The following papers have so far been announced:
| Peter Apostoli & Akira Kanda | Approximation Spaces of Abstract Sets & the Granularity of Mathematical Knowledge | |
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Axiomatic set theory was conceived in response to the paradoxes that shook the foundations of mathematics at the turn of the last century. As yet, however, there is no proof that the theory is consistent, and so the Problem of Foundations heretofore remained open. Our recent solution to the antinomies of abstract set theory, described below, establishes a surprising connection between the foundations of mathematics and quantum mechanics. The model of abstract set theory we obtained (Cantor-Frege-Gilmore set theory, or "CFG") shows that the mathematical continuum has a fundamental scale or "Planck length". Russell's contradiction is the naturally expected discontinuity that results from the attempt to resolve the continuum into parts smaller than this fundamental scale. This "granularity" of the set theoretic universe was predicted by Rough Set Theory, a method of topological information granulation of growing importance to computer science. CFG is a model of abstract rough set theory which unifies domain theory (a theory of impredicatively defined computable functions used in the semantics of programming languages) with issues at large in the newly emerging field of granular computation. Set theory is thought to underlie the entirety of mathematics and hence also computation; thus, the consequences and applications of CFG are both profound and wide-ranging. We present an approximation space (U,R) which is an infinite (hypercontinuum) solution to the domain equation |
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| John Bell | Russell's Paradox and Diagonalization in a Constructive Context | |
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One of the most familiar uses of the Russell paradox, or, at least, of the idea underlying it, is in proving Cantor's theorem that the cardinality of any set is strictly less than that of its power set. The other method of proving Cantor's theorem-employed by Cantor himself in showing that the set of real numbers is uncountable-is that of diagonalization. Typically, diagonalization arguments are used to show that function spaces are "large" in a suitable sense. Classically, these two methods are equivalent.
But constructively they are not: while the argument for Russell's paradox is perfectly constructive, (i.e. employs on intuitionistically acceptable principles of logic) the method of diagonalization fails to be so. In my paper I shall describe the ways in which these two methods diverge in a constructive setting. |
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| Ulrich Blau | The Significance of the Largest and Smallest Numbers for the Oldest Paradoxes | |
| Wilfried Buchholz | On Gentzen's consistency proofs for arithmetic | |
Gentzen has given three consistency proofs for arithmetic:
In (1) the consistency of number theory PA is proved by embedding PA into cut-free omega-arithmetic PAomega. The main feature of this embedding is an operator R on infinitary derivations (in omega-arithmetic) defined by (transfinite) recursion on well-founded trees. In (2), for each finite PA-derivation d certain reduced PA-derivations d[i] are defined by primitive recursion on the build up of d in such a way that the endsequent of d results from the endsequents of the d[i]'s by an inference of PAomega. By iterating these reduction steps, again every PA-derivation is translated into a PAomega-derivation of the same sequent. In (3) for every possible PA-derivation d of 0=1 a new PA-derivation red(d) of 0=1 is defined in so to speak "geometrical" terms by referring to the global structure of d. To each d an ordinal o(d) is assigned in such a way that o(red(d)) can be shown to be less than o(d). Now the core of our explanation is the following. (2) is related to (1) in so far as the definition of d[i] can be literally read off from the recursion equations for the operator R. The reduced derivation red(d) in (3) essentially coincides with d[0] from (2). Gentzen's ordinal assignment in (3) can be explained by combining the infinitary approach in (1) with Sch"utte-style computations on the ordinal lengths of infinitary derivations resulting from cut-eliminating transformations like R. |
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| Wolfgang Buschlinger | The use of semantical paradox for graph theoretical problems: False-Systems | |
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In the last 150 years an important mathematical theory evolved: graph theory. Furthermore, though grown up in the slums of topology, since the end of the 50s graph theory has been regarded not only as being important, but even as being fundamental.
In order not simply to claim but to justify its fundamental character, a method is given how graph theory can be reduced to a more basic theory, at least partially. Just as in the case of set theory, the reduction concentrates on logic, yet with semantics added to this. The main core of the investigation lies in sets of sentences that may refer onto other sentences of the same set, claiming these sentences to be true or false at the same time. Thereby, first of all a definition of basic graph theoretical terms is easily obtained. Thereafter, as long as non-paradoxical sets of sentences are examined, from logico-semantical considerations it is possible to derive both theorems of the theory of finite graphs as well as theorems of the theory of infinite graphs. En passent one also gets a sufficient condition for Yablo-style sets of sentences to be non-paradoxical. Yet, one aspect of graph theory cannot be caught by logic: the topological aspect. | ||
| Andrea Cantini | Relating KF to NF | |
| Appendix B of The Principles of Mathematics concludes with an unsolved antinomy, which has a semantical flavour, as it involves the very notions of proposition and truth. We consider Russell's contradiction within the context of an abstract impredicative theory of of truth. The theory is obtained by carrying out a Kripke-style construction within (a consistent fragment of) Quine's set theory NF. The resulting notion of truth has peculiarities, which contrast with the so-called KF-axioms (KF=Kripke-Feferman). | ||
| Charles Chihara | Shapiro Objection to the Constructibility Theory in the Light of Russell's Theory of Types | |
| This paper is a defense of the mathematical theory called "the constuctibility theory" that was put forward in my book Constructibility and Mathematical Existence. In particular, I take on an objection to this theory that was raised by Stewart Shapiro in his paper "Modality and Ontology" and that he has repeated with some minor changes in his recent book on mathematical structuralism. In my response to Shapiro's criticism, I shall compare the Constructibility Theory both to Russell's Theory of Types (which it resembles closely in some of its formal features) and also to the more standard set theoretical version of simple type theory. In the course of these discussions, I shall point out some of the lesser known features of Russell's theory and also clear up some widespread misconceptions and confusions about the Constructibility Theory. | ||
| Solomon Feferman | Typical Ambiguity: Having your cake and eating it too | |
