“Small” large cardinal notions in the language of ZFC are those large cardinal notions that are consistent with V = L. Besides their original formulation in classical set theory, we have a variety of analogue notions in systems of admissible set theory, admissible recursion theory, constructive set theory, constructive type theory, explicit mathematics and recursive ordinal notations (as used in proof theory). On the face of it, it is surprising that such distinctively set-theoretical notions have analogues in such disaparate and (...) relatively constructive contexts. There must be an underlying reason why that is possible (and, incidentally, why “large” large cardinal notions have not led to comparable analogues). My long term aim is to develop a common language in which such notions can be expressed and can be interpreted both in their original classical form and in their analogue form in each of these special constructive and semi-constructive cases. This is a program in progress. What is done here, to begin with, is to show how that can be done to a considerable extent in the settings of classical and admissible set theory (and thence, admissible recursion theory). The approach taken here is to expand the language of set theory to allow us to talk about (possibly partial) operations applicable both to sets and to operations and to formulate the large cardinal notions in question in terms of operational closure conditions; at the same time only minimal existence axioms are posited for sets. The resulting system, called Operational Set Theory, is a partial adaptation to the set-theoretical framework of the explicit mathematics framework Feferman (1975). The.. (shrink)

In the sole extended break from his life and varing in this way we can associate a sysied career in England, Alan Turing spent the tem of logic with any constructive ordinal. It may be asked whether such a years 1936–1938 doing graduate work at..

In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantor elaborated the nature of the infinite, and in doing so helped transform the face of twentieth-century mathematics. Feferman details the development of Cantorian concepts and the foundational difficulties they engendered. He argues (...) that the freedom provided by Cantorian set theory was purchased at a heavy philosophical price, namely adherence to a form of mathematical platonism that is difficult to support. -/- Beginning with a previously unpublished lecture for a general audience, Deciding the Undecidable, Feferman examines the famous list of twenty-three mathematical problems posed by David Hilbert, concentrating on three problems that have most to do with logic. Other chapters are devoted to the work and thought of Kurt Gödel, whose stunning results in the 1930s on the incompleteness of formal systems and the consistency of Cantors continuum hypothesis have been of utmost importance to all subsequent work in logic. Though Gödel has been identified as the leading defender of set-theoretical platonism, surprisingly even he at one point regarded it as unacceptable. -/- In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part--if not all--of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, Is Cantor Necessary?, is answered with a resounding no. -/- This volume of important and influential work by one of the leading figures in logic and the foundations of mathematics is essential reading for anyone interested in these subjects. (shrink)

Ambiguity is a property of syntactic expressions which is ubiquitous in all informal 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 (“eating it too”). The paper begins with a brief tour of Russell’s uses of typical ambiguity, including his treatment of the statement Cls ∈ Cls. This is generalized to a treatment in simple type theory of statements of the form A ∈ B where A and B are class expressions for which A is prima facie of the same or higher type than B. In order to treat mathematically more interesting statements of self membership we then formulate a version of typical ambiguity for such statements in an extension of Zermelo-Fraenkel set theory. Specific attention is given to how the“naive” theory of categories can thereby be accounted for. (shrink)

Both the constructive and predicative approaches to mathematics arose during the period of what was felt to be a foundational crisis in the early part of this century. Each critiqued an essential logical aspect of classical mathematics, namely concerning the unrestricted use of the law of excluded middle on the one hand, and of apparently circular \impredicative" de nitions on the other. But the positive redevelopment of mathematics along constructive, resp. predicative grounds did not emerge as really viable alternatives to (...) classical, set-theoretically based mathematics until the 1960s. Now we have a massive amount of information, to which this lecture will constitute an introduction, about what can be done by what means, and about the theoretical interrelationships between various formal systems for constructive, predicative and classical analysis. (shrink)

