I'd say most ideas are "supposed", as were "the Heavens" and as I pointed out, Bacon who introduced the word to English (and read and wrote Latin fluently)certainly considered it as containing the imaginary as well as the real. "All things seen and unseen" certainly includes the purely imaginary. Are not Father Christmas - and the gods - not unseen items? Are you arguing that they are not in the universe? The "world" includes the "world of idea" and you recall that I included everything that is and that will exist into my definition. That was precisely in order to meet the requirement which is stipulated in the Funk and Wagnalls definition that it is everything "existing" (including existing ideas). By stipulating a set of {things which will exist} we also include future ideas. Or do you believe that ideas don't exist, or come from some shadowy alternate Universe? Stop fighting it. It is a very useful concept. Study some set theory. Most implementation's require a universal set (This gets very complicated very quickly, so I am leaving it here for now).
Logics are a branch of philosophy. Yet the idea of the all inclusive universe is by no means restricted to logics. And we were talking metaphysics - for a while there anyway :-) Still are for that matter. Even if we are (temporarily I'm sure) stuck in a cosmological, ontological and epistemological loop.
I thought we had concluded that a "statement of truth" in search of a context was usually a "non-statement of truth"?
Yes. 5.43645645 + 2.1229098 = 7.55936625 is true (in base 10 and assuming "+" means addition and "=" implies equality) and useful as it solves an addition problem, speaks to the nature of numbers and of the symbols we use to manipulate them. It also proves that you can add, and I can check your results. Bingo! We are communicating meaningfully. All that from one little sum :-)
"Saints fly only in the eyes of their disciples." -- Hindu proverb
Awesome. I will be quoting it. Do you have a reference? This is also a truth. And knowing at least one Hindu "saint" too many (self proclaimed and more wannabe than actual IMO), and far too many of his disciples, I can attest to the belief if not the practice :-) Is it useful? Certainly. It speaks to many levels and by analogy tells us something very fundamental about people who are prepared to be disciples as well as about the so called "saints".
Wittgenstein's opinion, although only published posthumously, is so convincing that most philosophers (and all scientists I know), use the "test of utility" to determine whether a meaning can be ascribed to a statement. Apropos of something, it applies to statements like "the snark was a boojum" - if it is not meaningful, it cannot be useful, and thus is neither true nor false. Just meaningless.
Your examples re the circle are circular. They are based on sin and cos
(which are based on the relationship of the radius and circumference of a
circle which is defined by space-time, and thus they can be redefined in
terms other than the rations of circles. For example I can redefine them in
terms of e, seperately defined by the nature of numbers in our space time)
or upon the value of PI which is also based on a circle. To test your
descriptions, I do not need a meta language* at all. I can use simple
algebra and trigonometry (which you introduced and which are an intrinsic
component of mathematical language) to prove that your formulae simplify to
a circle (and by the way, in the first example, 0<=theta<=2*Pi is wrong. You
would leave out the point at 0/360 degrees thus you have not defined a
circle.). The only reason for not doing that here is the difficulty of
drawing a diagram in ASCII email. Barely doable, and a preposterous waste of
energy.
Add the requirement for threading, full XTML, complete character sets and dynamic discussions with shared whiteboards (preferably able to show who did what) to your email document requirement.
Your assertion that I am implying a Platonic ideal, does not convert what I have said into even hoping that there might be one. A platonic ideal would suppose the idea of a perfect circle, where even the definition of a "perfect circle" would only be a shadow of that perfection. /me shudders and unimagines Plato. And claiming that the "idea" or "supposition" or "unseen" or "invisible" or "definitional" set of class {circle} does not exist, does not remove it from the universe. You see, I think you are deliberately looking away from the point.
A circle (as defined) can only exist in this "real" universe because of the
shape of space-time.
If our space-time were different, the circle you draw would be different.
And even were it is possible to create a "circle", it would have different
attributes.
So it is space-time that defines a circle.
As such the circle - even the idea of a circle is defined by space-time.
Space-time exists only as an attribute of a universe.
Let me put as a syllogism:
Nothing which is not in a universe can interact with that universe
(definitional antecedent).
The definition of a circle is determined by the space-time which is an
attribute of that universe (logical antecedent).
The definition of a circle is in a universe (consequent).
Unless you can "break" one of the above points, then the final line of that syllogism must stand as the logical consequence.
