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  1. Roy T. Cook (2003). Review of J. Mayberry, The Foundations of Mathematics in the Theory of Sets. [REVIEW] British Journal for the Philosophy of Science 54 (2):347-352.
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  2. Howard Whitley Eves (1965). An Introduction to the Foundations and Fundamental Concepts of Mathematics. New York, Holt, Rinehart and Winston.
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  3. Laurence Goldstein (2004). The Barber, Russell's Paradox, Catch-22, God, Contradiction, and More. In Graham Priest, J. C. Beall & Bradley Armour-Garb (eds.), The Law of Non-Contradiction. Clarendon Press
  4. Geoffrey Hellman (2006). Pluralism and the Foundations of Mathematics. In ¸ Itekellersetal:Sp. 65--79.
    A plurality of approaches to foundational aspects of mathematics is a fact of life. Two loci of this are discussed here, the classicism/constructivism controversy over standards of proof, and the plurality of universes of discourse for mathematics arising in set theory and in category theory, whose problematic relationship is discussed. The first case illustrates the hypothesis that a sufficiently rich subject matter may require a multiplicity of approaches. The second case, while in some respects special to mathematics, raises issues of (...)
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  5. Geoffrey Hellman (2006). What is Categorical Structuralism? In Johan van Benthem, Gerhard Heinzman, M. Rebushi & H. Visser (eds.), The Age of Alternative Logics. Springer 151--161.
  6. Ignasi Jané (2010). Idealist and Realist Elements in Cantor's Approach to Set Theory. Philosophia Mathematica 18 (2):193-226.
    There is an apparent tension between the open-ended aspect of the ordinal sequence and the assumption that the set-theoretical universe is fully determinate. This tension is already present in Cantor, who stressed the incompletable character of the transfinite number sequence in Grundlagen and avowed the definiteness of the totality of sets and numbers in subsequent philosophical publications and in correspondence. The tension is particularly discernible in his late distinction between sets and inconsistent multiplicities. I discuss Cantor’s contrasting views, and I (...)
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  7. I. Jane (2010). Idealist and Realist Elements in Cantor's Approach to Set Theory. Philosophia Mathematica 18 (2):193-226.
    There is an apparent tension between the open-ended aspect of the ordinal sequence and the assumption that the set-theoretical universe is fully determinate. This tension is already present in Cantor, who stressed the incompletable character of the transfinite number sequence in Grundlagen and avowed the definiteness of the totality of sets and numbers in subsequent philosophical publications and in correspondence. The tension is particularly discernible in his late distinction between sets and inconsistent multiplicities. I discuss Cantor’s contrasting views, and I (...)
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  8. Ignagio Jane (2001). Reflections on Skolem's Relativity of Set-Theoretical Concepts. Philosophia Mathematica 9 (2):129-153.
    In this paper an attempt is made to present Skolem's argument, for the relativity of some set-theoretical notions as a sensible one. Skolem's critique of set theory is seen as part of a larger argument to the effect that no conclusive evidence has been given for the existence of uncountable sets. Some replies to Skolem are discussed and are shown not to affect Skolem's position, since they all presuppose the existence of uncountable sets. The paper ends with an assessment of (...)
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  9. Kevin C. Klement (2014). Early Russell on Types and Plurals. Journal for the History of Analytical Philosophy 2 (6):1-21.
    In 1903, in The Principles of Mathematics (PoM), Russell endorsed an account of classes whereupon a class fundamentally is to be considered many things, and not one, and used this thesis to explicate his first version of a theory of types, adding that it formed the logical justification for the grammatical distinction between singular and plural. The view, however, was short-lived; rejected before PoM even appeared in print. However, aside from mentions of a few misgivings, there is little evidence about (...)
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  10. Enrique V. Kortright (1994). Philosophy, Mathematics, Science and Computation. Topoi 13 (1):51-60.
    Attempts to lay a foundation for the sciences based on modern mathematics are questioned. In particular, it is not clear that computer science should be based on set-theoretic mathematics. Set-theoretic mathematics has difficulties with its own foundations, making it reasonable to explore alternative foundations for the sciences. The role of computation within an alternative framework may prove to be of great potential in establishing a direction for the new field of computer science.Whitehead''s theory of reality is re-examined as a foundation (...)
