In a recent paper [Hellman, 2003], we examined to what extent category theory (“CT”) provides an autonomous framework for mathematical structuralism. The upshot of that investigation was that, as it stands, while CT provides many valuable insights into mathematical structure---specific structures and structure in general---, it does not sufficiently address certain key questions of logic and ontology that, in our view, any structuralist framework needs to address. On the positive side, however, a theory of large domains was sketched as a (...) way of supplying answers to those key questions, answers intended to be friendly to CT both in demonstrating its autonomy vis-à-vis set theory and in preserving its “arrows only” methods of describing and interrelating structures and the insights that those methods provide. The “large domains”, hypothesized as logicomathematical possibilities, are intended as suitably rich background universes of discourse relative to which both category-and-topos theory and set theory can be developed side by side, without either emerging as “prior to” the other. Although those domains, as described, resemble natural models of set theory (on an iterative conception) or toposes suitably enriched with an equivalent of the Replacement Axiom, they are defined without set-membership as a primitive, and also without ‘function’ or ‘category’ or ‘functor’ as primitives; all that is required is a combination of ‘part/whole’ and plural quantification (in effect, the resources of monadic second-order logic). This background.. (shrink)

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 (...) ontological multiplicity and relativity encountered in the natural sciences as well. (shrink)

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 (...) conclude that his account of that distinction is only tenable if the definiteness of the set-theoretical universe is rejected. Partially supported by the Spanish CICYT, grant MTM 2008–03389. Earlier versions of this paper were presented at the VIII International Ontology Congress in San Sebastian, and at the Seminar in Logic and the Philosophy of Mathematics of the University of Bristol. I wish to thank the participants for their comments. I am also grateful to Joan Bertran, Joan Climent, and one anonymous referee for their careful reading of the paper and their helpful remarks and suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)

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 (...) conclude that his account of that distinction is only tenable if the definiteness of the set-theoretical universe is rejected. (shrink)

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 (...) the assumptions on which Skolem's argument rests from a present-day perspective. (shrink)

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 (...) why he abandoned this view. In this paper, I speculate as to what those reasons were, and evaluate them in light of recent work and interest in plural logic. (shrink)

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 (...) for the sciences. His theory does not simply attempt to add formal rigor to the sciences, but instead relies on the methods of the biological and social sciences to construct his world-view. Whitehead''s theory is a rich source of notions that are intended to explain every element of experience. It is a product of Whitehead''s earlier attempt to provide a mathematical foundation for the physical sciences and is still consistent with modern physics. (shrink)

This book will appeal to mathematicians and philosophers interested in the foundations of mathematics.

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, (...) composed of well defined objects.It is the extension of Greek notion of 'number' (arithmos) into Cantor's 'transfinite'. (shrink)

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).

THE FOUNDATIONS OF MATHEMATICS () PREFACE The object of this paper is to give a satisfactory account of the Foundations of Mathematics in accordance with ...

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 (...) not jointly, lack the problematic feature. (shrink)

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.

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.

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 (...) affects Frege's system. (shrink)

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 1 (...) 1-comprehension schema would already be inconsistent. In the present paper, we show that this is not the case. (shrink)

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.

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 (...) set theory, minimalism is unable to makeadequate progress in resolving the paradoxes, and must be replaced by adrastically different sort of theory. Such a theory, it is shown, must befundamentally non-minimalist. (shrink)

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.

Russell's paradox represents either of two interrelated logical antinomies. The most commonly discussed form is a contradiction arising in the logic of sets or classes. Some classes (or sets) seem to be members of themselves, while some do not. The class of all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself. However, suppose that we can form a class of all classes (or (...) sets) that, like the null class, are not included in themselves. The paradox arises from asking the question of whether this class is in itself. It is if and only if it is not. The other form is a contradiction involving properties. Some properties seem to apply to themselves, while others do not. The property of being a property is itself a property, while the property of being a cat is not itself a cat. Consider the property that something has just in case it is a property (like that of being a cat ) that does not apply to itself. Does this property apply to itself? Once again, from either assumption, the opposite follows. The paradox was named after Bertrand Russell, who discovered it in 1901. (shrink)

In their correspondence in 1902 and 1903, after discussing the Russell paradox, Russell and Frege discussed the paradox of propositions considered informally in Appendix B of Russell’s Principles of Mathematics. It seems that the proposition, p, stating the logical product of the class w, namely, the class of all propositions stating the logical product of a class they are not in, is in w if and only if it is not. Frege believed that this paradox was avoided within his philosophy (...) due to his distinction between sense (Sinn) and reference (Bedeutung). However, I show that while the paradox as Russell formulates it is ill-formed with Frege’s extant logical system, if Frege’s system is expanded to contain the commitments of his philosophy of language, an analogue of this paradox is formulable. This and other concerns in Fregean intensional logic are discussed, and it is discovered that Frege’s logical system, even without its naive class theory embodied in its infamous Basic Law V, leads to inconsistencies when the theory of sense and reference is axiomatized therein. (shrink)