Feferman, S. and G. Jäger, Systems of explicit mathematics with non-constructive μ-operator. Part I, Annals of Pure and Applied Logic 65 243-263. This paper is mainly concerned with the proof-theoretic analysis of systems of explicit mathematics with a non-constructive minimum operator. We start off from a basic theory BON of operators and numbers and add some principles of set and formula induction on the natural numbers as well as axioms for μ. The principal results then state: BON plus set (...) induction is proof-theoretically equivalent to Peano arithmetic PA; BON plus formula induction is proof-theoretically equivalent to the system <0 of second-order arithmetic. (shrink)

The paper starts with an examination and critique of Tarski’s wellknown proposed explication of the notion of logical operation in the type structure over a given domain of individuals as one which is invariant with respect to arbitrary permutations of the domain. The class of such operations has been characterized by McGee as exactly those definable in the language L∞,∞. Also characterized similarly is a natural generalization of Tarski’s thesis, due to Sher, in terms of bijections between domains. My main (...) objections are that on the one hand, the Tarski-Sher thesis thus assimilates logic to mathematics, and on the other hand fails to explain the notion of same logical operation across domains of different sizes. A new notion of homomorphism invariant operation over functional type structures (with domains M0 of individuals and {T, F} at their base) is introduced to accomplish the latter. The main result is that an operation is definable from.. (shrink)

Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, and there (...) are quite a few highly technical journals in logic, such as The Journal of Sym-. (shrink)

A notion of finitary inductively presented (f.i.p.) logic is proposed here, which includes all syntactically described logics (formal systems)met in practice. A f.i.p. theory FS0 is set up which is universal for all f.i.p. logics; though formulated as a theory of functions and classes of expressions, FS0 is a conservative extension of PRA. The aims of this work are (i)conceptual, (ii)pedagogical and (iii)practical. The system FS0 serves under (i)and (ii)as a theoretical framework for the formalization of metamathematics. The general approach (...) may be used under (iii)for the computer implementation of logics. In all cases, the work aims to make the details manageable in a natural and direct way. (shrink)

This is a survey of work on set-theoretical invariance criteria for logicality. It begins with a review of the Tarski-Sher thesis in terms, first, of permutation invariance over a given domain and then of isomorphism invariance across domains, both characterized by McGee in terms of definability in the language L∞,∞. It continues with a review of critiques of the Tarski-Sher thesis, and a proposal in response to one of those critiques via homomorphism invariance. That has quite divergent characterization results depending (...) on its formulation, one in terms of FOL, the other by Bonnay in terms of L∞,∞, both without equality. From that we move on to a survey of Bonnay’s work on similarity relations between structures and his results that single out invariance with respect to potential isomorphism among all such. Turning to the critique that calls for sameness of meaning of a logical operation across domains, the paper continues with a result showing that the isomorphism invariant operations that are absolutely definable with respect to KPU−Inf are exactly those definable in full FOL; this makes use of an old theorem of Manders. The concluding section is devoted to a critical discussion of the arguments for set-theoretical criteria for logicality. (shrink)

From 1931 until late in his life (at least 1970) Godel called for the pursuit of new axioms for mathematics to settle both undecided number-theoretical propositions (of the form obtained in his incompleteness results) and undecided set-theoretical propositions (in particular CH). As to the nature of these, Godel made a variety of suggestions, but most frequently he emphasized the route of introducing ever higher axioms of in nity. In particular, he speculated (in his 1946 Princeton remarks) that there might be (...) a uniform (though non-decidable) rationale for the choice of the latter. Despite the intense exploration of the \higher in nite" in the last 30-odd years, no single rationale of that character has emerged. Moreover, CH still remains undecided by such axioms, though they have been demonstrated to have many other interesting set-theoretical consequences. (shrink)

elaboration of the last part of my Tarski Lecture, “Truth unbound”, UC Berkeley, 3 April 2006, and of the lecture, “A nicer formal theory of non-hierarchical truth”, Workshop on Mathematical Methods in Philosophy, Banff , 18-23 Feb. 2007.