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Ambiguity is a property of syntactic expressions which is ubiquitous in all languages-natural, scientific and mathematical; the efficient use of language depends to an exceptional extent on this feature. Disambiguation is the process of separating out the possible meanings of ambiguous expressions. Ambiguity is typical if the process of disambiguation can be carried out in some systematic way. Russell made use of typical ambiguity in the theory of types in order to combine the assurance of its (apparent) consistency ("having the cake") with the freedom of the informal untyped theory of classes and relations (and "eating it too"). I shall begin with a brief tour of his own uses of typical ambiguity. Then I will take up cases that he did not consider, namely statements of the form: A is a member of B, where B is a class expression and A is an expression prima facie of the same or higher type than B. Three versions of typical ambiguity for such statements will be treated, respectively in the simple theory of types, Zermelo-Fraenkel set theory, and explicit mathematics. I will show how the "naive" theory of categories and other global theories of structures can thereby be accounted for (to some extent) in the latter two frameworks. |
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| Harvey Friedman | Research Program on Ways Out of Russell's Paradox | |
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We Conjecture that all intellectually sensible consistent formal systems are formally interpretable in current set theory (with large cardinals). It should follow that all intellectually sensible consistent formal systems are synonymous with a fragment of current set theory (with large cardinals). In this sense, all ways out of Russell's Paradox - including ways involving non set theoretic concepts - are Conjectured to be subsumed under current set theory (with large cardinals). We present a number of dramatic formal and informal conjectures that tend to support this Conjecture. Recent results concerning these conjectures suggest the viability of several new deep research directions in the foundations of mathematics. |
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| Sy Friedman | Completeness and Iteration in Modern Set Theory | |
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Set theory entered the modern era through the work of Goedel and Cohen, which provided set-theorists with the necessary tools to deal with the undecidability of a large number of mathematical questions. Through these methods, together with their generalisation into the context of large cardinals, set-theorists have had great success in determining the axiomatic strength of a wide range of problems, not only within set theory but also within other areas of mathematics.
Through this work a very attractive picture of the universe of sets is starting to emerge, allowing for the existence of inner models satisfying strong large cardinal axioms. We shall employ the principles of Completeness and Iteration to argue for the inevitability of this picture. |
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| Nicholas Griffin | The Prehistory of Russell's Paradox | |
| The paper traces the steps by which Russell came to his paradox and his immediate response to it, up until his famous letter communicating the paradox to Frege. It also considers more briefly the earlier set-theoretic paradoxes - Burali-Forti's paradox and Cantor's greatest cardinal paradox - and addresses the question of why Russell was so concerned about the paradoxes while Burali-Forti and Cantor were not. | ||
| Kai Hauser | Two Versions of Realism | |
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In his first major philosophical publication "Russell's Mathematical Logic" Gödel employs some of Russell's pronouncedly realist views as a platform for a radical form of mathematical realism according to which the objective existence of mathematical objects is justified on analogous grounds as the existence of physical bodies.
Realist convictions about sets and concepts also form the basis of Gödel's argument in "What is Cantor's Continuum Problem?" that the continuum hypothesis has a determinate truth value. But in the second edition, published nearly 20 years later, he remarks that the objective existence of objects of mathematical intuition is not decisive for the issue under discussion. This discussion becomes meaningful already by virtue of the psychological fact that we have intuitions which are sufficiently clear to produce the standard axioms of set theory as well as an open series of extensions. In my talk I will examine two questions.
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| Allen Hazen | Interpreting the 1925 Logic | |
| Russell, in the 1925 revision of Principia Mathematica, proposed a version of Type Theory distinct from both the Ramified Type Theory of the first edition and from the Simple Type Theory of Ramsey. Perhaps in part because of its intermediate strength- to much like classical set theory to interest most constructivists or finitists, to weak for the classically minded- it attracted very little attention, and Gödel in th 1944 appears to be the first person to have noted it in print. Russell's exposition is neither rigorous nor detailed, but there is enough evidence for a confident diagnosis of what he intended. It is like Ramified Type Theory in distinguishing among propositional functions by referrence to their complexity of definition. This distinction is analogous to the arithmetic and analytic hierarchies familiar in recursion theory and descriptive set theory. The novelty in the 1925 logic is to allow "simple" higher-order functions to take "complex" arguments, much as a class low in one of the definitional hierarchies can have as members sets which are higher in the hierarchy. | ||
| Geoffrey Hellman | Russell's Absolutism vs(?) Structuralism | |
| Russell seems consistently to have maintained, with Frege, an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed (or proposed) by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct (modulo type restrictions), not merely adequate for mathematics. And Dedekind's "structuralist" views are subjected to severe criticism in the Principles. It is interesting to compare this absolutist stance with modern versions of structuralism. It turns out to be rather difficult to avoid some vestiges of absolutism. Structuralism based on model theory, after all, standardly takes sets from a fixed universe as absolute objects, not themselves subject to structural interpretation. Indeed, an extension of Russell's criticism of Dedekind's structuralism seems quite telling against its modern incarnation, Shapiro-Resnik "ante rem" (hyperplatonist) structuralism. This leaves category theory and modal structuralism as remaining "non-absolutist" options. We will suggest that, perhaps surprisingly, they actually ought to be brought together. | ||
| Andreas Herberg-Rothe | Tarski's "Magic Trick." The unrecognized antinomies of the meta-meta-language | |
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To let the antinomy disappear as if by magic, Tarski implements the differentiation between object language and meta-language. The audience as well as he himself miss, however, that this differentiation is possible at the meta-meta level first. Tarski thinks he can interpret the meta-meta level in an ascending gradation as simple meta-language of preceding meta-language, which became object language now. Exactly this is impossible in the framework of his solution approach.