Influenced by G. E. Moore, Russell broke with Idealism towards the end of 1898; but in later years he characterized his meeting Peano in August 1900 as ?the most important event? in ?the most important year in my intellectual life?. While Russell discovered his paradox during his post-Peano period, the question arises whether he was already committed, during his pre-Peano Moorean period, to assumptions from which his paradox may be derived. Peter Hylton has argued that the pre-Peano Russell was thus (...) vulnerable to (at least one version of) Russell's paradox and hence that the paradox exposes a pre-existing difficulty in Russell's Moorean philosophy. Contrary to Hylton, I argue that the Moorean Russell adhered to views which insulated him against the paradox. Further, I argue that Russell became vulnerable to his paradox as a result of changes in his Moorean position occasioned, first, by his acceptance of Cantor's theory of the transfinite, and, second, by his correspondence with Frege. I conclude with some general comments regarding Russell's acceptance of naïve set theory. (shrink)

The papers collected in this volume represent the main body of research arising from the International Munich Centenary Conference in 2001, which commemorated ...

Working in the fragment of Martin-Löfs extensional type theory [12] which has products (but not sums) of dependent types, we consider two additional assumptions: firstly, that there are (strong) equality types; and secondly, that there is a type which is universal in the sense that terms of that type name all types, up to isomorphism. For such a type theory, we give a version of Russell's paradox showing that each type possesses a closed term and (hence) that all terms of (...) each type are provably equal. We consider the kind of category theoretic structure which corresponds to this kind of type theory and obtain a categorical version of the paradox. A special case of this result is the degeneracy of a locally cartesian closed category with a morphism which is generic in the sense that every other morphism in the category can be obtained from it via pullback. (shrink)

Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...) theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science. (shrink)

This paper is concerned with locating the specific assumption that led Frege into Russell's Paradox. His understanding of reflexive pronouns was weak, for one thing, but also, by assimilating concepts to functions he was misled into thinking one could invariably replace a two-place relation with a one-place property. /// Este trabajo se ocupa de localizar el supuesto específico que llevó a Frege a la Paradoja de Russell. Por una parte, su comprensión de los pronombres reflexivos era débil pero, por otra, (...) al asimilar los conceptos a las funciones pensó equivocadamente que siempre podíamos reemplazar una relación diádica con una propiedad monádica. (shrink)

Sobocinski in his paper on Leśniewski's solution to Russell's paradox (1949b) argued that Leśniewski has succeeded in explaining it away. The general strategy of this alleged explanation is presented. The key element of this attempt is the distinction between the collective (mereological) and the distributive (set-theoretic) understanding of the set. The mereological part of the solution, although correct, is likely to fall short of providing foundations of mathematics. I argue that the remaining part of the solution which suggests a specific (...) reading of the distributive interpretation is unacceptable. It follows from it that every individual is an element of every individual. Finally, another Leśniewskian-style approach which uses so-called higher-order epsilon connectives is used and its weakness is indicated. (shrink)

In a recent article, Emil Badici contends that the inclosure schema substantially fails as an analysis of the paradoxes of self-reference because it is question-begging. The main purpose of this note is to show that Badici's critique highlights a necessity condition for the success of dialectic about paradoxes. The inclosure argument respects this condition and remains solvent.

We provide an overview of consistent fragments of the theory of Frege’s Grundgesetze der Arithmetik that arise by restricting the second-order comprehension schema. We discuss how such theories avoid inconsistency and show how the reasoning underlying Russell’s paradox can be put to use in an investigation of these fragments.

In this paper, I consider two curious subsystems ofFrege's Grundgesetze der Arithmetik: Richard Heck's predicative fragment H, consisting of schema V together with predicative second-order comprehension (in a language containing a syntactical abstraction operator), and a theory T in monadic second-order logic, consisting of axiom V and 1 1-comprehension (in a language containing anabstraction function). I provide a consistency proof for the latter theory, thereby refuting a version of a conjecture by Heck. It is shown that both Heck and T (...) prove the existence of infinitely many non-logical objects (T deriving,moreover, the nonexistence of the value-range concept). Some implications concerning the interpretation of Frege's proof of referentiality and the possibility of classifying any of these subsystems as logicist are discussed. Finally, I explore the relation of T toCantor's theorem which is somewhat surprising. (shrink)

