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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
Foundational and mathematical uses of higher types
 REFLECTIONS ON THE FOUNDATIONS OF MATHEMATICS: ESSAY IN HONOR OF SOLOMON FEFERMAN
, 1999
"... In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles ..."
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In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles which generalize (and for n = 0 coincide with) the socalled `weak' König's lemma WKL (which has been studied extensively in the context of second order arithmetic) to logically more complex tree predicates. Whereas the second order context used in the program of reverse mathematics requires an encoding of higher analytical concepts like continuous functions F : X ! Y between Polish spaces X;Y , the more exible language of our systems allows to treat such objects directly. This is of relevance as the encoding of F used in reverse mathematics tacitly yields a constructively enriched notion of continuous functions which e.g. for F : IN ! IN can be seen (in our higher order context)
Does Mathematics Need New Axioms?
 American Mathematical Monthly
, 1999
"... this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called f ..."
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this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.
Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
Is the Continuum Hypothesis a definite mathematical problem?
"... [t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is gr ..."
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[t]he analysis of the phrase “how many ” unambiguously leads to a definite meaning for the question [“How many different sets of integers do their exist?”]: the problem is to find out which one of the א’s is the number of points of a straight line … Cantor, after having proved that this number is greater than א0, conjectured that it is א1. An equivalent proposition is this: any infinite subset of the continuum has the power either of the set of integers or of the whole continuum. This is Cantor’s continuum hypothesis. … But, although Cantor’s set theory has now had a development of more than sixty years and the [continuum] problem is evidently of great importance for it, nothing has been proved so far relative to the question of what the power of the continuum is or whether its subsets satisfy the condition just stated, except that … it is true for a certain infinitesimal fraction of these subsets, [namely] the analytic sets. Not even an upper bound, however high, can be assigned for the power of the continuum. It is undecided whether this number is regular or singular, accessible or inaccessible, and (except for König’s negative result) what its character of cofinality is. Gödel 1947, 516517 [in Gödel 1990, 178]
Foundations of Real Analysis and Computability Theory in NonAristotelian Finitary Logic
, 2005
"... This paper outlines new paradigms for real analysis and computability theory in the recently proposed nonAristotelian finitary logic (NAFL). Constructive real analysis in NAFL (NRA) is accomplished by a translation of diagrammatic concepts from Euclidean geometry into an extension (NPAR) of the NAF ..."
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This paper outlines new paradigms for real analysis and computability theory in the recently proposed nonAristotelian finitary logic (NAFL). Constructive real analysis in NAFL (NRA) is accomplished by a translation of diagrammatic concepts from Euclidean geometry into an extension (NPAR) of the NAFL version of Peano Arithmetic (NPA). Such a translation is possible because NPA proves the existence of every infinite proper class of natural numbers that is definable in the language of NPA. Infinite sets are not permitted in NPAR and quantification over proper classes is banned; hence Cantor’s diagonal argument cannot be legally formulated in NRA, and there is no ‘cardinality ’ for any collection (‘superclass’) of real numbers. Many of the useful aspects of classical real analysis, such as, the calculus of Newton and Leibniz, are justifiable in NRA. But the paradoxes, such as, Zeno’s paradoxes of motion and the BanachTarski paradox, are resolved because NRA admits only closed
ANALYSIS IN J2
, 2005
"... Abstract. This is an expository paper in which I explain how core mathematics, particularly abstract analysis, can be developed within a concrete countable set J2 (the second set in Jensen’s constructible hierarchy). The implication, wellknown to proof theorists but probably not to most mainstream ..."
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Abstract. This is an expository paper in which I explain how core mathematics, particularly abstract analysis, can be developed within a concrete countable set J2 (the second set in Jensen’s constructible hierarchy). The implication, wellknown to proof theorists but probably not to most mainstream mathematicians, is that ordinary mathematical practice does not require an enigmatic metaphysical universe of sets. I go further and argue that J2 is a superior setting for normal mathematics because it is free of irrelevant settheoretic pathologies and permits stronger formulations of existence results. Perhaps many mathematicians would admit to harboring some feelings of discomfort about the ethereal quality of Cantorian set theory. Yet draconian alternatives such as intuitionism, which holds that simple numbertheoretic statements like the twin primes conjecture may have no definite truth value, probably violate the typical working mathematician’s intuition far more severely than any vague unease he may feel about remote cardinals such as, say, ℵℵω. I believe that ordinary mathematical practice is actually most compatible with
On the notion of object A logical genealogy∗
"... Abstract. We argue that logic is not a uniform terrain where all truths lie on a par. We analyze the apparatus of firstorder classical logic with identity into two main ingredients: a deeper and wider component and, on top of it, a narrower component which consists of principles that articulate our ..."
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Abstract. We argue that logic is not a uniform terrain where all truths lie on a par. We analyze the apparatus of firstorder classical logic with identity into two main ingredients: a deeper and wider component and, on top of it, a narrower component which consists of principles that articulate our modern notion of object.
On the calculating power of Laplace’s demon (Part I)
, 2006
"... We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a s ..."
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We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a start on our programme by proposing a general logical framework for describing physical theories, and analysing several possible formulations of a simple Newtonian theory from the point of view of determinism. Our emphasis throughout is on clarifying the precise physical and metaphysical assumptions that typically underlie a claim that some physical theory is ‘deterministic’. A sequel paper is planned, in which we shall apply similar methods to the analysis of other physical theories. Along the way, we discuss some possible repercussions of this kind of investigation for both physics and logic. 1
Justin ClarkeDoane Monash University [Note: This is the penultimate draft of a paper that is forthcoming in Noûs.] Moral Epistemology: The Mathematics Analogy *
"... There is a long tradition of comparing moral knowledge to mathematical knowledge. Plato compared mathematical knowledge to knowledge of the Good. 1 In recent years, metaethicists have found the comparison to be illuminating. 2 Sometimes the comparison is supposed to show that moral realism is peculi ..."
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There is a long tradition of comparing moral knowledge to mathematical knowledge. Plato compared mathematical knowledge to knowledge of the Good. 1 In recent years, metaethicists have found the comparison to be illuminating. 2 Sometimes the comparison is supposed to show that moral realism is peculiarly problematic. For example, James Rachels writes, “[H]ow do we know moral facts?....In mathematics there are proofs….But moral facts are not accessible by…these familiar methods [1998, p. 3].” But other times the comparison is supposed to show that moral realism is no more problematic than mathematical realism. For example, Hilary Putnam writes, “[A]rguments for “antirealism ” in ethics are virtually identical with arguments for antirealism in the philosophy of mathematics; yet philosophers who resist those arguments in the latter case often capitulate in the former [2004, p. 1].”