I will talk here about three problems that have bothered me for a number of years, during which time I have experimented with a variety of solutions and encouraged others to work on them. I have raised each of them separately both in full and in passing in various contexts, but thought it would be worthwhile on this occasion to bring them to your attention side by side. In this talk I will explain the problems, together with some things that (...) have been tried in the past and some new ideas for their solution. (shrink)

This paper presents examples of infinite diagrams whose use is more or less essential for understanding and accepting various proofs in higher mathematics. The significance of these is discussed with respect to the thesis that every proof can be formalized, and a “pre” form of this thesis that every proof can be presented in everyday statements-only form.

(Click here for the papers published between 1996 and 2000, and here for more recent papers.) V. Lifschitz, On the semantics of STRIPS ," in: Reasoning about Actions and Plans , 1987, pp. 1-9. M. Gelfond and V. Lifschitz, The stable model semantics for logic programming ," in Logic Programming: Proceedings of the Fifth International Conference and Symposium , 1988, pp. 1070-1080.

This paper is mainly concerned with proof-theoretic analysis of some second-order systems of explicit mathematics with a non-constructive minimum operator. By introducing axioms for variable types we extend our first-order theory BON to the elementary explicit type theory EET and add several forms of induction as well as axioms for μ. The principal results then state: EET plus set induction is proof-theoretically equivalent to Peano arithmetic PA <0).

Does science justify any part of mathematics and, if so, what part? These questions are related to the so-called indispensability arguments propounded, among others, by Quine and Putnam; moreover, both were led to accept significant portions of set theory on that basis. However, set theory rests on a strong form of Platonic realism which has been variously criticized as a foundation of mathematics and is at odds with scientific realism. Recent logical results show that it is possible to directly formalize (...) almost all, if not all, scientifically applicable mathematics in a formal system that is justified simply by Peano Arithmetic (via a proof-theoretical reduction). It is argued that this substantially vitiates the indispensability arguments. (shrink)

Following a discussion of various forms of set-theoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unlimited” or “naive” category theory. The author proposed four criteria for such some years ago. The article describes how much had previously been accomplished on one approach to meeting those criteria, then takes care of one important obstacle that had (...) been met in that approach, and finally explains what remains to be done if one is to have a fully satisfactory solution. (shrink)

• Session on CF&FCT proposed by E. Landry; participants: G. Hellman, E. Landry, J.-P. Marquis and C. McLarty..

2. Various philosophical and semantical theories are candidates for axiomatization (but not all, e.g. coherence, pragmatic, fuzzy theories). NB: axiomatizations are not uniquely determined.

Ernest Nagel Lecture, Columbia University, Sept. 27, 2007.

The most prominent “schools” or programs for the foundations of mathematics that took shape in the first third of the 20th century emerged directly from, or in response to, developments in mathematics and logic in the latter part of the 19th century. The first of these programs, so-called logicism, had as its aim the reduction of mathematics to purely logical principles. In order to understand properly its achievements and resulting problems, it is necessary to review the background from that previous (...) period. (shrink)

The unfolding of schematic formal systems is a novel concept which was initiated in Feferman , Gödel ’96, Lecture Notes in Logic, Springer, Berlin, 1996, pp. 3–22). This paper is mainly concerned with the proof-theoretic analysis of various unfolding systems for non-finitist arithmetic . In particular, we examine two restricted unfoldings and , as well as a full unfolding, . The principal results then state: is equivalent to ; is equivalent to ; is equivalent to . Thus is proof-theoretically (...) equivalent to predicative analysis. (shrink)

This book brings together several directions of work in model theory between the late 1950s and early 1980s.