This paper dates from about 1994: I rediscovered it on my hard drive in the spring of 2002. It represents an early attempt to explore the connections between the Julius Caesar problem and Frege's attitude towards Basic Law V. Most of the issues discussed here are ones treated rather differently in my more recent papers "The Julius Caesar Objection" and "Grundgesetze der Arithmetik I 10". But the treatment here is more accessible, in many ways, providing more context and a better (...) sense of how this issue relates to broader issues in Frege's philosophy. (shrink)

Standard (classical) logic is not independent of set theory. Which formulas are valid in logic depends on which sets we assume to exist in our set-theoretical universe. Second-order logic is just set theory in disguise. The typically logical notions of validity and consequence are not well defined in second-order logic, at least as long as there are open issues in set theory. Such contentious issues in set theory as the axiom of choice, the continuum hypothesis or the existence of inaccessible (...) cardinals, can be equivalently transformed into question about the logical validity of pure sentences of second-order logic, where “pure” means that they only contain logical symbols and bound variables. Even standard first-order logic depends on the acceptance on infinite sets in our set-theoretical universe. Should we choose to admit only finite sets, the number of logically valid pure first-order formulas would increase dramatically and first-order logic would not be recursively enumerable any longer. (shrink)

Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction in numerical (...) economics could mean. Method: An algebraic simple model system is subjected to a deeper structure of underlying variables. With an algorithm simulating the steps in taking a limit of second order difference quotients the error terms are studied at the background of their algebraic expression. Results: With the algorithm that was applied to a simple quadratic polynomial system we found unstably amplified round-off errors. The possibility of numerical chaos is already known but not in such a simple system as used in our paper. The amplification of the errors implies that it is not possible with computer means to constructively show that the algebra and numerical analysis will ‘on the long run’ converge to each other and the error term will vanish. The algebraic vanishing of the error term cannot be demonstrated with the use of the computer because the round-off errors are amplified. In philosophical terms, the amplification of the round-off error is equivalent to the continuum hypothesis. This means that the requirement of (numerical) construction of mathematical objects is no safeguard against inference-only conclusions of qualities of (numerical) mathematical objects. Unstably amplified round-off errors are a same type of problem as the ordering in size of transfinite cardinal numbers. The difference is that the former problem is created within the requirements of constructive mathematics. This can be seen as the reward for working numerically constructive. (shrink)

John Burgess in a 2004 paper combined plural logic and a new version of the idea of limitation of size to give an elegant motivation of the axioms of ZFC set theory. His proposal is meant to improve on earlier work by Paul Bernays in two ways. I argue that both attempted improvements fail. I am grateful to Philip Welch, two anonymous referees, and especially Ignacio Jané for written comments on earlier versions of this paper, which have led to substantial (...) improvements. Thanks also to the participants in a discussion group at the University of Bristol, where an earlier version was presented. (shrink)

Does category theory provide a foundation for mathematics that is autonomous with respect to the orthodox foundation in a set theory such as ZFC? We distinguish three types of autonomy: logical, conceptual, and justificatory. Focusing on a categorical theory of sets, we argue that a strong case can be made for its logical and conceptual autonomy. Its justificatory autonomy turns on whether the objects of a foundation for mathematics should be specified only up to isomorphism, as is customary in other (...) branches of contemporary mathematics. If such a specification suffices, then a category-theoretical approach will be highly appropriate. But if sets have a richer `nature' than is preserved under isomorphism, then such an approach will be inadequate. (shrink)

In this paper, I pursue such a logical foundation for arithmetic in a variant of Zermelo set theory that has axioms of subset separation only for quantifier-free formulae, and according to which all sets are Dedekind finite. In section 2, I describe this variant theory, which I call ZFin0. And in section 3, I sketch foundations for arithmetic in ZFin0 and prove that certain foundational propositions that are theorems of the standard Zermelian foundation for arithmetic are independent of ZFin0.<br><br>An equivalent (...) theory of sets and an equivalent foundation for arithmetic was introduced by John Mayberry and developed by the current author in his doctoral thesis. In that thesis, the independence results mentioned above are proved using proof-theoretic methods. In this paper, I offer model-theoretic proofs of the central independence results using the technique of cumulation models, which was introduced by Steve Popham, a doctoral student of Mayberry<br>from the early 1980s. (shrink)

Recent interest in non-well-founded set theories has been concentrated on Aczel's anti-foundation axiom AFA. I compare this axiom with some others considered by Aczel, and argue that another axiom, FAFA, is superior in that it gives the richest possible universe of sets consistent with respecting the spirit of extensionality. I illustrate how using FAFA instead of AFA might result in an improvement to Barwise and Etchemendy's treatment of the liar paradox.