Hermit
P.S. A "metalanguage" is nothing special, it is simply a language about language. And may be a formal statement of the language in the language itself. For example, the metalanguage for EBNF is written in EBNF. (extended bauer normal form). In the same way a meta-set-theory can be built which excludes the universal set (or more correctly, makes set-theory the universe set but makes it inapplicable for any application but the discussion of set-theory - told you it gets complex)) and deals with set-theory about set-theory. Nothing wrong with that.
P.P.S. Anyone studying set theory should also study symbolic logic. It is a good idea to study symbolic logic first. Irving Copi, Symbolic Logic, fifth edition, Macmillan, 1979 is a very recommended starting point. I still use Richmond H. Thomason Symbolic Logic An Introduction, 1st edition, Macmillan, 1970 with people I am introducing to Sybolic Logic, Lamda Calculus and Set Theory.
Set Theory Bibliography (Thomas Forster's master bibliography for the entire subject, as updated by Paul West):
For those unfamiliar with the field, two places to start are the New Foundations Home Page (http://math.idbsu.edu/~holmes/holmes/nf.html) and Thomas Forster's book "Set Theory with a Universal Set". A new option is afforded by the recent appearance of Holmes's elementary text.
Recent Work
Holmes, M. R. [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.
Forster, Thomas [1997]
Quine's NF, 60 years on.
American Mathematical Monthly, vol. 104, no. 9 (November), pp. 838-845.
Esser, O. [1996]
Inconsistency of GPK + AFA.
Mathematical Logic Quarterly 42, pp. 104-108.
Dziergowski, D. [1995]
Models of intuitionistic TT and NF.
Journal of Symbolic Logic 60, pp. 640-653.
Holmes, M.R. [1995a]
The equivalence of NF-style set theories with "tangled" type theories; the
construction of omega-models of predicative NF (and more).
Journal of Symbolic Logic 60, pp. 178-189.
Holmes, M.R. [1995b]
Untyped lambda-calculus with relative typing.
Typed Lambda-Calculi and Applications (Proceedings of TLCA '95), Springer,
pp. 235-248.
Jech, T. [1995]
OTTER experiments in a system of combinatory logic
Journal of Automated Reasoning, 14, pp. 413-426.
Comprehensive Bibliography
Arruda, A. [1970a]
Sur les systèmes NFi de Da Costa.
Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris
(série A) 270, pp. 1081-1084.
Arruda, A. [1970b]
Sur les systèmes NF-omega.
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-omega.
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.
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. and Casalegno, P. [1985]
The consistency of some 4-stratified subsystems of NF including NF3.
Journal of Symbolic Logic 50, pp. 407-411.
Boffa, M. and Crabbé, M. [1975]
Les théorèmes 3-stratifiés de NF3.
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.
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.
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, vol 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, vol. 13: 179-201.
Cocchiarella, N.B. [1992b]
Conceptual realism versus Quine on classes and higher-order logic,
Synthese, vol. 90: 379-436.
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.
Crabbé, M. [1975]
Types ambigus.
Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris
(série A) 280, pp. 1-2.
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 Mathematicae 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^alpha.
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. [1992]
On NFU.
Notre Dame Journal of Formal Logic 33, pp 112-119.
Crabbé, M. [1994]
The Hauptsatz for stratified comprehension: a semantic proof.
Mathematical Logic Quarterly 40, pp, 481-489.
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 Ci, Ci*, Ci=, Di 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. [1965b]
On two systems of set theory.
Proc. Koningl. Nederl. Ak. v. Wetens. (serie A) 68, pp 95-99.
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. [1974]
Remarques sur les Calculs Cn, Cn*, Cn=, et Dn.
Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris
(série A) 278, pp. 818-821.
Dziergowski, D. [1991]
Intuitionistic typical ambiguity.
Archive for Mathematical Logic 31, pp. 171-182.
Dziergowski, D. [1993a]
Typical ambiguity and elementary equivalence.
Mathematical Logic Quarterly 39, pp. 436-446.
Dziergowski, D. [1993b]
Le théorème d'ambiguïté et son extension à la logique intuitionniste.
Dissertation doctorale. Université catholique de Louvain, Institut de
mathématique pure et appliquée.
Dziergowski, D. [1995]
Models of intuitionistic TT and NF.
Journal of Symbolic Logic 60, pp. 640-653.
Engeler, E. and Röhrli, H. [1969]
On the problem of foundations of category theory.
Dialectica 23, pp. 58-66.
Esser, O. [1996]
Inconsistency of GPK + AFA.
Mathematical Logic Quarterly 42, pp. 104-108.
Feferman, S. [1972]
Some formal systems for the unlimited theory of structures and categories.
Unpublished.