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  11. G. Landini (2006). The Ins and Outs of Frege's Way Out. Philosophia Mathematica 14 (1):1-25.
    Confronted with Russell's Paradox, Frege wrote an appendix to volume II of his Grundgesetze der Arithmetik. In it he offered a revision to Basic Law V, and proclaimed with confidence that the major theorems for arithmetic are recoverable. This paper shows that Frege's revised system has been seriously undermined by interpretations that transcribe his system into a predicate logic that is inattentive to important details of his concept-script. By examining the revised system as a concept-script, we see how Frege imagined (...)
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  12. Øystein Linnebo (2013). The Potential Hierarchy of Sets. Review of Symbolic Logic 6 (2):205-228.
    Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.
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  13. Juan José Luetich (2012). I Think, Therefore I Exist; I Belong, Therefore I Am. Transactions of The Luventicus Academy (3):1-4.
    The actions of perceiving and grouping are the two that the human being carries out when thinking in entities different from himself. In this article “The Mirror Problem” and “The Peer Problem”, which correspond respectively to self-perception and the perception of others, are studied. By solving these two problems, the thinker arrives to the following conclusions: “I exist” and “I am”.
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  14. J. P. Mayberry (2000). The Foundations of Mathematics in the Theory of Sets. Cambridge University Press.
    This book will appeal to mathematicians and philosophers interested in the foundations of mathematics.
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  15. John Mayberry (1994). What is Required of a Foundation for Mathematics? Philosophia Mathematica 2 (1):16-35.
    The business of mathematics is definition and proof, and its foundations comprise the principles which govern them. Modern mathematics is founded upon set theory. In particular, both the axiomatic method and mathematical logic belong, by their very natures, to the theory of sets. Accordingly, foundational set theory is not, and cannot logically be, an axiomatic theory. Failure to grasp this point leads obly to confusion. The idea of a set is that of an extensional plurality, limited and definite in size, (...)
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  16. John Mayberry (1977). The Consistency Problem for Set Theory: An Essay on the Cantorian Foundations of Mathematics (II). British Journal for the Philosophy of Science 28 (2):137-170.
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  17. Stephen Pollard (1996). Sets, Wholes, and Limited Pluralitiest. Philosophia Mathematica 4 (1):42-58.
    This essay defends the following two claims: (1) liraitation-of-size reasoning yields enough sets to meet the needs of most mathematicians; (2) set formation and mereological fusion share enough logical features to justify placing both in the genus composition (even when the components of a set are taken to be its members rather than its subsets).
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  18. Frank Plumpton Ramsey (1960). The Foundations of Mathematics and Other Logical Essays. Paterson, N.J.,Littlefield, Adams.
    THE FOUNDATIONS OF MATHEMATICS () PREFACE The object of this paper is to give a satisfactory account of the Foundations of Mathematics in accordance with..
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  19. Frank Plumpton Ramsey & D. H. Mellor (eds.) (1978). Foundations: Essays in Philosophy, Logic, Mathematics, and Economics. Humanties Press; Routledge.
  20. Michael D. Resnik (1999). Review of G. Boolos, Logic, Logic, and Logic. Philosophia Mathematica 7 (3):328-335.
Russell's Paradox
  1. Andrew Bacon (2013). Curry's Paradox and Omega Inconsistency. Studia Logica 101 (1):1-9.
    In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this paper I show that a number of logics are susceptible to a strengthened version of Curry's paradox. This can be adapted to provide a proof theoretic analysis of the omega-inconsistency in Lukasiewicz's continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of Lukasiewicz logic which individually, but (...)
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  2. Andrew Bacon (2013). Paradoxes of Logical Equivalence and Identity. Topoi (1):1-10.
    In this paper a principle of substitutivity of logical equivalents salve veritate and a version of Leibniz’s law are formulated and each is shown to cause problems when combined with naive truth theories.
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  3. Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.) (2013). Paraconsistency: Logic and Applications. Springer.
    A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change (...)
