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Advice on Abductive Logic
 Logic Journal of the IGPL
"... The action of thought is excited by the initiation of doubt and ceases when belief is attained; so that the production of belief is the sole function of thought. ..."
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The action of thought is excited by the initiation of doubt and ceases when belief is attained; so that the production of belief is the sole function of thought.
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
"... ..."
Transfer Principles in Set Theory
, 1997
"... CONTENTS PART A. HIGHLIGHTS. Introduction. A1. Two basic examples of transfer principles. A2. Some formal conjectures. A3. Sketch of some proofs. A4. Ramsey Cardinals. A5. Towards a new view of set theory. PART B. FULL LIST OF CLAIMS. (Based on 5/1996 abstract) 1. Transfer principles from N to On. ..."
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CONTENTS PART A. HIGHLIGHTS. Introduction. A1. Two basic examples of transfer principles. A2. Some formal conjectures. A3. Sketch of some proofs. A4. Ramsey Cardinals. A5. Towards a new view of set theory. PART B. FULL LIST OF CLAIMS. (Based on 5/1996 abstract) 1. Transfer principles from N to On. A. Mahlo cardinals. B. Weakly compact cardinals. C. Ineffable cardinals. D. Ramsey cardinals. E. Ineffably Ramsey cardinals. F. Subtle cardinals. G. From N to <On. H. Converses. 2. Transfer principles for general functions. A. Equivalence with Mahloness. B. Equivalence with weak compactness. C. Equivalence with ineffability. D. Equivalence with Ramseyness. E. Equivalence with ineffable Ramseyness. F. From N to <On. G. Converses. H. Some necessary conditions. 3. Transfer principles with arbitrary alternations of quantifiers. 4. Decidability of statements on N. 5. Decidability of statements on <On and On. NOTE: Talks are based on Part A only 2 PART A. HIGHLI
Independence and Large Cardinals
, 2010
"... The independence results in arithmetic and set theory led to a proliferation of mathematical systems. One very general way to investigate the space of possible mathematical systems is under the relation of interpretability. Under this relation the space of possible mathematical systems forms an intr ..."
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The independence results in arithmetic and set theory led to a proliferation of mathematical systems. One very general way to investigate the space of possible mathematical systems is under the relation of interpretability. Under this relation the space of possible mathematical systems forms an intricate hierarchy of increasingly strong systems. Large cardinal axioms provide a canonical means of climbing this hierarchy and they play a central role in comparing systems from conceptually distinct domains. This article is an introduction to independence, interpretability, large cardinals and their interrelations. Section 1 surveys the classic independence results in arithmetic and set theory. Section 2 introduces the interpretability hierarchy and describes some of its basic features. Section 3 introduces the notion of a large cardinal axiom and discusses some of the central examples. Section 4 brings together the previous themes by discussing the manner in which large cardinal axioms provide a canonical means for climbing the hierarchy of interpretability and serve as an intermediary in the comparison
MATHEMATICAL LOGIC: WHAT HAS IT DONE FOR THE PHILOSOPHY OF MATHEMATICS?
"... to discuss some claims concerning the relationship between mathematical logic and the philosophy of mathematics that repeatedly occur in his writings. Although I do not know to what extent they are representative of his present position, they correspond to widespread views of the logical community a ..."
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to discuss some claims concerning the relationship between mathematical logic and the philosophy of mathematics that repeatedly occur in his writings. Although I do not know to what extent they are representative of his present position, they correspond to widespread views of the logical community and so seem worth discussing anyhow. Such claims will be used as reference to make some remarks about the present state of relations between mathematical logic and the philosophy of mathematics. Kreisel’sviewsgreatlyinfluencedmeintheSixtiesandthe Seventies. His critical remarks on the foundational programs taught me that one could and should have an approach to the subject of mathematical logic less dogmatic, corporative and even thoughtless than the one the logical community sometimes used to have. This is even more true today when the professionalization of mathematical logic generates a flood of results but few new ideas and the lack of ideas leads to the sheer byzantinism of most current production in mathematical logic. In the past few years,however,IhavecometotheconclusionthatKreisel’scriticism hasnot been radical enough: his main worry seems to have been to preserve as much as possible to save the savable of the tradition of mathematical logic. His critical remarks have focused on the defects of the foundational schools, thus drawing attention away from the intrinsic defects of mathematical logic itself
Economic Dynamics and Computation Resurrecting the Icarus Tradition ∗
, 2003
"... "The [hydraulic] mechanism just described is the physical analogue of the ideal economic market. The elements which contribute to the determination of prices are represented each with its appropriate rôle and open to the scrutiny of the eye. We are thus enabled ..."
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"The [hydraulic] mechanism just described is the physical analogue of the ideal economic market. The elements which contribute to the determination of prices are represented each with its appropriate rôle and open to the scrutiny of the eye. We are thus enabled
London
"... My title embodies an ambiguity that I hope to make something of. In one sense, it suggests a thesis that Stephen Toulmin himself espouses or is committed to. In another, it suggests a thesis held by me, but occasioned, in whole or in part, by reflecting on Toulmin’s writings. Of course, the two sens ..."
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My title embodies an ambiguity that I hope to make something of. In one sense, it suggests a thesis that Stephen Toulmin himself espouses or is committed to. In another, it suggests a thesis held by me, but occasioned, in whole or in part, by reflecting on Toulmin’s writings. Of course, the two senses are not robustly disjoint. My principal purpose is to lend these theses some degree of favour, if not in every case theses of Toulmin’s own making, then perhaps of Toulmin’s example. Thesis one. The validity standard is nearly always the wrong standard for reallife reasoning. It is widely assumed that valid argument is nearly the best there is, improved upon only by argument that is sound. When made to note that actual reasoners hardly ever attain the validity standard, the received response is to make the best of a bad thing, insisting that, for beings like us, reasoning is best when it most closely approximates to the strict canons of deduction. Against this, cooler heads counsel that the validity standard is best only when a reasoner’s target is such as to call for it, as when, for example, one seeks a proof of a proposition of set theory. But even this is wrong. It is wrong in the sense that it fails to make clear how deeply the validity standard is
Kurt Gödel and Computability Theory
"... Abstract. Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the meth ..."
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Abstract. Although Kurt Gödel does not figure prominently in the history of computabilty theory, he exerted a significant influence on some of the founders of the field, both through his published work and through personal interaction. In particular, Gödel’s 1931 paper on incompleteness and the methods developed therein were important for the early development of recursive function theory and the lambda calculus at the hands of Church, Kleene, and Rosser. Church and his students studied Gödel 1931, and Gödel taught a seminar at Princeton in 1934. Seen in the historical context, Gödel was an important catalyst for the emergence of computability theory in the mid 1930s. 1
My fourty years on his shoulders
, 2007
"... Gödel's legacy is still very much in evidence. We will not attempt to properly discuss the full impact of his work and all of the ongoing important research programs that it suggests. This would require a book length manuscript. Indeed, there are several books discussing the Gödel legacy from m ..."
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Gödel's legacy is still very much in evidence. We will not attempt to properly discuss the full impact of his work and all of the ongoing important research programs that it suggests. This would require a book length manuscript. Indeed, there are several books discussing the Gödel legacy from many points of view, including, for example, [Wa87], [Wa96], [Da05], and the historically comprehensive five volume set [Go,8603]. In sections 27 we briefly discuss some research projects that are suggested by some of his most famous contributions. In sections 811 we discuss some highlights of a main recurrent theme in our own research, which amounts to an expansion of the Gödel incompleteness phenomenon in a critical direction.