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:
"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—see below).
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.
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.
Peter Aczel
[1988]
Non-Well-Founded Sets CSLI
This elegant booklet is included here principally for
sentimental reasons: it got put in and nobody wants to take it out.
It's not really got a great deal to do with Set Theory with a
Universal set.
Tin Adlešić and Vedran Čačić
A Modern Rigorous Approach to Stratification in NF/NFU
Logica Universalis 16(3), 451–468 (2022) https://doi.org/10.1007/
s11787-022-00310-y
Adlešić, T., Čačić, V.:
The
Cardinal Squaring Principle and an Alternative Axiomatization of NFU.
Bulletin of the Section of Logic 52(4), 551–581 (2023)
https://doi.org/10.18778/0138-0680.2023.25
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, 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.
Generally 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.
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}.
Beeson, Michael [2021]
The Church numbers in NF set theory
at
https://arxiv.org/abs/2108.09270
Beeson, Michael [2021]
Intuitionistic NF Set Theory
at https://arxiv.org/abs/2108.09270
These two offerings of Beeson are timely, readable and useful. There is a massive overlap with Forster's Tutorial on Constructive NF but Beeson has verified everything in them in LEAN.
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.
There is an English translation by Forster appended to the end of
Pabion's article below
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
(2009) pp 17—26.
Bowler, Nathan and Thomas Forster [2024]
Internal Automorphisms and Antimorphisms of
Models of NF
Journal of
Symbolic Logic
89 (2024) to appear
Nathan J. Bowler, Zuhair Al-Johar, M. Randall Holmes:
The Axiom Scheme of Acyclic Comprehension. Notre Dame J. Formal Log. 55
(1): 11-24 (2014)
Button,
Tim
Boolean Level Theory
This article is struggling through a submission process,
but we are confident that the result it claims is
correct, and that 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'', ``the
surjective image of a wellfounded set is a set'' 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.
Readers whose interests lie in Set Theory 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; Princeton have kindly supplied the editors with photocopies which we are gradually turning into an annotated pdf file. The files are not linked here because of copyright concerns. If there are Church scholars who want to see the stash of ruminations and our commentaries we would of course have to clear it with Princeton. Naturally they could always approach Princeton on their own account.
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.
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
Marcel says "Though not yet published, the [above] are
connected with stratification and positive stuff"
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.
Generally the editors have a low opinion of these contributions of Da Costa and we list them here solely because of our policy of `if in doubt leave it in''
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.
Dowek, Gilles. [2001]
Stratified Foundations as a theory modulo.
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.
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
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.
Esser,
Olivier E. and Forster, Thomas E. [2007]
Relaxing Stratification
Bull.
Belg. Math. Soc. Simon Stevin, vol 14, (2007), pp. 247-258.
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
Firestone, Clifford Dixon. [1947]
"Sufficient Conditions for the modelling of Axiomatic Set Theory"
Ph.D. thesis, Cornell University.
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 Føllesdal, 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/~tef10/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. [2023]
The Burali-Forti paradox..
The Reasoner 17 sept 2023, pp. 40-41.
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.
Thomas Forster and Nathan Bowler [2024]
Internal Automorphisms and Antimorphisms of
Models of NF
Journal of
Symbolic Logic
74 (2024) to appear
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 Randall Holmes [2024/5]
Synonymy Questions concerning the Quine
Systems.
Journal of Symbolic Logic to appear
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 52} (2011) pp 1—20.
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. [2012]
``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.
Forster, Thomas E, Adam Lewicki and Alice Vidrine
Category Theory with Stratified Set Theory
This article was accepted by the JSL subject to certain modifications. However the modifications never got executed — for reasons that are not entirely clear. This is a pity beco's it is actually a very nice article (if i say so myself — tf). However a pdf is available on arxiv:
Subjects: Category Theory (math.CT); Logic (math.LO)
Cite as: arXiv:1911.04704 [math.CT]
(or arXiv:1911.04704v2 [math.CT] for this version)
https://doi.org/10.48550/arXiv.1911.04704
Submission history
From: Alice Vidrine [view email]
[v1] Tue, 12 Nov 2019 07:28:29 UTC (27 KB)
[v2] Wed, 13 Nov 2019 22:45:41 UTC (27 KB)
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.