The meta-meta-language cannot be in one and the same respect a (simple) meta-language for preceding object language and at the same time express the difference between the meta-language and object language. Leveling of the difference between the meta-meta-language, in one respect of (2nd) meta-language with regard to preceding (1st) meta-language (which now could be an object language only) and in other respect as language expressing the difference between meta-language and object language, is one part of Tarski's magic trick. The second is to blur the difference between meta-language terms and object language expressions in the meta-language. Leveling of this difference is the prerequisite for stipulation of meta-languages of arbitrary ranking as simple object languages of ever-higher meta-language. This, however, is not possible because semantic terms refer to the relation of object language and objects whereas object language expressions refer in contrast only immediately to objects. The difference between semantic terms and object language expressions can work at the level of meta-meta-language first, as the difference between meta- and object language is phrased and semantic terms are built by semantic terms first at this level. The Liar can be formulated in the meta-meta-language. |
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| Tobias Hürter | Inconsistency in the real world | |
| This is a remark on the distinction between the theoretical and the pretheoretical notion of natural number. The first notion is number theory, the second notion is the one based on the our elementary abilities to count and to add and multiply natural numbers. I want to present a weak formal theory T capturing the strength of the first notion. In particular, T is not comitted to infinitely many natural numbers. I proceed to show that the second notion is independent from T in a strong sense: A situation is conceivable in which we can see from within T that there are only finitely many natural numbers. All this is based on work by Hugh Woodin. | ||
| Peter Hylton | Presupposition and Type Theory | |
| Andrew Irvine | Russell on Method | |
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Russell's youthful quest for certainty is both famous and well documented. "I wanted certainty in the kind of way in which people want religious faith", he says in his autobiography. Understood in this way, his foundationalism in epistemology, while in some respects groundbreaking, was also perfectly in keeping with his original epistemic goals. At least in principle, epistemic justification has as its aim both logical and scientific certainty. In contrast, Russell's views about philosophical method stress the uncertainty of not just philosophical knowledge, but logical and scientific knowledge as well. They stress the fallibility of all branches of justified belief. In his many discussions about what he calls the liberal or scientific outlook, Russell the foundationalist is nowhere to be seen. This apparent tension between Russell's views on epistemology and his views on method is instructive. According to one hypothesis, the tension need not be resolved since, ultimately, it never really arises: theories of epistemic justification, on the one hand, and theories of philosophical method, on the other, are properly understood as being distinct theories about fully distinct phenomena. But according to a competing hypothesis, the tension is real and needs to be resolved. On this view, the resolution comes only when Russell abandons his foundationalism in favour of an alternative conception of knowledge. |
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| Gerhard Jäger | On fixed point theories | |
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Following Poincare and Weyl, one way of avoiding problems in the foundations of mathematics is to restrict the buildup of sets to those which can be introduced by definitions which avoid the vicious circle principle. This approach has been later brought into precise mathematical form by Feferman and Schütte, who also determined the ordinal \Gamma_0 as the ordinal of predicative mathematics. On the other hand, a large variety of subsystems of second order arithmetic and set theory has been studied since then, whose analysis makes heavy use of methods of impredicative proof theory such as elaborate collapsing techniques. However, research of the last years has also shown that there exists an interesting variety of theories, so-called metapredicative systems, with the following characteristics:
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| Hans Kamp | Russell's Theory of Descriptions and Presuppositional Description Theories: How incompatible are they? | |
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Russell considered his Theory of Descriptions an important contribution to philosophy because of the solutions it offers to a variety of problems in epistemology, metaphysics, loigical theory and philosophy of language. Developments in the second half of the 20th century, beginning with Strawson's paper "On Referring", have made it increasingly implausible that the way in which definite descriptions behave in natural language could be adequately described without assuming that they produce presuppositional effects. Moreover, the general methodological benefits that could be derived from treating descriptions as non-presuppositional would, in the light of our current understanding of how natural language functions, be quite limited. For over the past decades overwhelming evidence has accumulated that presupposition is an extremely general phenomenon in natural language, and that languages like English (and for all we know all others) contain large numbers of expressions and constructions that all give rise to presuppositions. So, even if definite descriptions were not among these, the general problems connected with presupposition would be there in any case. However, when we look at the phenomenon of presupposition in natural language in the way which I belive one should, then at least some of the aspects of the Theory of Descriptions which make it so attractive are not lost (as earlier ways of thinking about presupposition may have suggested) but essentially preserved. I will outline a theory of presuppositrion based on this perspective, demonstrate how it works for a few cases involving (presuppositionally analysed) descriptions, and state what I see as the points of convergence of the treatment of descriptions offered by this theory and the Russellian account. The last part of the talk will be concerned with some remaining difficulties, connected with those cases which according to Rusell involve narrow scope. (Such as, among others, "The golden mountain does not exist.") |
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| Phillip Keller | What singletons could be | |
