary volume, Cahiers du Centre de Logique,**16**2009.

**Wang, H. [1950]**A formal system of logic.

Journal of Symbolic Logic**15**, pp. 25-32.**Wang, H. [1952]**Negative types.

MIND**61**, pp. 366-368.**Wang, H. [1953]**The categoricity question of certain grand logics.

Mathematische Zeitschrift**59**, pp. 47-56.This paper vanished from view almost immediately because it concerns the system NF + AC, which within months of this paper appearing was shown to be inconsistent. It may be that something useful can be saved of the ideas in it, but the matter seems never to have been investigated — tf.

**Weydert, E. [1989]**How to approximate the naïve comprehension scheme inside of classical logic.

Ph.D. thesis, Friedrich-Wilhelms-Universität Bonn.

Bonner mathematische Schriften**194**.**Whitehead, A.N. and Russell, B.A.W. [1910]**Principia Mathematica. Cambridge University Press.**Yasuhara, Mitsuru. [1984]**A finite axiomatisation of New Foundations within four types

Unpublished**Yasuhara, Mitsuru. [1984]**A consistency proof of Quine's New Foundations

Unpublished