Gabbay, Murdoch Jamie
The language of Stratified Sets is confluent and strongly normalising
lmcs:3681 - Logical Methods in Computer Science, May 23, 2018, 14, Issue 2.
Jamie has from time to time claimed a consistency proof for NF based on his work on nominal sets. However at the time of writing this is the only item that he wants us to cite.
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.
Gorbow, Paul C [1986]
Self-similarity in the foundations,
Ph.D. Thesis
Acta Philosophica Gothoburgensia 32
Gorbow, Paul C [1986]
Algebraic New Foundations,
Journal of Symbolic Logic 84 Number 2, June 2019
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 many!) finite axiomatisations of NF
Hatcher, W.S. [1968]
The Foundations of Mathematics (1968) W. B. Saunders 327pp. ISBN-10 : 0721645658 ISBN-13 : 978-0721645650
Chapter 7 is a nice introduction to the Quine systems, and gives a picture of the Mathematical community's take on them in the late 60's. [I still have to verify that it doesn't contain any original work]
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.
Adam Lewicki, Forster, Thomas E, and Alice Vidrine
Category Theory with Stratified Set Theory
This article was accepted by the JSL subject to certain modifications. However the modifications never got executed — for reasons that are not entirely clear. This is a pity beco's it is actually a very nice article (if i say so myself — tf). However a pdf is available on arxiv:
Subjects: Category Theory (math.CT); Logic (math.LO)
Cite as: arXiv:1911.04704 [math.CT]
(or arXiv:1911.04704v2 [math.CT] for this version)
https://doi.org/10.48550/arXiv.1911.04704
Submission history
From: Alice Vidrine [view email]
[v1] Tue, 12 Nov 2019 07:28:29 UTC (27 KB)
[v2] Wed, 13 Nov 2019 22:45:41 UTC (27 KB)
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 Holmes' 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
[2001]
Foundations of mathematics in polymorphic
type theory.
Topoi, 20, pp. 29-52.
Holmes writes: 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]
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.
M. Randall Holmes, Zuhair Al-Johar:
Acyclic Comprehension is equal to Stratified Comprehension. CoRR abs/2010.14949 (2020)
M. Randall Holmes, Zuhair Al-Johar, Nathan J. Bowler:
The Axiom Scheme of Acyclic Comprehension. Notre Dame J. Formal Log. 55(1): 11-24 (2014)
M Randall Holmes, Forster, Thomas
E. [2024/5]
Synonymy Questions concerning the Quine
Systems.
Journal of Symbolic Logic to appear
M. Randall Holmes,
Forster, Thomas E. and
[2024/5]
Permutation methods in NF
and NFU” (with Randall Holmes) in the NF 70th
anniversary volume Cahiers du Centre de
Logique 16 2009 pp 33–76..
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
Ph.D. thesis, Princeton.
This version has an introduction by Allen Hazen
Kemeny, J.G. [1950]
Type
theory vs. set theory Abstract of the above.
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—9, 1989.
Kisielewicz, Andrzej
[1998]
A very strong set theory?
Studia Logica
61 171—178, 1998.
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. [1994]
Stratified Languages and Permutation Models
Ph.D. thesis Techische Universität Berlin.
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.
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
(i) towards
the end of it he considers the possibility of having an inhomogeneous
ordered pair function in NF.
See section 10.3 pp 98ff. This discussion bears reading, even fifty years later.
He also
(ii) shows the consistency of some fragments of NF obtained by
dropping one or other of Hailperin's axioms. Section 10.4 thm 10.8 shows
the consistency of the fragment that drops P6.
We have not secured Lake's permission to link his thesis: no-one seems to know how to get hold of him!
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.
Nuno Maia and Matteo Nizzardo [2024]
Identity and Extensionality in
Boffa Set Theory
Philosophia Mathematica?? (1-3):115-123 (2024)
Mathias, A.R.D. [2001]
The strength of Mac Lane Set theory
Annals of Pure and Applied Logic 110 (1-3):107-234 (2001)
This is an important article. It doesn't contain much explicitly about NF but contains a wealth of relevant useful background. Essential reading.