| According to David Lewis (1991), membership is the only mystery set theory has left for us mereologists. In fact, it is only the special case of membership-in-a-singleton that troubled Lewis and needs to trouble us, if we take the subset relation to be a special case of the (antecedently understood) relation of part to whole. My task, then, is to explain what it means to say that a thing, a, is a member of a special sort of set, {a}, having a as its only member. My proposal, roughly, is that singletons, and sets in general, are a special sort of property, i.e. monadic universals in a broadly Armstrongian sense. {a}, in particular, is the property having all the properties of a, and membership is the relation of exemplifying such properties. I will call such properties "natures", the general idea being that they provide enough structure to enable mereology to go proxy for set theory. I will sketch, develop and make plausible my proposal and then defend it against its two main alternatives, developed by Armstrong and Bigelow. | ||
| Gregory Landini | Logicism's 'Insolubilia' and Their Solution by Russell's Substitutional Theory | |
| It is well known that Russell's 1903 Principles of Mathematics construed logic as a synthetic and a priori science of structure. Russell reified structures by adopting an ontology of propositions- mind and language independent 'states of affairs.' The thesis of Logicism advanced in the work held that the intuitions grounding all non-applied mathematics are logical intuitions of propositional structure. It has been widely reported that Logicism is dead and indeed that it died at Russell's own hands. Valiant as it was, the ramified type-theory of the 1910 Principia Mathematica did not save logicism from Russell's paradoxes (of classes and predication). The system offers no genuine "solution" of the paradoxes, and requires an infinity axiom and an axiom of reducibility- neither of which can be counted among the truths of pure logic. Logicism is dead. Or is it? What has not been widely reported is that prior to composing the system of Principia Mathematica, Russell did solve the paradoxes. Applying his 1905 theory of incomplete symbols to form definite descriptions of propositions, he invented a type-free intensional calculus which proxies a simple type-theory of attributes (and thereby a type-theory of classes and relation in extension.) This "no-classes" or "substitutional theory," as it came to be called, has only recently begun to be investigated. This paper shows that Russell acted too hastily when he abandoned substitution in 1908. The substitutional theory may well resurrect logicism just has Russell had originally hoped. | ||
| Shaughan Lavine | Objectivity: The Justification for Extrapolation | |
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Set theory can be obtained by extrapolating, in a mathematically precise sense, from the finite mathematics of indefinitely large sets. That extrapolation is the origin of and motivation for infinitary set theory. But on what grounds can it be argued that the extrapolation is not just a technical trick, but a justified move from knowledge of the indefinitely large to knowledge of the infinite? The application of mathematics to physics provides the link. Finite mathematics, with its context-dependent bounds of availability, provides a natural setting for a theory of measurements. Insofar as a successful theory of the correlations and behavior of some types of physical measurements is construed as a theory of the correlations and behavior of measurements of objective physical quantities, the theory of the objective quantities themselves must be obtained by replacing the epistemic bounds of measurement by the bounds of "measurement" of the physical quantities by the world, that is by bounds fixed and equal to each other once and for all, equal to what exists in the hypothesized objective world. But that change yields the extrapolated version of the measurement theory. Physics makes use, not only of measurements of values of individual physical quantities at points, but of measurements of values of field quantities throughout a region, that is, of the measurement of functions. Insofar as measurements may take on arbitrary values, the extrapolated mathematical theory will be one that allows arbitrary functions. The need to solve differential equations and to integrate over such functions then inevitably leads to full modern set theory. |
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| James Levine | When and How Did Russell Become Vulnerable to Russell's Paradox? | |
| 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 vulnerable to (at least a version of) it during his pre-Peano Moorean period. Peter Hylton has argued that he was 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 actually insulated him against the paradox and that it was only after his post-Peano acceptance of Cantor's theory of the transfinite that Russell rejected these Moorean views and became vulnerable to the paradox. I conclude with some general remarks concerning the relation between Russell's acceptance of the universal class (which occurs pre-Peano) and his acceptance of the unrestricted variable and naive set theory (which occurs only post-Peano). | ||
| Godehard Link | Bertrand Russell: The Invention of Logical Philosophy | |
Bertrand Russell was unique in bringing together the old disciplines of mathematics, logic, and philosophy to create a new field of study, logical philosophy, which in turn paved the way for modern mathematical logic and set theory. By way of introduction to the main themes of the conference, some major topics emerging from Russell's work are highlighted that may count as evidence for this development:
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| Bernard Linsky | The Resolution of Russell's Paradox in Principia Mathematica | |
| *20 of Principia Mathematica presents the "no-classes" theory of classes as a solution to the paradox. As has been pointed out by Gödel and others, there are some unclarities about how the contextual definitions of class expressions are to be applied. In Meaning and Necessity [1947] Rudolf Carnap pointed out that since identity produces an intensional context, even the elimination of class abstracts from identity statements must be done with care. Recently D.A.Martin has pointed out that the theory seems to produce too many classes of classes, again due to the combination of intensional functions with extensional classes. I propose a slight extension, or perhaps clarification, of the use of class variables in PM, and so resolve this difficulty. | ||
| Per Martin-Löf | The Doctrine of Types | |
| Russell's doctrine of types, as formulated in 1903 and 1908, will be elucidated by being put in relation to the doctrine of types that underlies constructive type theory. Particular attention will be paid to the distinction that the latter theory makes between sets, types and categories. | ||