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  4. Andrew Boucher, A Comprehensive Solution to the Paradoxes.
    A solution to the paradoxes has two sides: the philosophical and the technical. The paradoxes are, first and foremost, a philosophical problem. A philosophical solution must pinpoint the exact step where the reasoning that leads to contradiction is fallacious, and then explain why it is so.
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  5. Manuel Bremer, Frege's Basic Law V and Cantor's Theorem.
    The following essay reconsiders the ontological and logical issues around Frege’s Basic Law (V). If focuses less on Russell’s Paradox, as most treatments of Frege’s Grundgesetze der Arithmetik (GGA)1 do, but rather on the relation between Frege’s Basic Law (V) and Cantor’s Theorem (CT). So for the most part the inconsistency of Naïve Comprehension (in the context of standard Second Order Logic) will not concern us, but rather the ontological issues central to the conflict between (BLV) and (CT). These ontological (...)
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  6. Manuel Bremer (2010). Universality in Set Theories. Ontos.
    The book discusses the fate of universality and a universal set in several set theories.
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  7. Arthur W. Burks & Irving M. Copi (1950). Lewis Carroll's Barber Shop Paradox. Mind 59 (234):219-222.
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  8. Marcoen J. T. F. Cabbolet (2015). The Importance of Developing a Foundation for Naive Category Theory. Thought: A Journal of Philosophy 4 (4):237-242.
    Recently Feferman has outlined a program for the development of a foundation for naive category theory. While Ernst has shown that the resulting axiomatic system is still inconsistent, the purpose of this note is to show that nevertheless some foundation has to be developed before naive category theory can replace axiomatic set theory as a foundational theory for mathematics. It is argued that in naive category theory currently a ‘cookbook recipe’ is used for constructing categories, and it is explicitly shown (...)
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  9. David Gray Carlson, Russell's Paradox and Legal Positivism.
    In 1902, Bertrand Russell overturned set theory, which aspired to reduce all sets to their rules of recognition. These rules were to have logical priority to empirical sets posited by empirical human beings. As a result of Russell's Paradox, set theory gave up the hope of theorizing sets. This paper claims Russell's Paradox can be applied directly to jurisprudence. The result is that legal positivism (carefully defined as the claim that law can be reduced to rules of recognition) is invalid (...)
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  10. Hector-Neri Castañeda (1976). Ontology and Grammar: I. Russell's Paradox and the General Theory of Properties in Natural Language. Theoria 42 (1-3):44-92.
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  11. Nino Cocchiarella (2000). Russell's Paradox of the Totality of Propositions. Nordic Journal of Philosophical Logic 5 (1):25-37.
    Russell's "new contradiction" about "the totality of propositions" has been connected with a number of modal paradoxes. M. Oksanen has recently shown how these modal paradoxes are resolved in the set theory NFU. Russell's paradox of the totality of propositions was left unexplained, however. We reconstruct Russell's argument and explain how it is resolved in two intensional logics that are equiconsistent with NFU. We also show how different notions of possible worlds are represented in these intensional logics.
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  12. John Corcoran (1983). 1983 Review in Mathematical Reviews 83e:03005 Of: Cocchiarella, Nino “The Development of the Theory of Logical Types and the Notion of a Logical Subject in Russell's Early Philosophy: Bertrand Russell's Early Philosophy, Part I”. Synthese 45 (1980), No. 1, 71-115. MATHEMATICAL REVIEWS 83:03005.
    CORCORAN RECOMMENDS COCCHIARELLA ON TYPE THEORY. The 1983 review in Mathematical Reviews 83e:03005 of: Cocchiarella, Nino “The development of the theory of logical types and the notion of a logical subject in Russell's early philosophy: Bertrand Russell's early philosophy, Part I”. Synthese 45 (1980), no. 1, 71-115 .
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  13. John Corcoran (1973). Book Review:Philosophy of Logic Hilary Putnam. [REVIEW] Philosophy of Science 40 (1):131-.