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.
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}.
Newman, M. H. A. (``Max") [1943]
Stratified Systems of Logic,
Mathematical Proceedings of the Cambridge Philosophical Society 39 pp 69—83
One has the impression that this article fell dead-born from the press. It may well be worth investigating. The version linked here has appended to it a review by Church. It appears to be from the JSL Reviews section, but I cannot discover a date and I am not going to cite it separately.
R. Matteo Nizzardo and Nuno Maia
[2024]
Identity and
Extensionality in Boffa Set Theory
Philosophia Mathematica?? (1-3):115-123 (2024)
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. 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 also (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 thesis). 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 is one of the fundamental papers in NF studies, and also a wonderful example of a proof in set theory using truth-definitions. Every NFiste should read it.
Oswald, U. [1974]
Sur
les modeles d'un fragment de NF
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.
English translation by Thomas Forster
here
Appended to this translation is a translation of the Boffa-Crabbé paper above
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.
Pireva, Diamant [2023]
Ambiguity in Typed Set Theory and the Universal-Existential Conjecture
M.Phil thesis Victoria University of Wellington 2023
Prati, N. [1994]
A partial model of NF with E.
Journal of Symbolic Logic 59, pp.
1245-1253.
Prati, N. [2023]
The Consistency of NF.
submitted to the Journal of Symbolic Logic
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.)
Resnik, M.D. [1993]
A Set Theoretic approach to the Simple Theory of Types.
Theoria Dec 1969
This is of interest to NFistes because it is an early treatment of the idea that TST can be expressed in a one-sorted language, the language of set theory.
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
Rouvelas, P.
Strong ambiguity
MLQ Math. Log. Q. 68 (2022), no. 1, 110–117 link
Rouvelas. P.
Cantorian models of Predicative NF.
J. Symb. Log 87 2022
link
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.
Ryan-Smith, Calliope. [2023]
Stratifiable Formulae are not Context-Free
This is a write-up of a summer project that the author did for Forster. It does what it says on the tin.
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.
This is the fundamental article for permutation methods applied to NF
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.
Sheridan, Flash [2024]
``Closer Look at the Russell Paradox”
has been accepted by Logique et Analyse.
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. There is a computer-verified version of this proof on https://us.metamath.org/nfeuni/nchoice.html
Specker, E.P. [1958]
Dualität.
Dialectica 12, pp. 451-465.
There is an annotated
English translation by Forster of this important and elegant
article in Føllesdal, 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.
Tzouvaras, Athanassios
Combinatorics
related to NF consistency
Presented at the NF 70th
anniversary meeting, Cambridge 2007
Tzouvaras, Athanassios
A
reduction of the NF consistency problem
Presented at
the NF 70th anniversary meeting, Cambridge 2007 Also in JSL 72 (2007), pp 285-304.
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.
Alice Vidrine, Forster, Thomas E, Adam Lewicki
Category Theory with Stratified Set Theory
This article was accepted by the JSL subject to certain modifications. However the modifications never got executed — for reasons that are not entirely clear. This is a pity beco's it is actually a very nice article (if i say so myself — tf). However a pdf is available on arxiv:
Subjects: Category Theory (math.CT); Logic (math.LO)
Cite as: arXiv:1911.04704 [math.CT]
(or arXiv:1911.04704v2 [math.CT] for this version)
https://doi.org/10.48550/arXiv.1911.04704
Submission history
From: Alice Vidrine [view email]
[v1] Tue, 12 Nov 2019 07:28:29 UTC (27 KB)
[v2] Wed, 13 Nov 2019 22:45:41 UTC (27 KB)
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
The managers have electronic copy of these two works of Yasuhara, but we have not secured his permission to link them.
Zuhair Al-Johar, M. Randall Holmes:
Acyclic Comprehension is equal to Stratified Comprehension. CoRR abs/2010.14949 (2020)
Zuhair Al-Johar, M. Randall Holmes, Nathan J. Bowler:
The Axiom Scheme of Acyclic Comprehension. Notre Dame J. Formal Log. 55(1): 11-24 (2014)