| David McCarty | A Brother to Genius: the Philosophy and Mathematics of Paul Du Bois-Reymond | |
| Paul du Bois-Reymond was, during the second half of the 19th Century, among Europe's most influential and successful mathematicians. Today, in the theories of integration, Fourier series, convergence and the calculus of variations, one can find theorems, lemmas and tests that bear his name. Independently of Cantor, du Bois-Reymond introduced notions of dense and of nowhere dense sets and proved basic results governing them. Indepedently of Weierstraß, he constructed a continuous, nowhere differentiable function. In 1873, he disproved a conjecture of Dirichlet and Riemann by showing how to define a continuous function with divergent Fourier series at every point on a dense set. But he also contributed, and significantly, to the intellectual realms we would now call "foundations of mathematics" and "philosophy of mathematics." In the 1870s, he defined a clear notion of infinite magnitude or order and showed that there are infinitely many distinct such orders. He seems to have been the first to introduce and employ diagonal arguments, using them to show that, to use contemporary terminology, there are more than countably many infinite orders and that analysis contains notable paradoxes. Before Borel and Brouwer, Du Bois-Reymond considered the prospect of free choice sequences, that is, real numbers given by infinite sequences whose terms cannot be derived from any rule, and questioned the validity of the law of excluded third for statements concerning real numbers. Perhaps most importantly, he argued that our mathematical knowledge suffers such inherent limitation that certain claims concerning the nature of the continuum, claims that are relatively easy to grasp, will forever remain undecided. The talk is devoted to setting out the fundamental ideas behind du Bois-Reymond's philosophy of mathematics and describing their critical relations to the views of Cantor, Dedekind and Hilbert. | ||
| Vann McGee | The Many Lives of Ebenezer Wilkes Smith | |
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Peter Unger's famous Problem of the Many is anticipated in a passage in Russell's 1923 paper, "Vagueness." Although severely compressed, Russell's version adds something interesting to Unger's treatment. Unger's argument requires assumptions about the part-whole relation which, while highly credible, are vulnerable to doubt. Russell's version does not depend on any special mereological assumptions. It is argued that the Unger-Russell argument is not, as Unger supposes it to be, persuasive evidence that there are no clouds or persons. Instead, it is a compelling argument for the inscrutability of reference, with striking implications, among other things, for our understanding of de re propositional attitude attributions. |
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| Nikolay Milkov | Russell's Turn of August-1900 and his Paradox | |
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Before August-1900 Russell's logic was nothing but mereology. His turn of August-1900 led to the discarding of part-whole logic and to the acceptance of a kind of intensional logic instead. To be more precise, Russell accepted a technique of treating infinite collections with the help of a singular concept which he called the 'denoting phrase'. In this way he accepted a logic of different orders (types) which distinguishes between classes, individuals, etc. Russell embraced these new ideas with the hope that they would give him the resources for a paradox-free treatment of infinity number. Unfortunately, this move failed to eliminate the paradox of infinity: it only removed it from the realm of infinite classes (ordinal numbers) to that of class-inclusion (cardinal numbers). This point coheres with the fact that Russell's long-elaborated solution to his paradox (1905-8) was nothing but a setting aside of some of the ideas he had adopted with his August-1900 turn:
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| Sebastiano Moruzzi | Russell's Theory of Vagueness | |
| In this paper I argue that Russell's theory of vagueness, endorsed by Russell in the twenties, can be called an epistemic theory of vagueness only in a very peculiar sense. Russell's theory of vagueness should be better qualified as a semantic theory of vagueness, i.e. a theory which qualifies vagueness as a feature of the meaning of expressions, because of the causal theory of meaning that he held. The usual metaphysical doctrine underlying the epistemic conception of vagueness, i.e. realism, is not implied by the metaphysical doctrine (i.e. neutral monism) that Russell maintained at that time. Only an unjustified metaphysical realist assumption can allow for realism in Russell's philosophical system, and this would be quite incompatible with his refusal of bivalence consequent his analysis of the phenomena of vagueness. Vagueness seems then to jeopardise Russell's metaphysical position. | ||
| Yiannis Moschovakis | On the Meaning of Belief Statements | |
| Jan Mycielski | Russell's paradox and Hilbert's (much forgotten) view of set theory | |
| Karl-Georg Niebergall | Is ZF finitistically reducible? | |
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One of the central goals of proof theory is to reduce larger and larger portions of mathematics to theories which are foundationally distinguished, e.g. finitary. PRA is regarded as such a theory, and proof theoretical reducibility of T to a finitary theory is taken to show the finitistic reducibility of T. Independently, J. Mycielski has developed a method for obtaining for each 1-order theory T a counterpart FIN[T], which he considers to be "isomorphic" (or "equivalent") to T and be finitary. In particular, ZF turns out to be "equivalent" to FIN[ZF]. Does this result show that the proof theoretician´s efforts are obsolete (it may even contradict some of his claims)? Or is there a mere equivocation? Observation 1: ZF is proof theoretically reducible to FIN[ZF]. Mycielski defines T to be locally finite iff each theorem of T which is closed has a finite model. Observation 2: If each locally finite theory is finitary, then ZF is finitistically reducible. Of course, this observation does not imply that ZF IS finitistically reducible. For the premise of observation 2 and the reduction concept employed here might be rejected as inadequate to our preformal understanding. Since, personally, I do not think that ZF is finitistically reducible, I will follow this line of argument in the rest of the talk. |
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| Francesco Orilia | Logical Rules, Principles of Reasoning and Russell's Paradox | |