    Putnam, Hilary FPhilosophy of logic. Harper Essays in Philosophy. Harper Torchbooks, No. TB 1544. Harper & Row, Publishers, New York-London, 1971. v+76 pp. The author of this book has made highly regarded contributions to mathematics, to philosophy of logic and to philosophy of science, and in this book he brings his ideas in these three areas to bear on the traditional philosophic problem of materialism versus (objective) idealism. The book assumes that contemporary science (mathematical and physical) is largely correct as (...)
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  14. J. N. Crossley (1973). A Note on Cantor's Theorem and Russell's Paradox. Australasian Journal of Philosophy 51 (1):70 – 71.
    It is claimed that cantor had the technical apparatus available to derive russell's paradox some ten years before russell's discovery.
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  15. Arnold Cusmariu (1979). Russell's Paradox Re-Examined. Erkenntnis 14 (3):365-370.
    I attempt to rescue Frege's naive conception of a set according to which there is a set for every property by redefining the technical concept of degree of an open sentence. Instead of making degree a function of the number of free variables, I make it a function of free variable occurrences. What Russell proved, then, is that there is not a relation-in-extension for every relation-in-intension. In a brief paper it is not possible to discuss how redefining the function-argument correlation (...)
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  16. David Ellerman, On the Self-Predicative Universals of Category Theory.
    This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. The paper also shows how the always-self-predicative universals of category theory provide the "opposite bookend" to the never-self-predicative universals of iterative set theory and thus that the paradoxes arose from having (...)
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  17. Fernando Ferreira & Kai F. Wehmeier (2002). On the Consistency of the Δ11-CA Fragment of Frege's Grundgesetze. Journal of Philosophical Logic 31 (4):301-311.
    It is well known that Frege's system in the Grundgesetze der Arithmetik is formally inconsistent. Frege's instantiation rule for the second-order universal quantifier makes his system, except for minor differences, full (i.e., with unrestricted comprehension) second-order logic, augmented by an abstraction operator that abides to Frege's basic law V. A few years ago, Richard Heck proved the consistency of the fragment of Frege's theory obtained by restricting the comprehension schema to predicative formulae. He further conjectured that the more encompassing Δ₁¹-comprehension (...)
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  18. Harty Field (2004). The Consistency of the Naïve Theory of Properties. Philosophical Quarterly 54 (214):78 - 104.
    If properties are to play a useful role in semantics, it is hard to avoid assuming the naïve theory of properties: for any predicate Θ(x), there is a property such that an object o has it if and only if Θ(o). Yet this appears to lead to various paradoxes. I show that no paradoxes arise as long as the logic is weakened appropriately; the main difficulty is finding a semantics that can handle a conditional obeying reasonable laws without engendering paradox. (...)
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  19. Danny Frederick (forthcoming). The Unsatisfactoriness of Unsaturatedness. In Piotr Stalmaszczyk (ed.), Topics in Predication Theory. Volume 2: Philosophy of Language and Logic. Peter Lang GmbH
    Frege proposed his doctrine of unsaturatedness as a solution to the problems of the unity of the proposition and the unity of the sentence. I show that Frege’s theory is mystical, ad hoc, ineffective, paradoxical and entails that singular terms cannot be predicates. I explain the traditional solution to the problem of the unity of the sentence, as expounded by Mill, which invokes a syncategorematic sign of predication and the connotation and denotation of terms. I streamline this solution, bring it (...)
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  20. Harvey M. Friedman, From Russell's Paradox To.
    Russell’s way out of his paradox via the impredicative theory of types has roughly the same logical power as Zermelo set theory - which supplanted it as a far more flexible and workable axiomatic foundation for mathematics. We discuss some new formalisms that are conceptually close to Russell, yet simpler, and have the same logical power as higher set theory - as represented by the far more powerful Zermelo-Frankel set theory and beyond. END.
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  21. Michael Glanzberg (2003). Minimalism and Paradoxes. Synthese 135 (1):13 - 36.
    This paper argues against minimalism about truth. It does so by way of acomparison of the theory of truth with the theory of sets, and considerationof where paradoxes may arise in each. The paper proceeds by asking twoseemingly unrelated questions. First, what is the theory of truth about?Answering this question shows that minimalism bears important similaritiesto naive set theory. Second, why is there no strengthened version ofRussell's paradox, as there is a strengthened Liar paradox? Answering thisquestion shows that like naive (...)
