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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.
The metamathematics of ergodic theory
 THE ANNALS OF PURE AND APPLIED LOGIC
, 2009
"... The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theo ..."
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The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns and recent applications to combinatorics and number theory force us to reconsider the constructive character of the theory and its methods. This paper surveys some recent contributions to the metamathematical study of ergodic theory, focusing on the mean and pointwise ergodic theorems and the Furstenberg structure theorem for measure preserving systems. In particular, I characterize the extent to which these theorems are nonconstructive, and explain how prooftheoretic methods can be used to locate their “constructive content.”
Prospects for mathematical logic in the twentyfirst century
 BULLETIN OF SYMBOLIC LOGIC
, 2002
"... The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently. ..."
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The four authors present their speculations about the future developments of mathematical logic in the twentyfirst century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
Finite Trees And The Necessary Use Of Large Cardinals
, 1998
"... this paper, a tree T = (V,) is a partial ordering with a minimum element, where V is finite, and the ancestors of any x V are linearly ordered under . The minimum element of T is called the root of T, and is written r(T). A tree is said to be trivial if and only if it has exactly one vertex, which m ..."
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this paper, a tree T = (V,) is a partial ordering with a minimum element, where V is finite, and the ancestors of any x V are linearly ordered under . The minimum element of T is called the root of T, and is written r(T). A tree is said to be trivial if and only if it has exactly one vertex, which must be its root. V = V(T) represents the set of all vertices of the tree T = (V,). In a tree T, if x < y and for no z is x < z < y, then we say that y is a child of x and x is the parent of y. Every vertex has at most one parent. However, vertices may have zero or more children. We write p(x,T) for the parent of x in T. We use Ch(T) = V(T)\{r(T)} for the set of all children of T. We write T 1 T 2 if and only if i) r(T 1 ) = r(T 2 ); ii) for all x Ch(T 1 ), p(x,T 1 ) = p(x,T 2 ). This is a partial ordering on trees. Note that if T 1 T 2
COMPLEXITY CLASSES AS MATHEMATICAL AXIOMS
, 810
"... Abstract. Treating a conjecture, P #P ̸ = NP, on the separation of complexity classes as an axiom, an implication is found in three manifold topology with little obvious connection to complexity theory. This is reminiscent of Harvey Friedman’s work on finitistic interpretations of large cardinal axi ..."
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Abstract. Treating a conjecture, P #P ̸ = NP, on the separation of complexity classes as an axiom, an implication is found in three manifold topology with little obvious connection to complexity theory. This is reminiscent of Harvey Friedman’s work on finitistic interpretations of large cardinal axioms. 1.
A DEFENCE OF MATHEMATICAL PLURALISM
, 2004
"... We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context. ..."
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We approach the philosophy of mathematics via a discussion of the differences between classical mathematics and constructive mathematics, arguing that each is a valid activity within its own context.
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