| Roughly, the logical rules allow us to construct a deduction, e.g. modus ponens, universal quantifier elimination or lambda-conversion, while the principles of reasoning guide us in the goal-driven effort to rationally obtain new beliefs from given ones by means of deductions. Typical examples of the latter can be found in belief revision theories. Russell's paradox and similar puzzles arise from an inconsistent set of logical rules, involving both classical logic and lambda-conversion. Accordingly, typical attempts at a solution restrict either classical logic or lambda-conversion. I investigate a different strategy, by assuming that the set of logical rules making up our standard logical competence is in fact inconsistent and asking what the principles of reasoning and belief revision must be like for us to cope with this. In the resulting perspective, logical rules are nonmonotic-like, for in the light of a logical paradox the principles of reasoning must be allowed to dictate that some logical rule or other has an exception. One upshot of this is that our knowledge of logical truths turns out to be much more conjectural in nature than it is typically assumed to be. | ||
| Charles Parsons | Communication and the uniqueness of the natural numbers | |
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In an earlier paper (in Iyyun 39 (1990)) I discussed the doubts that could be raised about the common-sense view that the natural numbers are a unique structure and presented an argument from assumptions about linguistic communication to the conclusion that communicating speakers would be able to see, without appealing to set theory, that their "natural numbers" are isomorphic. In the present paper this issue will be revisited and a clearer and (let us hope) more convincing version of the argument presented. This
type of argument does not rule out the possibility that the speakers' natural numbers might be interpreted as nonstandard by a purely external interpreter. Recent writings by McGee, Lavine, and Field address similar issues and raise some further questions. Can one show uniqueness in a more outright manner? Can an argument of this kind be extended into set theory, on the model of Zermelo's quasi-categoricity theorem? These questions will be addressed to the extent that time permits. |
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| Jacek Pasniczek | Clark-Russell Paradox in Theories of Meinongian Objects | |
| Around the turn of XX-th century Russell strongly critisised Meinong's ontology accusing it of inconsistency. Although Russell was wrong as to alleged source of inconsistency, there are still some paradoxes which endanger this ontology. According to Meinong, an object must possess all properties involved in its description; in particular, 'the golden mountain' is golden and it is a mountain. Yet this object does not exist in the ordinary meaning of the word. Russell argued accordingly that a slightly different object 'the existent golden mountain' (1) exists as having assigned the property of existence, while on the other hand, (2) does not exist like 'the golden mountain". This inconsistency can be easily avoided by claiming that there are two diffrernt senses of 'existence' and consequently, that two sorts of predication should be associated with every property. Such a strategy is adopted by contemporary reconstructions of Meinong's theory of objects. However, if the theory still allows for some strong logical principles such as comprehension pronciples for defining objects and properties then one can prove, by building so called Clark-Russell paradox, that the theory is inconsistent. As it is showed in this paper, Clark-Russell paradox bears some close resemblance to the original Russell's paradox. | ||
| Volker Peckhaus & Reinhard Kahle | Hilbert's Paradox | |
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In 1903 Gottlob Frege sent a complimentary copy of the second volume of the "Grundgesetze der Arithmetik" to the Göttingen mathematician David Hilbert, containing the description of Russell's Paradox and Frege's admission that this paradox can be formulated in the system of the "Grundgesetze". In his response Hilbert declared that the paradox described had been known in Göttingen for a long time. He himself had found other, even more convincing examples four to five years ago, and after having informed Zermelo the latter found the one mentioned by Frege three to four years ago. It is well known that Zermelo indeed discovered a set-theoretical paradox in Cantor's theory, independently of Russell. But what were these contradictions Hilbert claimed to have found around 1898/1899? There are some further traces of Hilbert's Paradox in correspondences of the time. The most explicit hint can be found in Blumenthal’s biography of Hilbert where we read that Hilbert formulated the contradictory notion of the set of all sets which arise from union and mapping on themselves. Hilbert never published this contradiction, but a version can be found in his 1905 lecture course "Logische Principien des mathematischen Denkens". There Hilbert discusses the paradoxes of set theory mentioning Zermelo's paradox and a contradiction of "purely mathematical nature" which was never published, as Hilbert stressed, but known to set-theorists, especially to Georg Cantor. In the historical part of the lecture the story of Hilbert's Paradox will be told and it will be shown that Hilbert's axiomatic program was indeed affected by the paradoxes, contrary to the standard view. In the systematical part of the lecture Hilbert's Paradox will be described and reconstruced using modern tools. It will be shown that it is closer to Cantor's Paradox than to Russell's. |
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| Graham Priest | Paraconsistent Set Theory and Metatheory | |
Russell's paradox is generated by two things:
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| Adolf Rami | Empty Names and Russell's Name Claim | |
In general the aim of my paper is to present and investigate a dilemma of our intuitions about proper names on the background of two different solutions of Bertrand Russell of this dilemma and Russell's Philosophy of Logical Atomism. In detail this will be done in four steps.
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| Michael Rathjen | Anti-foundation and constructive set theory | |
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More often than not the blame for paradoxes like Russell's has been laid on some sort of circularity. It is not uncommon to find circular analyses of philosophical or linguistic phenomena rejected on grounds that they conflict with basic principles of mathematics in that the only coherent conception of set outlaws circularity. Developments in mathematical logic over the last thirty years, however, have shown that such criticism is often groundless.
