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21
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.
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.
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.”
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
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
Computer Theorem Proving in Math
"... We give an overview of issues surrounding computerverified theorem proving in the standard puremathematical context. ..."
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We give an overview of issues surrounding computerverified theorem proving in the standard puremathematical context.
Brief introduction to unprovability
"... Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. ..."
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Abstract The article starts with a brief survey of Unprovability Theory as of autumn 2006. Then, as an illustration of the subject's modeltheoretic methods, we reprove exact versions of unprovability results for the ParisHarrington Principle and the KanamoriMcAloon Principle using indiscernibles. In addition, we obtain a short accessible proof of unprovability of the ParisHarrington Principle. The proof employs old ideas but uses only one colouring and directly extracts the set of indiscernibles from its homogeneous set. We also present modified, abridged statements whose unprovability proofs are especially simple. These proofs were tailored for teaching purposes. The article is intended to be accessible to the widest possible audience of mathematicians, philosophers and computer scientists as a brief survey of the subject, a guide through the literature in the field, an introduction to its modeltheoretic techniques and, finally, a modeltheoretic proof of a modern theorem in the subject. However, some understanding of logic is assumed on the part of the readers. The intended audience of this paper consists of logicians, logicaware mathematicians andthinkers of other backgrounds who are interested in unprovable mathematical statements.
1 LECTURE NOTES ON ENORMOUS INTEGERS by
, 2001
"... Abstract. We discuss enormous integers and rates of growth after [PH77]. This breakthrough was based on a variant of the classical finite Ramsey theorem. Since then, examples have been given of greater relevance to a number of standard mathematical and computer science contexts, often involving even ..."
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Abstract. We discuss enormous integers and rates of growth after [PH77]. This breakthrough was based on a variant of the classical finite Ramsey theorem. Since then, examples have been given of greater relevance to a number of standard mathematical and computer science contexts, often involving even more enormous integers and rates of growth. 1.F(x 1,...,x k) = F(x 2,...,x k+1) N = the nonnegative integers. THEOREM 1.Let F:N k Æ{1,...,r}. There exists x 1 <... < x k+1 such that F(x 1,...,x k) = F(x 2,...,x k+1). This is an immediate consequence of a more general combinatorial theorem called Ramsey’s theorem, but it is much simpler to state. We call this adjacent Ramsey theory. There are inherent finite estimates here. THEOREM 1.2. For all k,r there exists t such that the following holds. Let F:N k Æ {1,...,r}. There exists x 1 <... < x k+1 £ t such that F(x 1,...,x k) = F(x 2,...,x k+1). QUESTION: What is the least such t = Adj(k,r)? THEOREM 1.3. Adj(k,1) = k. Adj(k,2) = 2k. THEOREM 1.4. Let k ≥ 5. Adj(k,3) is greater than an exponential stack of k2 1.5’s topped off with k1. E.g., Adj(6,3)> 10 173, Adj(7,3)> 10^10 172. THEOREM 1.5. Adj(k,r) is at most an exponential stack of k1 2’s topped off with a reasonable function of k and r. Our adjacent Ramsey theory from the 80’s is lurking in the background in [DLR95]. 2. THE ACKERMANN HIERARCHY There is a good notation for really big numbers up to a point. We use a streamlined version of the Ackerman hierarchy. 2 Let f:Z + Æ Z + be strictly increasing. We define the critical function f’:Z + Æ Z + of f by: f’(n) = the result of applying f n times at 1. Define f 1:Z + Æ Z + to be the doubling function, and f n+1:Z + Æ Z + be f n’. Thus f 1 is doubling, f 2 is exponentiation, f 3 is iterated exponentiation; i.e., f 3(n) = E*(n) = an exponential stack of n 2’s. f 4 is confusing. We can equivalently present this by the recursion equations f 1(n) = 2n, f k+1(1) = f k(1), f k+1(n+1) = f k(f k+1(n)), where k,n = 1. We define A(k,n) = f k(n). Note that A(k,1) = 2, A(k,2) = 4. For k ≥ 3, A(k,3)> A(k2,k2), and as a function of k, eventually strictly dominates each f n, n ≥ 1. A(3,4) = 65,536. A(4,3) = 65,536. A(4,4) = E*(65,536). And