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24
Local stability of ergodic averages
 Transactions of the American Mathematical Society
"... We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesguemeasure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages An ..."
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We consider the extent to which one can compute bounds on the rate of convergence of a sequence of ergodic averages. It is not difficult to construct an example of a computable Lebesguemeasure preserving transformation of [0, 1] and a characteristic function f = χA such that the ergodic averages Anf do not converge to a computable element of L2([0,1]). In particular, there is no computable bound on the rate of convergence for that sequence. On the other hand, we show that, for any nonexpansive linear operator T on a separable Hilbert space, and any element f, it is possible to compute a bound on the rate of convergence of (Anf) from T, f, and the norm ‖f ∗ ‖ of the limit. In particular, if T is the Koopman operator arising from a computable ergodic measure preserving transformation of a probability space X and f is any computable element of L2(X), then there is a computable bound on the rate of convergence of the sequence (Anf). The mean ergodic theorem is equivalent to the assertion that for every function K(n) and every ε> 0, there is an n with the property that the ergodic averages Amf are stable to within ε on the interval [n, K(n)]. Even in situations where the sequence (Anf) does not have a computable limit, one can give explicit bounds on such n in terms of K and ‖f‖/ε. This tells us how far one has to search to find an n so that the ergodic averages are “locally stable ” on a large interval. We use these bounds to obtain a similarly explicit version of the pointwise ergodic theorem, and show that our bounds are qualitatively different from ones that can be obtained using upcrossing inequalities due to Bishop and Ivanov. Finally, we explain how our positive results can be viewed as an application of a body of general prooftheoretic methods falling under the heading of “proof mining.” 1
The complexity of propositional proofs
 Bulletin of Symbolic Logic
"... Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorit ..."
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Abstract. Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes. Contents
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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Cited by 17 (5 self)
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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
Weak theories of nonstandard arithmetic and analysis
 Reverse Mathematics
, 2001
"... Abstract. A general method of interpreting weak highertype theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomialtime ..."
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Cited by 7 (5 self)
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Abstract. A general method of interpreting weak highertype theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomialtime computable arithmetic. A means of formalizing basic real analysis in such theories is sketched. §1. Introduction. Nonstandard analysis, as developed by Abraham Robinson, provides an elegant paradigm for the application of metamathematical ideas in mathematics. The idea is simple: use modeltheoretic methods to build rich extensions of a mathematical structure, like secondorder arithmetic or a universe of sets; reason about what is true in these enriched structures;
Saturated models of universal theories
 Annals of Pure and Applied Logic
"... A notion called Herbrand saturation is shown to provide the modeltheoretic analogue of a prooftheoretic method, Herbrand analysis, yielding uniform modeltheoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a ..."
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Cited by 7 (3 self)
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A notion called Herbrand saturation is shown to provide the modeltheoretic analogue of a prooftheoretic method, Herbrand analysis, yielding uniform modeltheoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a third approach, which is finitary but retains the semantic flavor of the modeltheoretic version. 1
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a pla ..."
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Cited by 6 (0 self)
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
Weyl’s predicative classical mathematics as a logicenriched type theory
, 2006
"... Abstract. In Das Kontinuum, Weyl showed how a large body of classical mathematics could be developed on a purely predicative foundation. We present a logicenriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definitio ..."
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Cited by 5 (5 self)
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Abstract. In Das Kontinuum, Weyl showed how a large body of classical mathematics could be developed on a purely predicative foundation. We present a logicenriched type theory that corresponds to Weyl’s foundational system. A large part of the mathematics in Weyl’s book — including Weyl’s definition of the cardinality of a set and several results from real analysis — has been formalised, using the proof assistant Plastic that implements a logical framework. This case study shows how type theory can be used to represent a nonconstructive foundation for mathematics. Key words: logicenriched type theory, predicativism, formalisation 1
"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how ..."
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Cited by 5 (2 self)
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
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.”