The talk will feature set theories with the so-called Axiom of Anti-Foundation, AFA. AFA provides a uniform tool for modelling many kinds of cicular phenomena. Particular emphasis will be put on constructive set theories as AFA allows for a constructive justification as well. In addition to being based on intuitionistic logic, constructive set theories adhere to Poincare's and Weyl's vicious circle principle which bans impredicative definitions as applied to sets. |
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| Francisco Rodriguez-Consuegra | Propositional ontology | |
| I will briefly indicate the role of propositional functions in Principia, then pointing out the way in which they continue to be introduced through propositions, in spite of the doubtful status of these last pseudo-entities, then trying to see if the notion of judgment, which is used by Russell to support propositions, can reasonably meet the requirements which are needed in a serious, coherent ontology. After a survey of some of Russell's different attempts to build up a convincing notion of proposition and judgment between 1910 and 1918, I will conclude that the propositional ontology of logical atomism was finally a failure, and so that the mathematical ontology usually associated to Russell’s logicism was also a failure. | ||
| Philippe de Rouilhan | Is the Ghost of the Tarski Hierarchy still with us? | |
| We all remember Kripke's observation made towards the end of his 1975 article on truth: "The ghost of the Tarski Hierarchy is still with us". Kripke had shown that a non-classical language containing a minimum of arithmetic could contain (a coded version of) its own truth predicate, but not that it could contain an "adequate" definition (of a coded version) of this predicate. What Kripke had not done, others have done subsequently, in particular Hintikka, in a privately circulated 72 page paper of 1991, the content of which is taken up again in his 1996 book, The Principles of Mathematics Revisited. Has "Tarski's curse" been thus "exorcized", as Hintikka claims, or is the ghost of the Tarski hierarchy again and still with us? | ||
| Helmut Schwichtenberg | Feasible programs from proofs | |
| We restrict induction and recursion on notation in all finite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types "box rho -> sigma" and formulas "box A -> B" as well as "all bar{x} A" with "complete" variables bar{x}, and by adding linear concepts to the lambda calculus (for object terms and proof terms). For the arithmetical system we define modified realizability and show that the programs extracted from proofs of Pi^0_2-theorems characterize the polynomial time computable functions. | ||
| Hartley Slater | The Uniform Solution | |
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This paper uses Bunt's formulation of Set Theory, (in his book 'Mass terms and Model Theoretic Semantics') to locate a hidden condition on the Naive Abstraction Axiom. On Bunt's account, Set Theory is a specialisation of his 'Ensemble Theory' which arises with ensembles which are the merge of their atomic parts (c.f. D.K. Lewis' 'Parts of Classes'), so the hidden condition is that the predicate in question is count. Russell's Paradox and the like are then avoided through the denial of this condition for the troublesome predicates.
The point is part of a larger analysis of the traditional paradoxes of self-reference which, for instance, sees Tarski's T-scheme as equally conditional: we can say that Ts iff p only on the condition that s has a determinate sense. So The Liar, and simliar difficulties, merely lead to the denial of that condition in some form. In this later respect the analysis of the semantic paradoxes bears some resmblance to McDonald's recent work (see J.P.L. 2000), but McDonald does not bring in the appropriate parallel with the set-theoretic paradoxes. Using Bunt's analysis of mass terms as a basis for our understanding of continuous phenomena, it is claimed, leads to a much more radical re-appraisal of Set Theory. For the count-mass distinction can be formulated in terms of those predicates which do, and those predicates which do not have a determinate number of instances. A number can be associated with some beef, but only given a further, arbitrary parameter, e.g. a unit of volume, or weight. So the attempt to give an arithmetical analysis of continua, as in classical Set Theory, must be abandoned (c.f. D. Bostock's 'Logic and Arithmetic' II). |
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| Holger Sturm | The relevance of Russells Paradox for a philosophical theory of properties | |
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Russell's paradox has become famous as the most serious challenge a foundational approach to mathematics must face, and, therefore, has mainly been considered as a set-theoretic paradox. As is well-known, however, Russell's paradox also has a twin brother in the realm of properties. There exist a number of formal theories based on different strategies of how to respond to the paradox within a type-free setting. The purpose of this talk is not to add a further player to the game of avoiding the paradox by presenting some new theory. Instead, I will question and discuss the meaning of the whole game. In particular, I will argue that the paradox, and, more generally, the whole phenomenon of self-instantiation, has been overestimated as an essential factor in a philosophical theory of properties. First, I will show that among the many tasks properties are supposed to fulfill there is only one for which the paradox is a challenge, namely in semantics, where properties are used as meaning-objects. Second, I will argue that even semantics provides no good reasons for the dominant position Russell's paradox has occupied so far. The argument is based on certain methodological assumptions regarding explanations in ontology and semantics. |
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| Robert Thomas | Mathematicians and mathematical objects | |
| The paper attempts to indicate why it is possible and even reasonable for mathematicians to be unconcerned with ontology of mathematical objects. The reason alleged is that such objects are not what pure mathematics is about. Rather, mathematics is about 'relations between objects' or 'types of relation', in the words of Poincaré and Russell, the so-called objects being mere pronouns. Notice is taken of the position that only what is real can be dependably reasoned about, a comfortable position wholly inadequate for dealing with mathematics. No ontological position is taken up, and doubt is cast on the meaningfulness of deciding whether a pronoun exists as distinct from the existence of what the pronoun may be used to refer to in applied mathematics. The device proposed for making the basically empirical Poincaré-Russell observation acceptable to philosophers is the pretence proposed by Mark Crimmins for serious semantic purposes in turn based on that proposed by Kendall Walton for aesthetic purposes. Finally, the view put forward is distinguished from postulationism, which most expressions of it, including Russell's, look a bit like. | ||
| Alasdair Urquhart | Russell's zig-zag theory 1903-05 | |
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During the years 1903 to 1905, Russell pursued a version of the foundations of logic that was strongly influenced by Frege's abortive attempt at solving Russell's paradox. The theory was type-free, based on the notion of function, and founded on the idea that the paradoxes could be solved by placing direct restrictions on the comprehension principle. The working out of the zig-zag theory required a kind of balancing act, in which Russell tried to preserve enough of the comprehension scheme to allow the derivation of ordinary mathematics, while avoiding contradictions. In the end, his efforts failed. In this talk, I survey some of this work, and compare it to modern systems such as illative combinatory logic, which can also be considered as attempts at neo-Fregean foundations for mathematics. |
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| Gabriel Uzquiano | Categoricity Theorems and Conceptions of Set | |
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There is a well-known "almost-categoricity" result for second-order set theory. Two models of second-order ZFC are not necessarily isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is more subtle for impure set theory, but there is a recent categoricity result due to Vann McGee. Two models of second-order ZFCU satisfying the additional axiom that the urelements form a set with the same universe of discourse are not necessarily isomorphic to each other, but the "pure sets" of one are isomorphic to the "pure sets" of the other. One reason this result is of interest is that we can appeal to this theory to ensure that there is a fixed truth value for each arbitrary, sentence of pure set theory.
