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Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
Set theory after Russell: the journey back to Eden. In One hundred years of Russell’s paradox, volume 6 of de Gruyter
, 2004
"... Does the phenomenon of formal independence in Set Theory fulfill the prophecy some might claim is the content of Russell’s discovery of the now famous Russell Paradox? This claim of course is that there can be no meaningful axiomatization of Set Theory because the concept of set is inherently vague, ..."
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Does the phenomenon of formal independence in Set Theory fulfill the prophecy some might claim is the content of Russell’s discovery of the now famous Russell Paradox? This claim of course is that there can be no meaningful axiomatization of Set Theory because the concept of set is inherently vague, moreover any choice of axioms is an
On the question of absolute undecidability
 Philosophia Mathematica
"... The incompleteness theorems show that for every sufficiently strong consistent formal system of mathematics there are mathematical statements undecided relative to the system.1 A natural and intriguing question is whether there are mathematical statements that are in some sense absolutely undecid ..."
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The incompleteness theorems show that for every sufficiently strong consistent formal system of mathematics there are mathematical statements undecided relative to the system.1 A natural and intriguing question is whether there are mathematical statements that are in some sense absolutely undecidable, that is, undecidable relative to any set of axioms that are justified. Gödel was quick to point out that his original incompleteness theorems did not produce instances of absolute undecidability and hence did not undermine Hilbert’s conviction that for every precisely formulated mathematical question there is a definite and discoverable answer. However, in his subsequent work in set theory, Gödel uncovered what he initially regarded as a plausible candidate for an absolutely undecidable statement. Furthermore, he expressed the hope that one might actually prove this. Eventually he came to reject this view and, moving to the other extreme, expressed the ∗I am indebted to John Steel and Hugh Woodin for introducing me to the subject and sharing their insights into Gödel’s program. I am also indebted to Charles Parsons for his work on Gödel, in particular, his 1995. I would like to thank Andrés Caicedo and Penelope Maddy for extensive and very helpful comments and suggestions. I would
Does Reductive Proof Theory Have A Viable Rationale?
 Erkenntnis
, 2000
"... The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly "universal " system or founda ..."
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The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly "universal " system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more justified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistencyproof rationale and the foundational reduction rationale. However, recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foundational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper, various reducti...
Mathematical Intuition vs. Mathematical Monsters
, 1998
"... Geometrical and physical intuition, both untutored and cultivated, is ubiquitous in the research, teaching, and development of mathematics. A number of mathematical “monsters”, or pathological objects, have been produced which⎯according to some mathematicians⎯seriously challenge the reliability of ..."
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Geometrical and physical intuition, both untutored and cultivated, is ubiquitous in the research, teaching, and development of mathematics. A number of mathematical “monsters”, or pathological objects, have been produced which⎯according to some mathematicians⎯seriously challenge the reliability of intuition. We examine several famous geometrical, topological and settheoretical examples of such monsters in order to see to what extent, if at all, intuition is undermined in its everyday roles.
Beyond the axioms: The question of objectivity in mathematics
"... I will be discussing the axiomatic conception of mathematics, the modern version of which is clearly due to Hilbert... ..."
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I will be discussing the axiomatic conception of mathematics, the modern version of which is clearly due to Hilbert...
Tarski’s Intuitive Notion of Set
"... Abstract. Tarski did research on set theory and also used set theory in many of his emblematic writings. Yet his notion of set from the philosophical viewpoint was almost unknown. By studying mostly the posthumously published evidence, his still unpublished materials, and the testimonies of some of ..."
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Abstract. Tarski did research on set theory and also used set theory in many of his emblematic writings. Yet his notion of set from the philosophical viewpoint was almost unknown. By studying mostly the posthumously published evidence, his still unpublished materials, and the testimonies of some of his collaborators, I try to offer here a first, global picture of that intuitive notion, together with a philosophical interpretation of it. This is made by using several notions of universal languages as framework, and by taking into consideration the evolution of Tarski’s thoughts about set theory and its relationship with logic and mathematics. As a result, his difficulties to reconcile nominalism and methodological Platonism are precisely located, described and much better understood. “I represent this very rude kind of antiPlatonism, one thing which I could describe as materialism, or nominalism with some materialistic taint, and it is very difficult for a man to live his whole life with this philosophical attitude, especially if he is a mathematician, especially if for some reasons he has a hobby which is called set theory, and worse –very difficult” (Tarski, Chicago, 1965) Tarski made important contributions to set theory, especially in the first years of his long and highly productive career. Also, it is usually accepted that set theory was the main instrument used by Tarski in his most significant contributions which had philosophical implications and presuppositions. In this connection we may mention the four definitions which are usually cited as supplying some sort of “conceptual analysis”, both methodologically (the first one) and from the point of view of the results obtained (the rest): (i) definable sets of real numbers; (ii) truth; (iii) logical consequence and (iv) logical notions. So we could reasonably conclude that for Tarski set theory was reliable as a working instrument, then presumably as a conceptual ground. As we shall see, there are some signs that the reason for this preference might have been its simple ontological structure as a theory, at least excluding the upper levels, the highest infinite. Nevertheless, very little was known about Tarski’s conception of set theory from the philosophical viewpoint, apart from some comments he made to his closest friends and collaborators, and the conjectures which could perhaps be
Absoluteness, truth, and quotients
"... The infinite in mathematics has two manifestations. Its occurrence in analysis has been satisfactorily formalized and demystified by the δ method of Bolzano, Cauchy and Weierstrass. It is of course the ‘settheoretic infinite ’ that concerns me here. Once the existence of an infinite set is accept ..."
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The infinite in mathematics has two manifestations. Its occurrence in analysis has been satisfactorily formalized and demystified by the δ method of Bolzano, Cauchy and Weierstrass. It is of course the ‘settheoretic infinite ’ that concerns me here. Once the existence of an infinite set is accepted, the axioms of set theory imply the existence of a transfinite hierarchy of larger and larger orders of infinity. I shall review some wellknown facts about the influence of these axioms of infinity ([28]) to the everyday mathematical practice and point out to some, as of yet not understood, phenomena at the level of the thirdorder arithmetic. Technical details from both set theory and operator algebras are kept at the bare minimum. In the Appendix I include definitions of arithmetical and analytical hierarchies in order to make this paper more accessible to nonlogicians. In this paper I am taking a position intermediate between pluralism and nonpluralism (as defined in [34]) with an eye for applications outside of set theory. Acknowledgments This paper is partly based on my talks at the ‘Truth and Infinity ’ workshop (IMS, 2011) and the ‘Connes Embedding Problem ’ workshop (Ottawa, 2008). I would like to thank the organizers of both meetings. Another driving force for this paper—and much of my work—originated in conversations with functional analysts, too numerous to list here, over the past several
Nonconstructive and Deciding the Undecidable
, 2007
"... Provided by the author(s) and NUI Galway in accordance with publisher policies. Please cite the published ..."
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Provided by the author(s) and NUI Galway in accordance with publisher policies. Please cite the published