Forster, T.E. [1976]
N.F.
Ph.D. thesis, University of Cambridge.
Forster, T.E. [1982]
Axiomatising set theory with a universal set.
Cahiers du Centre de Logique (Louvain-la-neuve) 4, pp. 61-76.
Forster, T.E. [1983a]
Quine's New Foundations, an introduction.
Cahiers du Centre de Logique (Louvain-la-neuve) 5. 100 pp.
Forster, T.E. [1983b]
Further consistency and independence results in NF obtained by the
permutation method.
Journal of Symbolic Logic 48, pp. 236-238.
Forster, T.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 "Phi-hat" in this paper is
faulty.)
Forster, T.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, T.E. [1987b]
Term models for weak set theories with a universal set.
Journal of Symbolic Logic 52, pp. 374-387.
Forster, T.E. [1989]
A second-order theory without a (second-order) model.
Zeitschrift für mathematische Logik und Grundlagen der Mathematik 35, pp.
285-286.
Forster, T.E. [1992a]
On a problem of Dzierzgowski.
Bulletin de la Société Mathématique de Belgique (série B) 44, pp. 207-214.
Forster, T.E. [1992b]
Set Theory with a Universal Set.
Cambridge University Press.
Forster, T.E. [1993]
A semantic characterisation of the well-typed formulae of lambda-calculus.
Theoretical Computer Science 110, pp 405-408.
Forster, T.E. [1995]
Set Theory with a Universal Set, second edition.
Cambridge University Press.
Forster, T.E. and Kaye, R. [1991]
End-extensions preserving power set.
Journal of Symbolic Logic 56, pp. 323-328.
(Errata in Forster [1992b], p. 139; repeated in Forster [1995], p. 152.)
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, 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.
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.
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, R. [1972]
Sur les modèles de NF.
Comptes Rendus hebdomadaires des séances de l'Académie des Sciences de Paris
(série A) 275, p. 567.
Hinnion, R. [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, R. [1975]
Sur la théorie des ensembles de Quine.
Ph.D. thesis, ULB Brussels.
Hinnion, R. [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.
Hinnion, R. [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, R. [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, R. [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, R. [1982]
NF et l'axiome d'universalité.
Cahiers du Centre de Logique (Louvain-la-neuve) 4, pp. 45-59.
Hinnion, R. [1986]
Extensionality in Zermelo-Fraenkel set theory.
Zeitschrift für mathematische Logik und Grundlagen der Mathematik 32, pp.
51-60.
Hinnion, R. [1989]
Embedding properties and anti-foundation in set theory.
Zeitschrift für mathematische Logik und Grundlagen der Mathematik 35, pp.
63-70.
Hinnion, R. [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.
Holmes, M.R. [1991a]
Systems of combinatory logic related to Quine's 'New Foundations.'
Annals of Pure and Applied Logic 53, pp. 103-133.
Holmes, M.R. [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, M.R. [1992]
Modelling fragments of Quine's 'New Foundations.'
Cahiers du Centre de Logique (Louvain-la-Neuve) 7, pp. 97-112.
Holmes, M.R. [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, M.R. [1994]
The set theoretical program of Quine succeeded (but nobody noticed).
Modern Logic 4, pp. 1-47.
Holmes, M.R. [1995a]
The equivalence of NF-style set theories with "tangled" type theories; the
construction of omega-models of predicative NF (and more).
Journal of Symbolic Logic 60, pp. 178-189.
Holmes, M.R. [1995b]
Untyped lambda-calculus with relative typing.
Typed Lambda-Calculi and Applications (Proceedings of TLCA '95), Springer,
pp. 235-248.
Jamieson, M.W. [1994]
Set theory with a Universal Set.
Ph.D. thesis, University of Florida. 114pp.
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. [199?]
The quantifier complexity of NF.
To appear.
Kemeny, J.G. [1950]
Type theory vs. set theory (abstract).
Journal of Symbolic Logic 15, p. 78.
Kirmayer, G. [1981]
A refinement of Cantor's theorem.
Proceedings of the American Mathematical Society 83, p. 774.
Körner, F. [1994]
Cofinal indiscernibles and some applications to New Foundations.
Mathematical Logic Quarterly 40, pp. 347-356.
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.
Lake, J. [1974]
Some topics in set theory.
Ph.D. thesis, Bedford College, London University.
Lake, J. [1975]
Comparing type theory and set theory.
Zeitschrift für mathematische Logik und Grundlagen der Mathematik 21, pp.
355-356.
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.
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.
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