We investigate whether similar categoricity results obtain for weaker axiomatizations of set theory with urelements. George Boolos has persuasively argued that the axioms of standard set theory are motivated by a mixture of two conceptions of set: The iterative conception of set, on which sets are formed in stages of a cumulative hierarchy; and the limitation of size doctrine, on which some objects form a set if and only if they are not too numerous. We discuss whether there are categoricity results for second-order axiomatizations of set theory that are in line with the iterative conception of set. A similar question arises for second-order axiomatizations of the limitation of size doctrine. We consider the case of New V, a modification of Frege's Basic Law V developed by Boolos that captures the idea of limitation of size. |
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| Albert Visser | Referring Expressions and the Context Principle | |
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Frege's context principle is that "Only in the context of a sentence does a word have meaning." The principle is notoriously hard to interpret. The dictum allows readings which are trivially false. E.g., in one sense, a word is meaningful by definition -or else it wouldn't be a word-, so words certainly do not need sentences to be meaningful. Dummett proposes an interpretation, of which an important ingredient is the idea that the sentence is the minimal unit with which we can perform a linguistic act.
In my lecture, I will argue that the context principle -under a sufficiently interesting reading- is false. I claim that saying a word, under the right kind of conditions, is indeed a first class citizen of the realm of linguistic acts. I will illustrate my claim by looking at the meanings of definite and indefinite referring expressions. What you do when you say such an expression is, roughly, to pick up a discourse referent from the previous discourse, respectively to introduce a new discourse referent into the discourse. In the second half of my talk I will take up the question of the proper logico-semantical treatment of definite and indefinite referring expressions in the light of the previous discussion. My proposal is to use a form of dynamic semantics, in the tradition of Kamp's Discourse Represention Theory, of Heim's File Change Semantics and of Groenendijk and Stokhof's Dynamic Predicate Logic. The specific approach I propose is Context Modification Logic. |
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| Russell Wahl | Russell, Realism, Ramification and Reducibility | |
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I examine Russell's ramified theory of types, looking at the various recent attempts to give an interpretation to it, some of them realist, which treat the theory of types as applying to structured intensional entities, some nominalist which do not treat functions as entities at all. I argue on the basis of such papers as "The Paradoxes of the Liar", and "Fundamentals" (1907) that Russell intended a nominalist reading in that as he was developing the theory of types he thought of the ramified theory as applying to ways of collecting together assertions about individuals, and not as a theory of types of entities in the world. This reading is also supported by the informal semantics given in Chapter II of the Introduction.
However, this interpretation appears to conflict with Russell's own realism concerning universals. I argue that it is best to distinguish propositional functions from universals although this solution is not without its difficulties. At least one understanding of the nominalist reading will yield an interpretation on which the axiom of reducibility is false. The classic counterexamples involve either a nominalist reading or a strict realist reading holding that first level properties are logically independent. There is a way, though, of understanding propositional functions which results in the axiom being true on all interpretations. On a logical realist view of propositional functions as structured intensional entities, too, there is a reading which would yield the axiom of reducibility as a necessary truth. |
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| Kai Wehmeier | Russell's Paradox in Consistent Fragments of Frege's Grundgesetze | |
| Russell's paradox arises within the system of Frege's Grundgesetze through the interplay between the second-order comprehension principle and the infamous basic law V. By weakening either of these principles, consistent subsystems of Frege's theory can be obtained. We consider such theories as result from restricting the second-order comprehension schema. Interestingly, the reasoning underlying Russell's paradox can there be used to show, among other things, that there are objects that are not the extensions of any concept. | ||
| Alan Weir | In Defence of Naïve Set Theory | |
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In the first, negative part I argue that standard resolutions of the Russell paradox are self-refuting. Such solutions are explicitly or implicitly hierarchical and the general complaint is that in order to set up these hierarchies without falling into superparadox one must be outside the hierarchies, but there is no such vantage point; I look at a number of cases in a little more detail. The only alternative to embracing naïve set theory, I urge, is a highly unattractive theoretical nihilism.
In the second more positive part I analyse the antinomies as resulting from a combination of
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| Philip Welch | On Gupta-Belnap Revision Theories of Truth, Kripkean Fixed Points, And The Next Stable Set | |
We consider various concepts connected to the Revision Theory of truth of Belnap and Gupta. We show that their concept of the class of revision theoretic definitions over a structure M, yields the sets of the first stable set over M (for M countable acceptable). The sets of integers thus definable over the two rather different structures of
It is easy to ask questions of concerning sequences of their truth sets over the natural numbers which are independent of ZFC. For their notion of "fully varied" revision theory, membership of the stable truth set they define is itself highly non-absolute: it is Pi^1_3. We should argue that a theory of truth that requires non-ZFC-absolute truth sets as being somewhat problematic. We give:
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| Hugh Woodin | Set Theory after Russell; The journey back to Eden | |
| Is Russell's Paradox a prophecy that Set Theory is destined to fail? I shall discuss some developments in Set Theory which support the claim that the answer to this question is emphatically, "No". I shall focus first on Second Order Number Theory, in the guise of the Theory of the Projective Sets and then I shall focus the problem resolving the Continuum Hypothesis. I shall finish with some recent conjectures which if true show that one can in fact define precisely the (wellordered) hierarchy of Large Cardinal Axioms. |