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  22. Laurence Goldstein (2004). The Barber, Russell's Paradox, Catch-22, God, Contradiction and More: A Defence of a Wittgensteinian Conception of Contradiction. In Graham Priest, Jc Beall & Bradley Armour-Garb (eds.), The law of non-contradiction: new philosophical essays. Oxford University Press 295--313.
    outrageous remarks about contradictions. Perhaps the most striking remark he makes is that they are not false. This claim first appears in his early notebooks (Wittgenstein 1960, p.108). In the Tractatus, Wittgenstein argued that contradictions (like tautologies) are not statements (Sätze) and hence are not false (or true). This is a consequence of his theory that genuine statements are pictures.
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  23. Santos Gon?alo (forthcoming). Numbers and Everything. Philosophia Mathematica.
    I begin by drawing a parallel between the intuitionistic understanding of quantification over all natural numbers and the generality relativist understanding of quantification over absolutely everything. I then argue that adoption of an intuitionistic reading of relativism not only provides an immediate reply to the absolutist's charge of incoherence but it also throws a new light on the debates surrounding absolute generality.
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  24. K. Grelling & L. Nelson (1907). Bemerkungen Zu den Paradoxien von Russell Und Burali-Forti. Abhandlungen Der Fries'schen Schule (Neue Serie) 2:300-334.
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  25. Reinhardt Grossmann (1972). Russell's Paradox and Complex Properties. Noûs 6 (2):153-164.
    The author argues that the primary lesson of the so-Called logical and semantical paradoxes is that certain entities do not exist, Entities of which we mistakenly but firmly believe that they must exist. In particular, Russell's paradox teaches us that there is no such thing as the property which every property has if and only if it does not have itself. Why should anyone think that such a property must exist and, Hence, Conceive of russell's argument as a paradox rather (...)
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  26. Claire Hill (2000). Husserl, Frege and 'the Paradox'. Manuscrito 23 (2):101-132.
    In letters that Husserl and Frege exchanged during late 1906 and early 1907, when it is thought that Frege abandoned his attempts to solve Russell's paradox, Husserl expressed his views about the "paradox". Studied here are three deep-rooted differences between their approaches to pure logic present beneath the surface in these letters. These differences concern Husserl's ideas about avoiding paradoxical consequences by shunning three potentially para-dox producing practices. Specifically, he saw the need for: 1) correctly drawing the line between meaning (...)
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  27. F. Graf Hoensbroech (1939). On Russell's Paradox. Mind 48 (191):355-358.
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  28. Leon Horsten & Øystein Linnebo (2016). Term Models for Abstraction Principles. Journal of Philosophical Logic 45 (1):1-23.
    Kripke’s notion of groundedness plays a central role in many responses to the semantic paradoxes. Can the notion of groundedness be brought to bear on the paradoxes that arise in connection with abstraction principles? We explore a version of grounded abstraction whereby term models are built up in a ‘grounded’ manner. The results are mixed. Our method solves a problem concerning circularity and yields a ‘grounded’ model for the predicative theory based on Frege’s Basic Law V. However, the method is (...)
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  29. Carlo Ierna (2016). The Reception of Russell’s Paradox in Early Phenomenology and the School of Brentano: The Case of Husserl’s Manuscript A I 35α. In Guillermo E. Rosado Haddock (ed.), Husserl and Analytic Philosophy. De Gruyter 119-142.
  30. Luca Incurvati & Julien Murzi (forthcoming). Maximally Consistent Sets of Instances of Naive Comprehension. Mind.
    Paul Horwich (1990) once suggested restricting the T-Schema to the maximal consistent set of its instances. But Vann McGee (1992) proved that there are multiple incompatible such sets, none of which, given minimal assumptions, is recursively axiomatizable. The analogous view for set theory---that Naïve Comprehension should be restricted according to consistency maxims---has recently been defended by Laurence Goldstein (2006; 2013). It can be traced back to W.V.O. Quine(1951), who held that Naïve Comprehension embodies the only really intuitive conception of set (...)
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