(Click here for selected papers published before 1996, and here for papers published after 2000.) V. Lifschitz, Foundations of logic programming ," in Principles of Knowledge Representation , CSLI Publications, 1996, pp. 69-127. E. Giunchiglia, N. Kartha and V. Lifschitz, Representing action: indeterminacy and ramifications ," Artificial Intelligence , Vol. 95, 1997, pp. 409-443. V. Lifschitz, On the logic of causal explanation ," Artificial Intelligence , Vol. 96, 1997, pp. 451-465. V. Lifschitz, Two components of an (...) action language ," Annals of Mathematics and Artificial Intelligence , Vol. 21, 1997, pp. 305-320. V. Lifschitz, Situation calculus and causal logic ," in Proceedings of the Sixth International Conference on Principles of Knowledge Representation and Reasoning, 1998, pp. 536-646. (shrink)

In addition to this being the centenary of Kurt Gödel’s birth, January marked 75 years since the publication (1931) of his stunning incompleteness theorems. Though widely known in one form or another by practicing mathematicians, and generally thought to say something fundamental about the limits and potentialities of mathematical knowledge, the actual importance of these results for mathematics is little understood. Nor is this an isolated example among famous results. For example, not long ago, Philip Davis wrote me about what (...) he calls The Paradox of Irrelevance: “There are many math problems that have achieved the cachet of tremendous significance, e.g. Fermat, 4 color, Kepler’s packing, Gödel, etc. Of Fermat, I have read: ‘the most famous math problem of all time.’ Of Gödel, I have read: ‘the most mathematically significant achievement of the 20th century.’ … Yet, these problems have engaged the attention of relatively few research mathematicians—even in pure math.” What accounts for this disconnect between fame and relevance? Before going into the question for Gödel’s theorems, it should be distinguished in one respect from the other examples mentioned, which in any case form quite a mixed bag. Namely, each of the Fermat, 4 color, and Kepler’s packing problems posed a stand-out challenge following extended efforts to settle them; meeting the challenge in each case required new ideas or approaches and intense work, obviously of different degrees. By contrast, Gödel’s theorems were simply unexpected, and their proofs, though requiring novel techniques, were not difficult on the scale of things. Setting that aside, my view of Gödel’s incompleteness theorems is that their relevance to mathematical logic (and its offspring in the theory of computation) is paramount; further, their philosophical relevance is significant, but in just what way is far from settled; and finally, their mathematical relevance outside of logic is very much unsubstantiated but is the object of ongoing, tantalizing efforts.. (shrink)

The concept of the (full) unfolding of a schematic system is used to answer the following question: Which operations and predicates, and which principles concerning them, ought to be accepted if one has accepted ? The program to determine for various systems of foundational significance was previously carried out for a system of nonfinitist arithmetic, ; it was shown that is proof-theoretically equivalent to predicative analysis. In the present paper we work out the unfolding notions for a basic schematic system (...) of finitist arithmetic, , and for an extension of that by a form of the so-called Bar Rule. It is shown that and are proof-theoretically equivalent, respectively, to Primitive Recursive Arithmetic, , and to Peano Arithmetic,. (shrink)

What Gödel accomplished in the decade of the 1930s before joining the Institute changed the face of mathematical logic and continues to influence its development. As you gather from my title, I’ll be talking about the most famous of his results in that period, but first I want to indulge in some personal reminiscences. In many ways this is a sentimental journey for me. I was a member of the Institute in 1959-60, a couple of years after receiving my PhD (...) at the University of California in Berkeley, where I had worked with Alfred Tarski, another great logician. The subject of my dissertation was directly concerned with the method of arithmetization that Gödel had used to prove his theorems, and my main concern after that was to study systematic ways of overcoming incompleteness. Mathematical logic was going through a period of prodigious development in the 1950s and 1960s, and Berkeley and Princeton were two meccas for researchers in that field. For me, the prospect of meeting with Gödel and drawing on him for guidance and inspiration was particularly exciting. I didn’t know at the time what it took to get invited. Hassler Whitney commented for an obituary notice in 1978 that “it was hard to appoint a new member in logic at the Institute because Gödel could not prove to himself that a number of candidates shouldn’t be members, with the evidence at hand.” That makes it sound like the problem for Gödel was deciding who not to invite. Anyhow, I ended up being one of the lucky few. (shrink)

The determinism-free will debate is perhaps as old as philosophy itself and has been engaged in from a great variety of points of view including those of scientific, theological, and logical character. This chapter focuses on two arguments from logic. First, there is an argument in support of determinism that dates back to Aristotle, if not farther. It rests on acceptance of the Law of Excluded Middle, according to which every proposition is either true or false, no matter whether the (...) proposition is about the past, present or future. In particular, the argument goes, whatever one does or does not do in the future is determined in the present by the truth or falsity of the corresponding proposition. The second argument coming from logic is much more modern and appeals to Gödel's incompleteness theorems to make the case against determinism and in favour of free will, insofar as that applies to the mathematical potentialities of human beings. The claim more precisely is that as a consequence of the incompleteness theorems, those potentialities cannot be exactly circumscribed by the output of any computing machine even allowing unlimited time and space for its work. The chapter concludes with some new considerations that may be in favour of a partial mechanist account of the mathematical mind. (shrink)

What is predicativity? While the term suggests that there is a single idea involved, what the history will show is that there are a number of ideas of predicativity which may lead to different logical analyses, and I shall uncover these only gradually. A central question will then be what, if anything, unifies them. Though early discussions are often muddy on the concepts and their employment, in a number of important respects they set the stage for the further developments, and (...) so I shall give them special attention. NB. Ahistorically, modern logical and set-theoretical notation will be used throughout, as long as it does not conflict with original intentions. (shrink)

applied, for instance, to developing a decision support system for the Space Shuttle INogueira et al., 2001] and to graph-theoretic problems arising in..

A theorem obtained by van Benthem for preservation of formulas under Chu transforms between Chu spaces is strengthened and derived from a general many-sorted interpolation theorem. The latter has been established both by proof-theoretic and model-theoretic methods; there is some discussion as to how these methods compare and what languages they apply to. In the conclusion, several further questions are raised.

In a recent paper, Ferraris, Lee and Lifschitz conjectured that the concept of a stable model of a first-order formula can be used to treat some answer set programming expressions as abbreviations. We follow up on that suggestion and introduce an answer set programming language that defines the mean- ing of counting and choice by reducing these constructs to first-order formulas. For the new language, the concept of a safe program is defined, and its semantic role is investigated. We (...) compare the new language with the concept of a disjunc- tive program with aggregates introduced by Faber, Leone and Pfeifer, and discuss the possibility of implementing a frag- ment of the language by translating it into the input language of the answer set solver DLV. The language is also compared with cardinality constraint programs defined by Syrjänen. (shrink)

When I was a teenager growing up in Los Angeles in the early 1940s, my dream was to become a mathematical physicist: I was fascinated by the ideas of relativity theory and quantum mechanics, and I read popular expositions which, in those days, besides Einstein’s The Meaning of Relativity, was limited to books by the likes of Arthur S. Eddington and James Jeans. I breezed through the high-school mathematics courses (calculus was not then on offer, and my teachers barely understood (...) it), but did less well in physics, which I should have taken as a reality check. On the philosophical side I read a mixed bag of Bertrand Russell, John Dewey and Alfred Korzybski (the missionary for “General Semantics” in Science and Sanity, a mish-mash of the theory of types, non-. (shrink)

The concept of a temporal phylogenetic network is a mathematical model of evolution of a family of natural languages. It takes into account the fact that languages can trade their characteristics with each other when linguistic communities are in contact, and also that a contact is only possible when the languages are spoken at the same time. We show how computational methods of answer set programming and constraint logic programming can be used to generate plausible conjectures about contacts between prehistoric (...) linguistic communities, and illustrate our approach by applying it to the evolutionary history of Indo-European languages. (shrink)

in Handbook of Knowledge representation, Elsevier, 2008.